Testing normality in any dimension by Fourier methods in a multivariate Stein equation
TT ESTING NORMALITY IN ANY DIMENSION BY F OURIERMETHODS IN A MULTIVARIATE S TEIN EQUATION
A P
REPRINT
Bruno Ebner
Institute of Stochastics,Karlsruhe Institute of Technology (KIT),Englerstr. 2, D-76133 Karlsruhe.
Norbert Henze
Institute of Stochastics,Karlsruhe Institute of Technology (KIT),Englerstr. 2, D-76133 Karlsruhe.
David Strieder
Karlsruher Str. 62a,D-69126 Heidelberg, [email protected]
July 7, 2020 A BSTRACT
We study a novel class of affine invariant and consistent tests for multivariate normality. The testsare based on a characterization of the standard d -variate normal distribution by means of the uniquesolution of an initial value problem connected to a partial differential equation, which is motivatedby a multivariate Stein equation. The test criterion is a suitably weighted L -statistic. We derive thelimit distribution of the test statistic under the null hypothesis as well as under contiguous and fixedalternatives to normality. A consistent estimator of the limiting variance under fixed alternatives aswell as an asymptotic confidence interval of the distance of an underlying alternative with respectto the multivariate normal law is derived. In simulation studies, we show that the tests are strongin comparison with prominent competitors, and that the empirical coverage rate of the asymptoticconfidence interval converges to the nominal level. We present a real data example, and we outlinetopics for further research. Statistical inference for a data set starts with assumptions on the underlying stochastic mechanism which determinesthe generation of the data. In most classical models for multidimensional data, such as multivariate linear regressionmodels or multivariate analysis of variance, the assumption of multivariate normality of the underlying random vectorsis inherent. Hence, prior to any serious statistical inference, one should check this assumption. To be specific, let
X, X , X , . . . be a sequence of independent identically distributed (i.i.d.) d -dimensional (column) vectors that aredefined on a common probability space (Ω , A , P ) . We make the basic standing assumptions that the distribution P X of X is absolutely continuous with respect to d -dimensional Lebesgue measure. In what follows, we denote by N d ( µ, Σ) the d -variate normal distribution with expectation vector µ and covariance matrix Σ , and we write N d := { N d ( µ, Σ) : µ ∈ R d , Σ ∈ R d × d positive definite } for the class of all non-degenerate d -variate normal distributions. The unit matrix of order d will be denoted by I d .The problem of matter is testing the hypothesis H : P X ∈ N d , MSC 2010 subject classifications.
Primary 62H15 Secondary 62G20
Key words and phrases
Test for multivariate normality; affine invariance; consistency; characteristic function; weighted L -statistic; multivariate Stein equation a r X i v : . [ m a t h . S T ] J u l esting normality in any dimension by Fourier methods in a multivariate Stein equationbased on X , . . . , X n , against general alternatives. The purpose of this paper is to introduce and study a novel classof affine invariant and consistent tests based on a partial differential equation (PDE) that determines the characteristicfunction of the multivariate standard normal law. We write ∇ for the gradient operator and consider for f ∈ L ( R d ) the initial value problem of the PDE (cid:26) ( t + ∇ ) f ( t ) = 0 , t ∈ R d ,f (0) = 1 . (1)Note that the operator Af ( x ) = ( x + ∇ ) f ( x ) is a multivariate Stein operator in the following sense: For a centredrandom vector X with E [ XX (cid:124) ] = I d , which has a differentiable density with full support R d , we have E [ Af ( X )] = E [ Xf ( X ) + ∇ f ( X )] = 0 for each function f with existing derivatives in every direction and for which all occurringexpectations exist, if and only if X has the normal distribution N d (0 , I d ) , see Theorem 3.5 in [43] as well as [37, 40, 52]for more information on the multivariate Stein lemma. Here and in the following the symbol (cid:124) means transpositionof column vectors and matrices. In the spirit of the Stein-Tikhomirov method, see [1, 19], and hence using thecharacteristic functions { exp( i t (cid:124) x ) , t ∈ R d } as test functions, a simple calculation shows the equivalence of the Steinequation to the initial value problem in (1). In the case d = 1 the same initial value problem was motivated by a fixedpoint of the zero bias transform in [17]. For more information on the zero bias transform, see [21, 50]. Theorem 1.1.
The characteristic function ψ ( t ) = exp (cid:18) − (cid:107) t (cid:107) (cid:19) , t ∈ R d , (2) of the d -variate standard normal distribution N d (0 , I d ) is the only solution of (1) . Proof. If f ∈ L ( R d ) is an arbitrary solution of (1), the product rule yields ∇ (cid:18) exp (cid:18) (cid:107) t (cid:107) (cid:19) f ( t ) (cid:19) = exp (cid:18) (cid:107) t (cid:107) (cid:19)(cid:18) tf ( t ) + ∇ f ( t ) (cid:19) = 0 . In view of f (0) = 1 , we have exp( (cid:107) t (cid:107) / f ( t ) = 1 , and the assertion follows.According to Theorem 1.1, the characteristic function (CF) of the d -variate standard normal distribution is the onlyCF satisfying ∇ ψ ( t ) = − tψ ( t ) . Our test statistic will be based on this equation. To achieve affine invariance of thetest statistic with respect to full rank affine transformations of X , . . . , X n , let Y n,j := S − / n ( X j − X n ) , j = 1 , ..., n, denote the so-called scaled residuals, where X = n − (cid:80) nj =1 X j and S n := n − (cid:80) nj =1 ( X j − X n )( X j − X n ) (cid:124) standfor the sample mean and the sample covariance matrix of X , . . . , X n , respectively. The matrix S − / n is the uniquesymmetric positive definite square root of S − n . To ensure almost sure invertibility of S n , we tacitly assume n ≥ d + 1 in what follows, see [15]. Writing ψ n ( t ) = 1 n n (cid:88) j =1 exp( i t (cid:124) Y n,j ) , t ∈ R d , (3)for the empirical CF of Y n, , . . . , Y n,n , our test statistic is T n,a = n (cid:90) R d (cid:107)∇ ψ n ( t ) + tψ ( t ) (cid:107) C w a ( t ) d t. (4)Here, w a ( t ) = exp (cid:0) − a (cid:107) t (cid:107) (cid:1) , a > , is a suitable weight function that depends on a positive parameter a , and (cid:107) · (cid:107) C denotes the complex Euclidean vector norm. Rejection of H is for large values of T n,a . With this approach, we obtaina flexible class of genuine tests for multivariate normality, all of which are motivated by the result of Theorem 1.1.Clearly, we propose a new approach to a well-known and widely studied problem, for a survey of affine invariant testsof multivariate normality, see [28], and for recent developments with an emphasis on L type statistics, see [18]. Welist a short overview of different approaches: [13, 14, 32, 46, 55] consider tests connected to the empirical characteristicfunction, while [29, 30, 31] are based on the empirical moment generating function. The most classical approach isto consider measures of multivariate skewness and kurtosis, see, e.g., [12, 34, 42, 44], although inconsistency of thosemeasures with regard to elliptically symmetric alternatives are known, see [4, 5, 25, 26]. Generalizations of testsfor univariate normality, as in [35, 53, 57], the examination of nonlinearity of dependence, see [11, 16], canonical2esting normality in any dimension by Fourier methods in a multivariate Stein equationcorrelations, see [56], and the notion of energy, see [54], are other approaches to this testing problem. Empiricalcompetitive Monte Carlo studies can be found in [18, 58].The rest of this paper unfolds as follows: In Section 2, we give a representation of T n,a that is amenable for compu-tational purposes. Moreover, we derive limits of T n,a , after suitable affine transformations, as a → ∞ and a → ,that hold elementwise on the underlying probability space. Section 3 deals with the limit distribution of T n,a underthe null hypothesis, and Section 4 considers the limit behavior of T n,a both under contiguous and fixed alternatives to H . Section 5 presents the results of a simulation study, and Section 6 exhibits a real data example. Section 7 containsa brief summary, and it indicates topics for further research. For the sake of readability, some of the proofs have beendeferred to Appendix A.Throughout the paper, we use the following notation: The symbol D = means equality in distribution, and P −→ and a . s . −→ stand for convergence in probability and almost sure convergence, respectively. Moreover, D −→ is shorthand forconvergence in distribution for random elements in whatever space (which will be clear from the context). If not statedotherwise, each limit refers to n → ∞ , and each unspecified integral is over R d . The stochastic Landau symbols o P (1) and O P (1) refer to convergence to zero in probability and stochastic boundedness, respectively. In this section, we provide some information on the test statistic T n,a defined in (4). The first result shows that T n,a allows for a simple representation that is amenable to computational purposes. Moreover, since this representationshows that T n,a depends on X , . . . , X n only via Y (cid:124) n,i Y n,j , i, j ∈ { , . . . , n } , the statistic T n,a is affine invariant. Theorem 2.1.
We have T n,a = n (cid:18) πa + 1 (cid:19) d d a + 1) − (cid:18) π a + 1 (cid:19) d n (cid:88) j =1 (cid:107) Y n,j (cid:107) a + 1 exp (cid:18) − (cid:107) Y n,j (cid:107) a + 2 (cid:19) (5) + 1 n (cid:18) πa (cid:19) d n (cid:88) i,j =1 Y (cid:124) n,i Y n,j exp (cid:18) − (cid:107) Y n,i − Y n,j (cid:107) a (cid:19) . Note that this representation is implemented in the R package mnt , see [9]. The proof of Theorem 2.1 is given inAppendix A.We now consider the elementwise limits (on the underlying probability space) of T n,a for fixed n as a → ∞ and a → . It will bee seen that the class of tests based on T n,a is ’closed at the boundaries’ a → ∞ and a → in thesense that, after suitable affine transformations, there are well-defined ’limit statistics’. Our first result refers to thelimit a → ∞ . Theorem 2.2.
Elementwise on the underlying probability space (Ω , A , P ) , we have lim a →∞ a d +2 nπ d T n,a = (cid:101) b ,d + 2 b ,d . (6) Here, b ,d = n − (cid:80) ni,j =1 ( Y (cid:124) n,i Y n,j ) is Mardia’s celebrated measure of multivariate skewness, see [42], and (cid:101) b ,d = n − (cid:80) ni,j =1 Y (cid:124) n,i Y n,j (cid:107) Y n,i (cid:107) (cid:107) Y n,j (cid:107) is a measure of multivariate skewness introduced by Móri, Rohatgi, andSzékely, see [44]. Proof.
Invoking (5), it follows that a d +2 nπ d T n,a = (cid:18) aa + 1 (cid:19) d +1 ad − an (cid:18) aa + (cid:19) d +1 n (cid:88) j =1 (cid:107) Y n,j (cid:107) exp (cid:18) − (cid:107) Y n,j (cid:107) a + 2 (cid:19) + a n n (cid:88) i,j =1 Y (cid:124) n,i Y n,j exp (cid:18) − (cid:107) Y n,i − Y n,j (cid:107) a (cid:19) =: A n − B n + C n (cid:18) aa + 1 (cid:19) d +1 = (cid:18) a (cid:19) − d − = 1 − (cid:18) d (cid:19) a + O ( a − ) (7)as a → ∞ and exp( − x ) = 1 − x + 12 x + O ( x ) (8)as x → , and we employ the identities (cid:80) nj =1 Y n,j = 0 , (cid:80) nj =1 (cid:107) Y n,j (cid:107) = nd as well as n (cid:88) i,j =1 Y (cid:124) n,i Y n,j (cid:107) Y n,i − Y n,j (cid:107) = − n (cid:88) i,j =1 ( Y (cid:124) n,i Y n,j ) = − n d, n (cid:88) i,j =1 Y (cid:124) n,i Y n,j (cid:107) Y n,i − Y n,j (cid:107) = 2 n (cid:101) b ,d + 4 n b ,d − n (cid:88) i,j =1 ( Y (cid:124) n,i Y n,j ) (cid:107) Y n,j (cid:107) , n (cid:88) i,j =1 ( Y (cid:124) n,i Y n,j ) (cid:107) Y n,j (cid:107) = n n (cid:88) j =1 (cid:107) Y n,j (cid:107) . to obtain A n = ad/ − d / − d/ o (1) as a → ∞ . Likewise, B n = 1 n (cid:18) a − (cid:18) d (cid:19) (cid:19) n (cid:88) j =1 (cid:107) Y n,j (cid:107) (cid:18) − (cid:107) Y n,j (cid:107) a + 2 (cid:19) + o (1)= (cid:18) da − d − d (cid:19) − n n (cid:88) j =1 (cid:107) Y n,j (cid:107) + o (1) ,C n = a n n (cid:88) i,j =1 Y (cid:124) n,i Y n,j (cid:18) − (cid:107) Y n,i − Y n,j (cid:107) a + (cid:107) Y n,i − Y n,j (cid:107) a (cid:19) + o (1)= da (cid:16)(cid:101) b ,d + 2 b ,d − n n (cid:88) j =1 (cid:107) Y n,j (cid:107) (cid:17) + o (1) . Upon combining, the assertion follows.Notice that the right hand side of (6) is a linear combination of two time-honored measures of multivariate skewness.Notably, the same linear combination showed up not only for the class of BHEP tests (see Theorem 2.1 of [27]), butalso as a limit of a related test statistic in connection with a test for multivariate normality based on a partial differentialequation for the moment generating function of the normal distribution, see [31].Regarding the limit of T n,a as a → , we have the following result. Theorem 2.3.
Elementwise on the underlying probability space, we have lim a → na d (cid:18)(cid:16) aπ (cid:17) d T n,a − d (cid:19) = d − d +1 n n (cid:88) j =1 (cid:107) Y n,j (cid:107) exp (cid:18) − (cid:107) Y n,j (cid:107) (cid:19) . Proof.
From the representation (5), it follows that T n,a π d = nd a + 1) d +1 − (cid:18) a + 1 (cid:19) d +1 n (cid:88) j =1 (cid:107) Y n,j (cid:107) exp (cid:32) − (cid:107) Y n,j (cid:107) a + 2 (cid:33) + 1 na d n (cid:88) i,j =1 Y (cid:124) n,i Y n,j exp (cid:32) − (cid:107) Y n,i − Y n,j (cid:107) a (cid:33) = A n,a − B n,a + C n,a lim a → A n,a = nd/ and lim a → B n,a = 2 d +1 (cid:80) nj =1 (cid:107) Y n,j (cid:107) exp (cid:0) −(cid:107) Y n,j (cid:107) / (cid:1) , elementwise on theunderlying probability space. To tackle C n,a , the relation (cid:80) nj =1 (cid:107) Y n,j (cid:107) = nd yields C n,a = 1 na d n (cid:88) j =1 (cid:107) Y n,j (cid:107) + 1 na d n (cid:88) i (cid:54) = j Y (cid:124) n,i Y n,j exp (cid:32) − (cid:107) Y n,i − Y n,j (cid:107) a (cid:33) = da d + 1 na d n (cid:88) i (cid:54) = j Y (cid:124) n,i Y n,j exp (cid:32) − (cid:107) Y n,i − Y n,j (cid:107) a (cid:33) , and the assertion follows.Interestingly, Theorem 2.3 means that for (very) small values of a , rejection of H for large values of T n,a is essentiallyequivalent to the rejection of H for small values of n n (cid:88) j =1 (cid:107) Y n,j (cid:107) e −(cid:107) Y n,j (cid:107) / . This statistic, upon expanding the exponential function, comprises even powers of (cid:107) Y n,j (cid:107) and is thus related to Mar-dia’s measure of multivariate kurtosis, which is defined by b ,d = n − (cid:80) nj =1 (cid:107) Y n,j (cid:107) , see [42]. T n,a In this section we derive the limit distribution of T n,a under the hypothesis H . In view of affine invariance, we assumewithout loss of generality that X has the standard normal distribution N d (0 , I d ) in what follows. The starting point isan alternative representation of T n,a , namely T n,a = (cid:90) (cid:107) Z n ( t ) (cid:107) w a ( t ) d t, (9)where Z n ( t ) = 1 √ n n (cid:88) j =1 (cid:18) Y n,j (cid:0) cos( t (cid:124) Y n,j ) + sin( t (cid:124) Y n,j ) (cid:1) − tψ ( t ) (cid:19) . (10)This assertion follows from straightforward calculations using (cid:90) cos( t (cid:124) Y n,j ) sin( t (cid:124) Y n,i ) w a ( t ) d t = 0 , (cid:90) cos( t (cid:124) Y n,j ) t (cid:124) Y n,j w a ( t ) d t = 0 . (11)Writing L := L ( R d , B d , w a ( t ) d t ) for the separable Hilbert space of (equivalence classes of) functions f : R d → R that are square integrable with respect to w a ( t ) d t , we regard Z n as a random element of the Hilbert space H = L ⊗ · · · ⊗ L . Putting f = ( f , . . . , f d ) , g = ( g , . . . , g d ) , the space H is equipped with the inner product (cid:104) f, g (cid:105) H := (cid:104) f , g (cid:105) L + . . . + (cid:104) f d , g d (cid:105) L and the norm (cid:107) f (cid:107) H = (cid:104) f, f (cid:105) / H . Notice that we have T n,a = (cid:90) (cid:107) Z n ( t ) (cid:107) w a ( t ) d t = (cid:107) Z n (cid:107) H . The main theorem of this section is as follows:
Theorem 3.1.
Under H , there is a centred Gaussian random element Z of H having covariance matrix kernel K ( s, t ) = (cid:0) I d − ( s − t )( s − t ) (cid:124) (cid:1) ψ ( s − t ) (12) + (cid:16) ss (cid:124) + tt (cid:124) − ts (cid:124) − st (cid:124) − I d + s (cid:124) t ( ss (cid:124) + tt (cid:124) − st (cid:124) − I d ) − s (cid:124) ts (cid:124) t st (cid:124) (cid:17) ψ ( s ) ψ ( t ) , s, t ∈ R d , such that Z n D −→ Z in H , where Z n is the random element defined in (10) . Since the proof of Theorem 3.1 is long and tedious, it is deferred to Appendix A. From Theorem 3.1 and the continuousmapping theorem, we obtain the following result. 5esting normality in any dimension by Fourier methods in a multivariate Stein equation
Corollary 3.2.
Under H , we have T n,a D −→ (cid:107) Z (cid:107) H = (cid:90) (cid:107) Z ( t ) (cid:107) w a ( t ) d t. It is well-known that the distribution of T ∞ ,a := (cid:107) Z (cid:107) H is that of T ∞ ,a D = (cid:80) ∞ j =1 λ j ( a ) N j , where N , N , . . . is asequence of i.i.d. standard normal random variables, and λ ( a ) , λ ( a ) , . . . are the positive eigenvalues associated withthe integral operator K f ( s ) := (cid:90) K ( s, t ) f ( t ) w a ( t ) d t, s ∈ R d , (13) f ∈ H . In view of the complexity of K ( s, t ) , we did not succeed in obtaining closed-form expressions for theseeigenvalues. In our simulation study presented in Section 5, we use approximate critical values for T n,a that have beenobtained by means of simulations. Some information on the limit null distribution, however, is given by the followingresult. Theorem 3.3.
We have E [ T ∞ ,a ] = (cid:16) πa (cid:17) d d − (cid:16) πa + 1 (cid:17) d (cid:0) a + (8 d + 48) a + (12 d + 40) a + d + 10 d + 16 (cid:1) d a + 1) . Proof.
From Fubini’s theorem, it follows that E [ T ∞ ,a ] = (cid:82) E (cid:107) Z ( t ) (cid:107) w a ( t ) d t . Moreover, writing tr for trace, we have E (cid:107) Z ( t ) (cid:107) = E [ Z ( t ) (cid:124) Z ( t )] = tr (cid:0) E [ Z ( t ) Z ( t ) (cid:124) ] (cid:1) = tr (cid:0) K ( t, t ) (cid:1) = d − (cid:16) d + d (cid:107) t (cid:107) − (cid:107) t (cid:107) + (cid:107) t (cid:107) (cid:17) exp (cid:0) − (cid:107) t (cid:107) (cid:1) . Since (cid:90) (cid:107) t (cid:107) e − a (cid:107) t (cid:107) d t = (cid:16) πa (cid:17) d d a ( d + 2) and (cid:90) (cid:107) t (cid:107) e − a (cid:107) t (cid:107) d t = (cid:16) πa (cid:17) d d a ( d + 6 d + 8) , the assertion follows by straightforward computations.In the univariate case, which is deliberately not excluded from our study, we have been able to calculate the first fourcumulants of T ∞ ,a . By the methods presented in Chapter 5 of [51] the m th cumulant of T ∞ ,a is derived by κ m ( a ) = 2 m − ( m − (cid:90) R h m ( t, t ) w a ( t ) d t. Here, h ( s, t ) = K ( s, t ) , and h m ( s, t ) := (cid:82) R h m − ( s, u ) K ( u, t ) w a ( u ) d u if m ≥ . In order to calculate κ m ( a ) , m ∈ { , , , } , we used the computer algebra system Maple, see [41]. The formulae for κ ( a ) and κ ( a ) are givenin the appendix.For κ ( a ) and κ ( a ) , we obtain κ ( a ) = (cid:90) R (cid:0) − (1 + t − t + t − t ) (cid:1) exp( − at ) d t = (cid:114) πa − (cid:114) πa + 1 − (cid:114) πa + 1 12( a + 1) + (cid:114) πa + 1 34( a + 1) − (cid:114) πa + 1 1516( a + 1) = ( − a − a − a − (cid:114) πa + 1 + 16 (cid:114) πa ( a + 1) a + 1) a E [ T ∞ ,a ] [ T ∞ ,a ] β ( a ) β ( a ) T ∞ ,a , d = 1 q \ a T ∞ ,a in the case d = 1 and κ ( a ) = 7260811 π a + 2) / (4 a + 8 a + 3) / √ a (2 a + 3) ( a + 1) (cid:18)(cid:18)(cid:18) a + 153607260811 a + 1080327260811 a + 4738567260811 a + 14492167260811 a + 32632327260811 a + 55599087260811 a + 72543487260811 a + a + 55359067260811 a + 160113367636 a + 525375929043244 a + 6017409116172976 a + 26673329043244 a + 2211329043244 √ a (cid:19) √ a + 2+ 1024 ( a + 3 / ( a + 1) ( a + 1 / (cid:0) a + 2 a + 3 (cid:1) (cid:19)(cid:112) a + 8 a + 3 − √ a + 27260811 (cid:18) a + 1024803453 a + 1104114779 a + 36544803453 a + 121054803453 a + 297018803453 a + 556163803453 a + 807017803453 a + 912747803453 a + a + 545801803453 a + 281319803453 a + 106779803453 a + 28293803453 a + 9616397 a + 372803453 √ a (cid:19)(cid:19) . From these cumulants, we obtain the expectation, the variance as well as the skewness β and the kurtosis β of T ∞ ,a for the case d = 1 (see Table 1), since E [ T ∞ ,a ] = κ ( a ) , Var [ T ∞ ,a ] = κ ( a ) , β ( a ) = κ ( a ) κ ( a ) / , β ( a ) = 3 + κ ( a ) κ ( a ) . By complete analogy with [17, 24], we can now approximate the distribution of T ∞ ,a by that of a member of the systemof Pearson distributions which has the same first four moments as T ∞ ,a . To this end, we used the statistic software R ,see [47], and the package PearsonDS , see [7]. Table 2 shows the quantiles of the fitted Pearson distribution, whichserve as approximations of the corresponding quantiles of the distribution of T ∞ ,a . T n,a under alternatives In this section, we assume that H does not hold, and we will derive limit distributions for T n,a both under contiguousand fixed alternatives to H . To define the setting for a triangular array of contiguous alternatives, we assume that, for7esting normality in any dimension by Fourier methods in a multivariate Stein equationeach n ≥ d + 1 , X n, , ..., X n,n are i.i.d. d -variate random vectors having Lebesgue density f n ( x ) = ϕ ( x ) (cid:16) g ( x ) √ n (cid:17) , x ∈ R d . Here, ϕ ( x ) = (2 π ) − d/ exp( −(cid:107) x (cid:107) / , x ∈ R d , is the density of the distribution N d (0 , I d ) , and g is a boundedmeasurable function satisfying (cid:82) g ( x ) ϕ ( x ) d x = 0 . Notice that f n is nonnegative for sufficiently large n due tothe boundedness of g . To derive the limit distribution of T n,a under this sequence of alternatives, we employ therepresentation (9), which comprises the random element Z n as defined in (10). For repeated later use, we putCS + ( s, t ) = cos( s (cid:124) t ) + sin( s (cid:124) t ) , CS − ( s, t ) = cos( s (cid:124) t ) − sin( s (cid:124) t ) , s, t ∈ R d . (14) Theorem 4.1.
Under the sequence of alternatives ( X n, , ..., X n,n ) n ≥ d +1 , we have Z n D −→ Z + c in H . Here, Z n is defined in (10) , Z is the centred Gaussian random element of H figuring in Theorem 3.1, and the shiftfunction c ( · ) is given by c ( t ) = (cid:90) Z ∗∗ ( x, t ) g ( x ) ϕ ( x ) d x, t ∈ R d , (15) where Z ∗∗ ( x, t ) = x CS + ( t, x ) − (cid:0) t + x + (2I d − tt (cid:124) ) 12 ( xx (cid:124) − I d ) t − t (cid:124) xt (cid:1) ψ ( t ) , x, t ∈ R d . Proof.
We write λ d for d -dimensional Lebesgue measure, and we put P ( n ) := ⊗ ( ϕλ d ) , Q ( n ) := ⊗ ( f n λ d ) . Further-more, let L n := d Q ( n ) / d P ( n ) . The boundedness of g and a Taylor expansion then give log( L n ( X n, , ..., X n,n )) = n (cid:88) j =1 log (cid:16) g ( X n,j ) √ n (cid:17) = n (cid:88) j =1 (cid:16) g ( X n,j ) √ n − g ( X n,j ) n (cid:17) + o P ( n ) (1) . (16)In the following we write σ = (cid:82) g ( x ) ϕ ( x ) d x < ∞ . Since, under P ( n ) , expectation and variance of the sum figuringin (16) converge to − σ / and σ , respectively, the Lindeberg–Feller central limit theorem and Slutsky’s lemma yield log( L n ) D −→ N (cid:16) − σ , σ (cid:17) under P ( n ) . (17)Notice that the boundedness of g ensures the validity of the Lindeberg condition. In view of Le Cam’s first lemma(see, e.g., [39], p. 297), the probability measures Q ( n ) and P ( n ) are mutually contiguous. According to Theorem 3.1,the auxiliary process Z ∗ n introduced in (38) is tight under P ( n ) and thus, in view of contiguity, also under Q ( n ) . Let { e k , k ≥ } be an arbitrary complete orthonormal system of H . It remains to show that, for each (cid:96) ≥ , we have Π (cid:96) ( Z n ) D −→ Π (cid:96) ( Z + c ) under Q ( n ) , where Π (cid:96) denotes the orthogonal projection onto the linear subspace of H spannedby e , . . . , e (cid:96) . We first consider Π (cid:96) ( Z ∗ n ) = (cid:96) (cid:88) j =1 (cid:104) Z ∗ n , e j (cid:105) H e j , where Z ∗ n is given in (38), with the only difference that X j is throughout replaced with X n,j . In view of Theorem 3.1,the asymptotic distribution of Z ∗ n under P ( n ) is centred Gaussian with a covariance operator K given by the covariancematrix kernel K ( s, t ) , whence (cid:104) Z ∗ n , e j (cid:105) H D −→ N (0 , (cid:104) K e j , e j (cid:105) H ) under P ( n ) . In view of (17) we have (cid:0) (cid:104) Z ∗ n , e (cid:105) H , ..., (cid:104) Z ∗ n , e (cid:96) (cid:105) H , log( L n ) (cid:1) (cid:124) D −→ N (cid:96) +1 (cid:32) (0 , . . . , , − σ / (cid:124) , (cid:34) Σ (cid:101) c (cid:101) c (cid:124) σ (cid:35) (cid:33) under P ( n ) for each (cid:96) ≥ . Here, Σ := (cid:0) (cid:104) K e i , e j (cid:105) H (cid:1) ≤ i,j ≤ (cid:96) ∈ R (cid:96) × (cid:96) and (cid:101) c = (cid:0)(cid:101) c , ..., (cid:101) c (cid:96) (cid:1) (cid:124) ∈ R (cid:96) , where, by Fubini’stheorem, (cid:101) c j := lim n →∞ E (cid:2) (cid:104) Z ∗ n , e j (cid:105) H , log( L n ) (cid:3) = (cid:104) c, e j (cid:105) H , and c is given in (15). According to Le Cam’s thirdLemma (see, e.g., [39], p. 300), it follows that (cid:0) (cid:104) Z ∗ n , e (cid:105) H , ..., (cid:104) Z ∗ n , e (cid:96) (cid:105) H (cid:1) (cid:124) D −→ N (cid:96) ( (cid:101) c, Σ) under Q ( n ) . Since, for thecentred Gaussian random element figuring in Theorem 3.1, we have (cid:0) (cid:104) Z + c, e (cid:105) H , ..., (cid:104) Z + c, e (cid:96) (cid:105) H (cid:1) (cid:124) D = N (cid:96) ( (cid:101) c, Σ) , (cid:0) (cid:104) Z ∗ n , e (cid:105) H , ..., (cid:104) Z ∗ n , e (cid:96) (cid:105) H (cid:1) (cid:124) D −→ (cid:0) (cid:104) Z + c, e (cid:105) H , ..., (cid:104) Z + c, e (cid:96) (cid:105) H (cid:1) (cid:124) (18)under Q ( n ) . Now, let Ψ : R (cid:96) → H be defined by Ψ( x ) := (cid:80) (cid:96)j =1 x j e j , x = ( x , ..., x (cid:96) ) (cid:124) . The continuous mappingtheorem and (18) then yield Π (cid:96) ( Z ∗ n ) = Ψ (cid:16)(cid:0) (cid:104) Z ∗ n , e (cid:105) H , ..., (cid:104) Z ∗ n , e (cid:96) (cid:105) H (cid:1) (cid:124) (cid:17) D −→ Ψ (cid:16)(cid:0) (cid:104) Z + c, e (cid:105) H , ..., (cid:104) Z + c, e (cid:96) (cid:105) H (cid:1) (cid:124) (cid:17) = Π (cid:96) ( Z + c ) under Q ( n ) . In view of the tightness of Z ∗ n unter Q ( n ) we conclude Z ∗ n D −→ Z + c under Q ( n ) . The assertion nowfollows from Slutsky’s lemma since, in view of 41 and 42, (cid:107) Z n − Z ∗ n (cid:107) H is asymptotically negligible under P ( n ) andthus, because of contiguity, also under Q ( n ) .As a corollary, we have the following result. Corollary 4.2.
Under the conditions of Theorem 4.1, we have T n,a D −→ (cid:107) Z + c (cid:107) H = (cid:90) (cid:13)(cid:13) Z ( t ) + c ( t ) (cid:13)(cid:13) w a ( t ) d t. We now consider fixed alternatives to H , and we suppose that the underlying distribution, in addition to being abso-lutely continuous, satisfies E (cid:107) X (cid:107) < ∞ . In view of affine invariance, we assume E [ X ] = 0 and E [ XX (cid:124) ] = I d . Ourfirst result is a strong limit of T n,a /n as n → ∞ . Theorem 4.3. If E (cid:107) X (cid:107) < ∞ , we have T n,a n a . s . −→ ∆ a , where ∆ a := (cid:90) (cid:107) µ ( t ) − tψ ( t ) (cid:107) w a ( t ) d t (19) and µ ( t ) = E [ X CS + ( t, X )] . Proof.
Invoking (9), we have n − T n,a = (cid:107) n − / Z n (cid:107) H , where Z n is given in (10). Putting Z n ( t ) = n − / (cid:80) nj =1 (cid:0) X j CS + ( t, X j ) − tψ ( t ) (cid:1) , the strong law of large numbers in Hilbert spaces yields (cid:107) n − / Z n (cid:107) H a . s . −→ ,and thus it remains to prove (cid:107) n − / ( Z n − Z n ) (cid:107) H a . s . −→ . To this end, notice that √ n (cid:0) Z n ( t ) − Z n ( t ) (cid:1) = 1 n n (cid:88) j =1 (cid:16) X j (cid:0) CS + ( t, Y n,j ) − CS + ( t, X j ) (cid:1) + ∆ n,j CS + ( t, Y n,j ) (cid:17) . Since CS + ( t, Y n,j ) = CS + ( t, X j ) + ε n,j ( t ) + η n,j ( t ) , where max( | ε n,j ( t ) | , | η n,j ( t ) | ) ≤ (cid:107) t (cid:107)(cid:107) ∆ n,j (cid:107) , it follows that (cid:13)(cid:13)(cid:13) n n (cid:88) j =1 X j (cid:0) CS + ( t, Y n,j ) − CS + ( t, X j ) (cid:1)(cid:13)(cid:13)(cid:13) ≤ n n (cid:88) j =1 (cid:107) X j (cid:107)(cid:107) t (cid:107)(cid:107) ∆ n,j (cid:107)≤ (cid:107) t (cid:107) n − / max j =1 ,...,n (cid:107) X j (cid:107) n / max j =1 ,...,n (cid:107) ∆ n,j (cid:107) . Since E (cid:107) X (cid:107) < ∞ , Theorem 5.2 of [6] yields n − / max j =1 ,...,n (cid:107) X j (cid:107) a . s . −→ , and from Proposition A.1 of [14],we have n / max j =1 ,...,n (cid:107) ∆ n,j (cid:107) a . s . −→ . Consequently, (cid:107) n − (cid:80) nj =1 X j (cid:0) CS + ( t, Y n,j ) − CS + ( t, X j ) (cid:1) (cid:107) H a . s . −→ .Furthermore, (cid:107) n − (cid:80) nj =1 ∆ n,j CS + ( t, Y n,j ) (cid:107) ≤ n − (cid:80) nj =1 (cid:107) ∆ n,j (cid:107) . Since the right hand side converges to 0 P -almost surely according to Proposition A.1 of [14], it follows that (cid:107) n − (cid:80) nj =1 ∆ n,j CS + ( t, Y n,j ) (cid:107) H a . s . −→ . Theremaining assertion (cid:107) n − / ( Z n − Z n ) (cid:107) H a . s . −→ now follows from the triangle inequality.As a corollary, we obtain the following result. Corollary 4.4.
The test for multivariate normality based on T n,a is consistent against each alternative distributionsatisfying E (cid:107) X (cid:107) < ∞ . Proof.
Let ψ X ( t ) = E [exp( i t (cid:124) X )] be the CF of X . By straightforward calculations, we have ∆ a = (cid:90) (cid:107)∇ ψ X ( t ) − ∇ ψ ( t ) (cid:107) C w a ( t ) d t, where ∆ a is given in (19). Since ∆ a = 0 if and only if X D = N d (0 , I d ) (recall the standing assumption E [ X ] = 0 and E [ XX (cid:124) ] = I d ), the assertion follows.Notice that, for each a > , ∆ a may be regarded as a measure of deviation from normality. The following result shedssome more light on ∆ a . Theorem 4.5. If E (cid:107) X (cid:107) < ∞ then, under the standing assumptions E [ X ] = 0 and E [ XX (cid:124) ] = I d , we have lim a →∞ a (cid:16) aπ (cid:17) d ∆ a = E [ X (cid:124) X (cid:107) X (cid:107) (cid:107) X (cid:107) ] + 2 E [( X (cid:124) X ) ] , (20) as well as lim a → π − d ∆ a = d − d +1 E (cid:20) (cid:107) X (cid:107) exp (cid:18) − (cid:107) X (cid:107) (cid:19)(cid:21) . Proof.
Straightforward calculations give ∆ a = I a, − I a, + I a, , where I a, = (cid:90) E [ X CS + ( t, X )] (cid:124) E [ X CS + ( t, X )] w a ( t ) d tI a, = 2 (cid:90) E [ X CS + ( t, X )] (cid:124) tψ ( t ) w a ( t ) d t, I a, = (cid:90) t (cid:124) tψ ( t ) w a ( t ) d t. Using addition theorems for the sine and the cosine function as well as (11) and (35),(36) und (37), it follows that I a, = (cid:16) πa (cid:17) d E (cid:104) X (cid:124) X exp (cid:16) − (cid:107) X − X (cid:107) a (cid:17)(cid:105) ,I a, = 2 (cid:16) π a + 1 (cid:17) d E (cid:104) (cid:107) X (cid:107) a + 1 exp (cid:16) − (cid:107) X (cid:107) a + 2 (cid:17)(cid:105) , I a, (cid:16) πa + 1 (cid:17) d d a + 2 . The Taylor expansions (7) und (8), together with E [ X ] = 0 , E [ XX (cid:124) ] = I d and E (cid:107) X (cid:107) < ∞ then yield a (cid:16) aπ (cid:17) d I a, = a E [ X (cid:124) X ] − a E (cid:104) X (cid:124) X (cid:107) X − X (cid:107) (cid:105) + E (cid:104) X (cid:124) X (cid:107) X − X (cid:107) (cid:105) + O ( a − )= ad E [ X (cid:124) X (cid:107) X (cid:107) (cid:107) X (cid:107) ] + 216 E [( X (cid:124) X ) ] − E (cid:107) X (cid:107) + O ( a − ) ,a (cid:16) aπ (cid:17) d I a, = a (cid:16) aa + (cid:17) d +1 E (cid:104) (cid:107) X (cid:107) exp (cid:16) − (cid:107) X (cid:107) a + 2 (cid:17)(cid:105) = (cid:16) ad − d − d (cid:17) − E (cid:107) X (cid:107) + O ( a − ) ,a (cid:16) aπ (cid:17) d I a, = a (cid:16) aa + 1 (cid:17) d +1 d ad − d − d O ( a − ) . Upon summarizing, the assertion follows. The second statement is proved following similar arguments.We remark in passing that the first term on the right hand side of (20) is the population measure of multivariateskewness in the sense of Móri, Rohatgi, and Székely [44], and E [( X (cid:124) X ) ] is population skewness in the sense ofMardia [42]. Thus, Theorem 4.5 can be regarded as the ’population counterpart’ of Theorems 2.2 and 2.3.[3] observed that, in the context of goodness-of-fit testing of a general parametric hypothesis (cid:101) H (say), weighted L -statistics have a normal limit under fixed alternatives to (cid:101) H . To state such a theorem in our case, we first introducesome notation. Again, we write ψ X ( t ) = E [exp( i t (cid:124) X )] for the CF of X and put ψ ± X ( t ) := Re ψ X ( t ) ± Im ψ X ( t ) , w ( t, X ) = X CS + ( t, X ) − Xψ + X ( t ) − t (cid:124) X ∇ ψ + X ( t ) + 12 (cid:0) ( XX (cid:124) + I d ) ∇ ψ − X ( t ) − E [ XX (cid:124) CS − ( t, X )]( XX (cid:124) − I d ) t (cid:1) . (21)Moreover, let L ( s, t ) := E (cid:2) w ( s, X ) w ( t, X ) (cid:124) (cid:3) , s, t ∈ R d . (22)We then have the following result. 10esting normality in any dimension by Fourier methods in a multivariate Stein equation Theorem 4.6. If E (cid:107) X (cid:107) < ∞ , we have √ n (cid:16) T n,a n − ∆ a (cid:17) D −→ N(0 , σ a ) , where σ a := 4 (cid:90) (cid:90) z ( s ) (cid:124) L ( s, t ) z ( t ) w a ( s ) w a ( t ) d s d t. (23) Here, z ( t ) := µ ( t ) − tψ ( t ) , (24) and L ( s, t ) is defined in (22) . Proof.
The basic observation is that, with Z n defined in (10) and z ( t ) := µ ( t ) − tψ ( t ) , we have √ n (cid:16) T n,a n − ∆ a (cid:17) = √ n (cid:0) (cid:107) n − / Z n (cid:107) H − (cid:107) z (cid:107) H (cid:1) = √ n (cid:104) n − / Z n − z, z + n − / Z n − z (cid:105) H = 2 (cid:104) Z n − √ nz, z (cid:105) H + n − / (cid:107) Z n − √ nz (cid:107) H . (25)Letting V n ( t ) := Z n ( t ) − √ nz ( t ) = n − / (cid:80) nj =1 (cid:0) Y n,j CS + ( t, Y n,j ) − µ ( t ) (cid:1) , the next step is to show that V n D −→ V in H (26)for some centred Gaussian random element V of H having covariance matrix kernel L ( s, t ) given in (22). The proof of(26) is completely analogous to that of Theorem 3.1 and is therefore omitted. In view of (26), the second summand in(25) is o P (1) , and the first converges in distribution to (cid:104) V, z (cid:105) H by the continuous mapping theorem. The distributionof (cid:104) V, z (cid:105) H is the normal distribution N (0 , σ a ) .Using Slutsky’s lemma, Theorem 4.6 yields the following asymptotic confidence interval for ∆ a . Corollary 4.7.
For α ∈ (0 , , let z − α/ denote the (1 − α/ -quantile of the standard normal distribution. If (cid:98) σ n,a is a consistent sequence of estimators for σ a , and if σ a > , then I n,a,α := (cid:104) T n,a n − (cid:98) σ n,a √ n z − α/ , T n,a n + (cid:98) σ n,a √ n z − α/ (cid:105) is an asymptotic confidence interval with level − α for ∆ a . A necessary and sufficient condition for σ a > is that the function R d (cid:51) s (cid:55)→ (cid:82) L ( s, t ) z ( t ) w a ( t ) d t does not vanish λ d -almost everywhere, see Remark 1 of [3].To construct a consistent sequence of estimators for σ a , we replace z ( s ) , z ( t ) and L ( s, t ) figuring in (23) with suit-able empirical counterparts. In view of (22) and (21) and the fact that ∇ ψ + X ( t ) = E [ X CS − ( t, X )] , ∇ ψ − X ( t ) = − E [ X CS + ( t, X )] , let L n ( s, t ) := 1 n n (cid:88) j =1 W n,j ( s ) W n,j ( t ) (cid:124) , (27)where W n,j ( t ) := Y n,j CS + ( t, Y n,j ) − Y n,j Ψ ,n ( t ) − t (cid:124) Y n,j Ψ ,n ( t ) − ( Y n,j Y (cid:124) n,j + I d )Ψ ,n ( t ) − Ψ ,n ( t )( Y n,j Y (cid:124) n,j − I d ) t, (28)and Ψ ,n ( t ) := 1 n n (cid:88) j =1 CS + ( t, Y n,j ) , Ψ ,n ( t ) := 1 n n (cid:88) j =1 Y n,j CS − ( t, Y n,j ) , (29) Ψ ,n ( t ) := 1 n n (cid:88) j =1 Y n,j CS + ( t, Y n,j ) , Ψ ,n ( t ) := 1 n n (cid:88) j =1 Y n,j Y (cid:124) n,j CS − ( t, Y n,j ) . (30)Furthermore, let z n ( t ) := 1 n n (cid:88) j =1 Y n,j CS + ( t, Y n,j ) − tψ ( t ) . (31)We then have the following result. 11esting normality in any dimension by Fourier methods in a multivariate Stein equation d \ a ( −√ , √ d (0 , / √ d (0 , √ /π ) d ∆ a Theorem 4.8.
Let (cid:98) σ n,a := 4 (cid:90) (cid:90) z n ( s ) (cid:124) L n ( s, t ) z n ( t ) w a ( s ) w a ( t ) d s d t, where L n ( s, t ) and z n ( t ) are defined in (27) and (31) , respectively. If E (cid:107) X (cid:107) < ∞ , then ( (cid:98) σ n,a ) is a consistentsequence of estimators for σ a , i.e., we have (cid:98) σ n,a P −→ σ a . Moreover, (cid:98) σ n,a = (cid:88) i,j =1 (cid:98) σ i,jn,a , (32) where (cid:98) σ i,jn,a is given in (47) . Since the proof of Theorem 4.8 is long and tedious, it is deferred to Appendix A. We stress that the representation (32)does not comprise any integral, which means that (cid:98) σ n,a is a feasible estimator.We close this section with an example that illustrates the feasibility of the asymptotic confidence interval. To this end,we consider the following standardized symmetric alternatives to normality. Firstly, let X D = U ( −√ , √ d have theuniform distribution on the cube ( −√ , √ d . In this case, we have ϕ X ( t ) = d (cid:89) i =1 sin( √ t i ) √ t i , ∇ ϕ X ( t ) ( j ) = 3 cos( √ t j ) t j − √ √ t j )3 t j d (cid:89) i (cid:54) = j sin( √ t i ) √ t i , where ∇ ϕ ( t ) ( j ) is the j th component of ∇ ϕ ( t ) . Secondly, we consider a Laplace distribution with i.i.d. marginals,denoted by Laplace (0 , / √ d , for which ϕ X ( t ) = d (cid:89) i =1
22 + t i , ∇ ϕ X ( t ) ( j ) = − t j (2 + t j ) d (cid:89) i (cid:54) = j
22 + t i . Finally, let X have a logistic distribution with i.i.d. marginals, denoted by Logistic (0 , /π ) d . In this case, we obtain ϕ X ( t ) = d (cid:89) i =1 √ t i sinh( √ t i ) , ∇ ϕ X ( t ) ( j ) = √ √ t j ) − t j cosh( √ t j )sinh( √ t j ) d (cid:89) i (cid:54) = j √ t i sinh( √ t i ) . In each case, ∆ a has been computed by numerical integration. The resulting values are displayed in Table 3.By means of a Monte Carlo study, we estimated the probability of coverage of the confidence interval I n,a,α figuringin corollary 4.7 for a ∈ { . , , , } , d ∈ { , } , and the sample sizes n ∈ { , , , , , } . The nominallevel is . , and the number of replications is . Simulations have been carried out with the statistic software R,see [47]. In particular, we used the package extraDistr , see [59], to generate variates from the Laplace distribution.The results are displayed in Table 4. As one can see the empirical coverage is converging to the nominal level, whileit is obviously slower in higher dimensions. For larger values of the tuning parameter a the confidence interval tendsto be too wide, so we conjecture that an improvement of the asymptotic interval can be found.12esting normality in any dimension by Fourier methods in a multivariate Stein equation This section presents the results of a Monte Carlo study, with the aim to compare the power of the proposed testwith respect to that of prominent competitors against selected alternatives. We used the statistic software R , see [47],and we employed the package MonteCarlo , see [38], which allows for parallel computing. In addition, we used thepackage expm , see [22], for the standardization of the data. Critical values for the test statistic have been estimatedby means of extensive simulations (100000 replications), and they are displayed in Table 5 for the weight parameters a ∈ { . , , , , , ∞} and the sample sizes n ∈ { , , } . Throughout, the level of significance is α = 0 . .For the sake of comparison, Table 5 displays the approximate critical values of T ∞ ,a in the special case d = 1 , whichhave been obtained in Section 3 by choosing a distribution of the Pearson family by equating the first four moments.As already mentioned in Section 2, the test statistic T n, ∞ is a linear combination of skewness in the sense of Mardia[42] and skewness in the sense of Móri, Rohatgi und Székely [44], and it equals the statistic HV ∞ of Henze–Visagie,see [31]. In the univariate case d = 1 , we compared the power of our novel test statistics with several competitors, which are • the Cramér–von Mises test (CvM), • the Anderson–Darling test (AD), • the Shapiro–Wilk test (SW), • the Baringhaus–Henze–Epps–Pulley test (BHEP), • the Henze–Visagie test (HV).The first three of these tests are well-known. The CvM-test and the AD-test have been implemented with the R -package nortest , see [23], which contains the functions cvm.test and ad.test , and for the SW-test we used thefunction shapiro.test of the stats -package. The test statistics BHEP and HV will be explained in (33) and (34),respectively.For the BHEP-test and the HV-test, critical values have been simulated with replications. These values andthose of Table 5 for the novel test statistics have been employed to assess the power of the various tests against severalalternatives. Table 6 exhibits percentages of rejection based on 100000 replications. An asterisk denotes power of100% and the best performing test for each alternative is marked in boldface. The choice of alternatives orients itselftowards those used in [31]. The acronym NMix1 denotes a mixture of the normal distributions N (0 , and N (3 , with weights . and . , respectively.The novel tests outperform the selected competitors for the t -distribution, the χ (15) -distribution and the distributionNMix1, and they keep up with the other procedures against the remaining alternatives. For most of the alternatives,power does not change much with varying the weight parameter a . A notable exception is the uniform distributionU ( −√ , √ , against which power breaks down for larger tuning parameters, a feature shared by the HV-test. For the dimensions d = 2 , d = 3 and d = 5 , we compared the novel test statistic with the following procedures: • the test of Baringhaus–Henze–Epps–Pulley (BHEP), • the test of Henze–Zirkler (HZ), • the test of Henze–Visagie (HV), • the energy test (EN).A recent synopsis of tests for multivariate normality is given in [18]. Just as the novel procedure, the BHEP-test (see[32]) is based on the empirical characteristic function (ECF). More precisely, it employs the test statisticBHEP a = (cid:90) | ψ n ( t ) − ψ ( t ) | ϕ a ( t ) d t, (33)where ϕ a ( t ) = (2 πa ) − d/ exp( −(cid:107) t (cid:107) / (2 a )) , and ψ n ( t ) and ψ ( t ) are given in (3) and (2), respectively. An alterna-tive representation for BHEP a isBHEP a = 1 n (cid:88) i,j =1 exp (cid:16) − a (cid:13)(cid:13) Y n,i − Y n,j (cid:13)(cid:13) (cid:17) − a ) − d n n (cid:88) j =1 exp (cid:18) − a (cid:13)(cid:13) Y n,j (cid:13)(cid:13) a ) (cid:19) + (1 + 2 a ) − d . a = 1 .The test HZ of Henze–Zirkler (cf. [33]) originates if we choose a = 1 / √ d + 1) n/ d +4 in the BHEP test. The R -package HZ , see [36], contains the function mvn , which calculates the statistic of the HZ-test.The recent test of Henze–Visagie, see [31], is the ’moment generating function analog’ of our novel test statistic. Itemploys the test statistic HV a = n (cid:90) (cid:107)∇ M n ( t ) − tM n ( t ) (cid:107) w a ( t ) d t, where M n ( t ) = n − (cid:80) nj =1 exp( t (cid:124) Y n,j ) is the empirical moment generating function of the scaled residuals. Analternative representation of HV a isHV a = 1 n (cid:16) πa (cid:17) d n (cid:88) i,j =1 exp (cid:16) (cid:107) Y n,i + Y n,j (cid:107) a (cid:17)(cid:16) Y (cid:124) n,i Y n,j + (cid:107) Y n,i + Y n,j (cid:107) (cid:16) a − a (cid:17) + d a (cid:17) . (34)In our comparative study, we put a = 5 , as recommended in [31].The rationale of the energy test of Székely and Rizzo, see [54], is based on the fact that, if X and Y are independentintegrable d -dimensional random vectors and X (cid:48) , Y (cid:48) denote independent copies of X and Y , respectively, then E (cid:107) X − Y (cid:107) − E (cid:107) X − X (cid:48) (cid:107) − E (cid:107) Y − Y (cid:48) (cid:107) ≥ . Here, equality holds if and only if X D = Y . The statistic of the energy test for multivariate normality isEN = n (cid:16) n n (cid:88) j =1 E (cid:107) (cid:101) Y n,j − Z (cid:107) − E (cid:107) Z − Z (cid:107) − n n (cid:88) i,j =1 E (cid:107) (cid:101) Y n,i − (cid:101) Y n,j (cid:107) (cid:17) . Here, (cid:101) Y n,j = (cid:112) n/ ( n − Y n,j , and Z , Z are i.i.d. with the normal distribution N d (0 , I d ) , which are also independentof Y n, , . . . , Y n,n . To calculate EN, notice that E (cid:107) Z − Z (cid:107) = 2Γ( d +12 ) / Γ( d ) and E (cid:107) a − Z (cid:107) = √ d +12 )Γ( d ) + (cid:114) π ∞ (cid:88) k =0 ( − k k !2 k (cid:107) a (cid:107) k +2 (2 k + 1)(2 k + 2) 2Γ( d +12 )Γ( k + )Γ( k + d + 1) . The R -package energy [48] contains the function mvnorm.etest to calculate EN. Note that all of the mentionedprocedures are also implemented in the R -package mnt , see [9].Just as done in the case d = 1 , we first simulated critical values with 100000 replications. With the same number ofreplications, we then simulated the power of the tests under discussion against selected alternatives. Again, the choiceof alternatives orients itself towards those used in [31]. Tables 7, 8 and 9 display percentages of rejection of H fordimensions d = 2 , d = 3 and d = 5 , respectively, and an asterisk again denotes power 100%. To generate pseudorandom numbers, we used the R -packages mvtnorm , see [20], and PearsonDS , see [7]. Suppressing the dimension d , the distribution NMix1 is a mixture of the normal distributions N d (0 , I d ) and N d (3 , I d ) with mixing proportions . and . , respectively. Here, stands for the d -dimensional vector that contains in each component. Likewise,NMix2 denotes a mixture of the normal distributions N d (0 , I d ) and N d (0 , B d ) with mixing proportions 0.1 and 0.9,respectively. Here, B d is a d × d -matrix with for each diagonal entry and . for each off-diagonal entry.The novel tests outperform their competitors for some alternatives, notably for the χ -, the Γ -, and the NMix-distribution, but they can also keep up for the other alternatives. However, just as in the univariate case, power isextremely low against the uniform distribution U ( −√ , √ , a feature shared by the HV-test. Based on the results ofthis simulation study, we recommend the choice a = 5 for the tuning parameter, since it leads to competitive poweragainst nearly each of the alternatives considered. The Black-Scholes-Merton model is a stochastic model for the dynamics of a financial market that contains derivativeinvestment instruments. One of the basic assumptions of this model is the normality of the log returns of stocksand indexes. To test the hypothesis joint normality of log returns of several indexes, we consider the five stockindexes Standard & Poor 500 (^GSPC), Dow Jones Industrial Average (^DJI), NASDAQ Composite (^IXIC), DAXPerfomance Index (^GDAXI), and EURO STOXX 50 (^STOXX50E), over a period of 50 trading days, starting July1st, 2017. The data (daily closing prices of the stocks) were obtained by means of the R -package quantmod , see [49].14esting normality in any dimension by Fourier methods in a multivariate Stein equation −0.02 −0.01 0.00 0.01 0.02 − . . . . GDAXI −0.01 0.00 0.01 0.02
STOXX50E −0.03 −0.02 −0.01 0.00 0.01
GSPC −0.04 −0.02 0.00 0.02
IXIC −0.03 −0.02 −0.01 0.00 0.01 − . . . . − . . . . − . − . . − . − . . . −0.03 −0.02 −0.01 0.00 0.01 − . − . . DJI
Figure 1: 2D projections of the log returns of the indexes .To model the independence assumption between the realisations, we ignored a time span of 10 trading days betweeneach of the five dimensional observations. Figure 1 shows a plot of the two-dimensional projections of the log returns.For each value a ∈ { . , , , , } of the weight parameter a , we performed a Monte Carlo simulation based on100000 replications, in order to estimate the p-value of the observations. The empirical p-values are displayed inTable 10. As can be seen, the hypothesis of a multivariate normality of the log returns of the selected stock prices isrejected at the 1%-level, for each of the choices of the weight parameter a . We propose a novel class of tests of normality based on an initial value problem connected to a multivariate Steinequation, which characterises the multivariate standard normal law. We derived asymptotic theory under the null hy-pothesis as well as under contiguous and fixed alternatives. Moreover, we proved consistency against each alternativedistribution that satisfies a weak moment condition, and we provided insights into the structure of the behaviour ofthe test statistic under fixed alternatives by calculating asymptotic confidence intervals for ∆ a , and by providing aconsistent estimator for the limiting variance σ a . Monte Carlo simulations show that the methods operate as expected,and that the new family of tests is a strong class of competitors to established procedures.A first open question for further research is to find explicit formulae or numerical stable approximations for the eigen-values λ j ( a ) , j = 1 , , . . . connected to the integral operator K in (13). We also leave as an open problem thecalculation of higher cumulants of T ∞ ,a for dimensions d > . Results of this kind would open ground to efficientapproximation methods for the computation of critical values that avoid Monte Carlo simulations and efficiency state-ments, since the largest eigenvalue has a crucial influence on the approximate Bahadur efficiency, see [2, 45]. An15esting normality in any dimension by Fourier methods in a multivariate Stein equationpromising new field of interest in connection with tests of multivariate normality is to consider their behaviour inhigh-dimensional settings, i.e., to answer the question whether one can find a suitable rescaling and shifting of the teststatistic to obtain a non trivial limit distribution under a suitable limiting regime, under which, e.g., n, d → ∞ suchthat d/n → τ ∈ [0 , ∞ ] . For first results, see [10]. As a starting point, we conjecture that for a sequence ( n d ) d ∈ N ,where n d ≥ d + 1 and n d = o (cid:16)(cid:0) a a +1 (cid:1) − d (cid:17) , we have under H as d → ∞ (cid:16) aπ (cid:17) d T n d ,a d a . s . −→ . Finally, it would be of interest to consider a related family of test statistics, which is given by S n,a = n (cid:90) R d (cid:107)∇ ψ n ( t ) + tψ n ( t ) (cid:107) C w a ( t ) d t. Thus, the theoretical CF in T n,a has been replaced by the empirical counterpart. Note that in the univariate case, thisfamily is extensively studied in [17], but the generalisation to higher dimensions is still open. We conjecture thatsimilar results as derived in Sections 2 to 4 hold for S n,a . Acknowledgment
The authors thank Yvik Swan for sharing his knowledge of multivariate Stein operators and Stein characterisations.
A Proofs
A.1 Proof of Theorem 2.1
Proof.
Putting t = ( t , . . . , t d ) (cid:124) ∈ R d and Y n,j = ( Y (1) n,j , . . . , Y ( d ) n,j ) (cid:124) , some algebra (using symmetry and the additiontheorem for the cosine function) yields T n,a = n (cid:90) (cid:13)(cid:13) ∇ ψ n ( t ) + tψ ( t ) (cid:13)(cid:13) C w a ( t ) d t = n (cid:90) (cid:13)(cid:13)(cid:13) n n (cid:88) j =1 i Y n,j exp (cid:0) i t (cid:124) Y n,j (cid:1) + tψ ( t ) (cid:13)(cid:13)(cid:13) C w a ( t ) d t = n (cid:90) (cid:13)(cid:13)(cid:13) n n (cid:88) j =1 (cid:110) tψ ( t ) − Y n,j sin( t (cid:124) Y n,j ) + i Y n,j cos( t (cid:124) Y n,j ) (cid:111)(cid:13)(cid:13)(cid:13) C w a ( t ) d t = n (cid:90) d (cid:88) k =1 (cid:40)(cid:18) n n (cid:88) j =1 t ( k ) ψ ( t ) − Y ( k ) n,j sin( t (cid:124) Y n,j ) (cid:19) + (cid:18) n n (cid:88) j =1 Y ( k ) n,j cos( t (cid:124) Y n,j ) (cid:19) (cid:41) w a ( t ) d t = n (cid:90) d (cid:88) k =1 (cid:110) t ( k ) t ( k ) exp (cid:0) − (cid:107) t (cid:107) ) − n n (cid:88) j =1 t ( k ) ψ ( t ) Y ( k ) n,j sin( t (cid:124) Y n,j )+ 1 n n (cid:88) i,j =1 Y ( k ) n,j Y ( k ) n,i cos( t (cid:124) ( Y n,i − Y n,j )) (cid:111) w a ( t ) d t. We thus have T n,a = n (cid:90) (cid:26) (cid:107) t (cid:107) exp (cid:0) − ( a + 1) (cid:107) t (cid:107) (cid:1) − n n (cid:88) j =1 t (cid:124) Y n,j sin( t (cid:124) Y n,j ) exp (cid:16) − (cid:16) a + 12 (cid:17) (cid:107) t (cid:107) (cid:17) + 1 n n (cid:88) i,j =1 Y (cid:124) n,i Y n,j cos (cid:0) t (cid:124) ( Y n,i − Y n,j ) (cid:1) exp (cid:0) − a (cid:107) t (cid:107) (cid:1)(cid:27) d t. (cid:90) (cid:107) t (cid:107) exp (cid:0) − a (cid:107) t (cid:107) (cid:1) d t = (cid:18) πa (cid:19) d d a , (35) (cid:90) cos( t (cid:124) c ) exp (cid:0) − a (cid:107) t (cid:107) (cid:1) d t = (cid:18) πa (cid:19) d exp (cid:18) − (cid:107) c (cid:107) a (cid:19) , (36) (cid:90) t (cid:124) c sin( t (cid:124) c ) exp (cid:0) − a (cid:107) t (cid:107) (cid:1) d t = (cid:18) πa (cid:19) d (cid:107) c (cid:107) a exp (cid:18) − (cid:107) c (cid:107) a (cid:19) , (37)the assertion follows readily. A.2 Proof of Theorem 3.1
Proof.
Recall that, in view of invariance, there is no loss of generality if we assume X D = N d (0 , I d ) . With the notationin (14), Z n defined in (10) takes the form Z n ( t ) = 1 √ n n (cid:88) j =1 (cid:0) Y n,j CS + ( t, Y n,j ) − tψ ( t ) (cid:1) . To prove Theorem 3.1, we use a central limit theorem for Hilbert space valued random elements, see, e.g., Theorem 2.7of [8]. Since Z n does not comprise independent summands, we approximate Z n by a sum of i.i.d. random elementsof H . To this end, we introduce the auxiliary random elements (cid:101) Z n ( t ) := 1 √ n n (cid:88) j =1 (cid:0) ( X j + ∆ n,j ) CS + ( t, X j ) − tψ ( t ) + X j CS − ( t, X j ) t (cid:124) ∆ n,j (cid:1) ,Z ∗ n ( t ) := 1 √ n n (cid:88) j =1 (cid:16) X j CS + ( t, X j ) − (cid:0) t + X j + (2 I d − tt (cid:124) ) 12 ( X j X (cid:124) j − I d ) t − t (cid:124) X j t (cid:1) ψ ( t ) (cid:17) (38) =: 1 √ n n (cid:88) j =1 Z ∗∗ j ( t ) (say), where ∆ n,j = Y n,j − X j = ( S − n − I d ) X j − S − n X n . (39)The proof of Theorem 3.1 comprises 3 steps. We show Z ∗ n D −→ Z in H , (40) (cid:107) Z n − (cid:101) Z n (cid:107) H P −→ , (41) (cid:107) (cid:101) Z n − Z ∗ n (cid:107) H P −→ . (42)The assertion then follows from Slutsky’s lemma. To prove (40), notice that Z ∗∗ , Z ∗∗ , . . . is a sequence of i.i.d.random elements of H . These elements are centred, since E [ Z ∗∗ ( t )] = E (cid:104) X CS + ( t, X ) − (cid:0) t + X + (2 I d − tt (cid:124) ) 12 ( XX (cid:124) − I d ) t − t (cid:124) Xt (cid:1) ψ ( t ) (cid:105) = E (cid:2) X CS + ( t, X ) − tψ ( t ) (cid:3) = 0 , t ∈ R d . The covariance matrix kernel E (cid:2) Z ∗ n ( s ) Z ∗ n ( t ) (cid:124) (cid:3) = E (cid:2) Z ∗∗ ( s ) Z ∗∗ ( t ) (cid:124) (cid:3) = K ( s, t ) (say), where s, t ∈ R d , is given by K ( s, t ) = E (cid:104)(cid:16) X CS + ( s, X ) − (cid:0) s + X + (2 I d − ss (cid:124) ) 12 ( XX (cid:124) − I d ) s − s (cid:124) Xs (cid:1) ψ ( s ) (cid:17)(cid:16) X CS + ( t, X ) − (cid:0) t + X + (2 I d − tt (cid:124) ) 12 ( XX (cid:124) − I d ) t − t (cid:124) Xt (cid:1) ψ ( t ) (cid:17) (cid:124) (cid:105) . E [ X ] = 0 and E [ XX (cid:124) ] = I d , tedious but straightforward calculations yield K ( s, t ) = E (cid:2) XX (cid:124) CS + ( s, X ) CS + ( t, X ) (cid:3) − sψ ( s ) E (cid:2) X (cid:124) CS + ( t, X ) (cid:3) − ψ ( s ) E (cid:2) XX (cid:124) CS + ( t, X ) (cid:3) − ψ ( s ) E (cid:2)(cid:0) (2 I d − ss (cid:124) ) 12 ( XX (cid:124) − I d ) − s (cid:124) X (cid:1) sX (cid:124) CS + ( t, X ) (cid:3) − E (cid:2) X CS + ( s, X ) (cid:3) t (cid:124) ψ ( t ) + st (cid:124) ψ ( s ) ψ ( t ) − E (cid:2) XX (cid:124) CS + ( s, X ) (cid:3) ψ ( t ) + I d ψ ( s ) ψ ( t )+ E (cid:2)(cid:0) (2 I d − ss (cid:124) ) 12 ( XX (cid:124) − I d ) − s (cid:124) X (cid:1) sX (cid:124) (cid:3) ψ ( s ) ψ ( t ) − E (cid:2) X CS + ( s, X ) t (cid:124) (cid:0) (2 I d − tt (cid:124) ) 12 ( XX (cid:124) − I d ) − t (cid:124) X (cid:1) (cid:124) (cid:3) ψ ( t )+ E (cid:2) Xt (cid:124) (cid:0) (2 I d − tt (cid:124) ) 12 ( XX (cid:124) − I d ) − t (cid:124) X (cid:1) (cid:124) (cid:3) ψ ( s ) ψ ( t )+ E (cid:2)(cid:0) (2 I d − ss (cid:124) ) 12 ( XX (cid:124) − I d ) − s (cid:124) X (cid:1) st (cid:124) (cid:0) (2 I d − tt (cid:124) ) 12 ( XX (cid:124) − I d ) − t (cid:124) X (cid:1) (cid:124) (cid:3) ψ ( s ) ψ ( t ) . Since the occurring expectations are given by E (cid:2) CS + ( t, X ) (cid:3) = ψ ( t ) , E (cid:2) X CS + ( t, X ) (cid:3) = tψ ( t ) , E (cid:2) X CS − ( t, X ) (cid:3) = − tψ ( t ) , E (cid:2) XX (cid:124) CS + ( t, X ) (cid:3) = ( I d − tt (cid:124) ) ψ ( t ) , E (cid:2) XX (cid:124) CS − ( t, X ) (cid:3) = ( I d − tt (cid:124) ) ψ ( t ) , E (cid:2) s (cid:124) XXX (cid:124) CS + ( t, X ) (cid:3) = (cid:0) s (cid:124) t ( I d − tt (cid:124) ) + st (cid:124) + ts (cid:124) (cid:1) ψ ( t ) , E (cid:104) XX (cid:124) CS + ( s, X ) CS + ( t, X ) (cid:105) = E (cid:104) XX (cid:124) (cid:0) sin( s + t ) + cos( s − t ) (cid:1)(cid:105) = (cid:0) I d − ( s − t )( s − t ) (cid:124) (cid:1) ψ ( s − t ) , E (cid:2) s (cid:124) XXX (cid:124) (cid:3) = 0 ∈ R d × d , E (cid:2) ( XX (cid:124) − I d ) st (cid:124) ( XX (cid:124) − I d ) (cid:3) = ts (cid:124) + s (cid:124) t I d , some algebra shows that K ( s, t ) takes the form given in (12). Thus, by the central limit theorem in Hilbert spaces,(40) follows. To prove (41), notice that cos( t (cid:124) Y n,j ) = cos( t (cid:124) X j ) − sin( t (cid:124) X j ) t (cid:124) ∆ n,j + ε n,j ( t ) , sin( t (cid:124) Y n,j ) = sin( t (cid:124) X j ) + cos( t (cid:124) X j ) t (cid:124) ∆ n,j + η n,j ( t ) , where max( | ε n,j ( t ) | , | η n,j ( t ) | ) ≤ (cid:107) t (cid:107) (cid:107) ∆ n,j (cid:107) . (43)Hence CS + ( t, Y n,j ) = CS + ( t, X j ) + CS − ( t, X j ) t (cid:124) ∆ n,j + ε n,j ( t ) + η n,j ( t ) , and some algebra gives Z n ( t ) − (cid:101) Z n ( t ) = 1 √ n n (cid:88) j =1 (cid:0) ( X j + ∆ n,j )( ε n,j ( t ) + η n,j ( t )) + ∆ n,j CS − ( t, X j ) t (cid:124) ∆ n,j (cid:1) . Putting A n = 1 √ n n (cid:88) j =1 (cid:107) X j (cid:107)(cid:107) ∆ n,j (cid:107) , B n = 1 √ n n (cid:88) j =1 (cid:107) ∆ n,j (cid:107) , C n = 1 √ n n (cid:88) j =1 (cid:107) ∆ n,j (cid:107) , (43) and the Cauchy–Schwarz inequality yield (cid:107) Z n ( t ) − (cid:101) Z n ( t ) (cid:107) ≤ A n (cid:107) t (cid:107) + B n (cid:107) t (cid:107) + C n (cid:107) t (cid:107) . By Theorem 5.2 of [6], we have n − / max j =1 ,...,n (cid:107) X j (cid:107) a . s . −→ . Invoking Proposition A.1 of [14], according towhich n / max j =1 ,...,n (cid:107) ∆ n,j (cid:107) a . s . −→ and (cid:80) nj =1 (cid:107) ∆ n,j (cid:107) = O P (1) , it is readily seen that each of the expressions A n , B n and C n converges to zero in probability as n → ∞ . In view of (cid:107) Z n − (cid:101) Z n (cid:107) H ≤ (cid:90) (cid:0) A n (cid:107) t (cid:107) + B n (cid:107) t (cid:107) + C n (cid:107) t (cid:107) (cid:1) w a ( t ) d t A n ( t ) = 1 √ n n (cid:88) j =1 (cid:16) ∆ n,j CS + ( t, X j ) + (cid:0) X j + 12 ( X j X (cid:124) j − I d ) t (cid:1) ψ ( t ) (cid:17) ,B n ( t ) = 1 √ n n (cid:88) j =1 (cid:16) X j CS − ( t, X j ) t (cid:124) ∆ n,j + (cid:0) ( I d − tt (cid:124) ) 12 ( X j X (cid:124) j − I d ) t − t (cid:124) X j t (cid:1) ψ ( t ) (cid:17) . Using the triangle inequality, some calculations give (cid:107) (cid:101) Z n − Z ∗ n (cid:107) H ≤ (cid:107) A n (cid:107) H + (cid:107) B n (cid:107) H , and thus (42) follows ifwe can show that (cid:107) A n (cid:107) H = o P (1) and (cid:107) B n (cid:107) H = o P (1) . We only prove (cid:107) A n (cid:107) H = o P (1) , since the reasoning for (cid:107) B n (cid:107) H = o P (1) is completely similar. From the definition of ∆ n,j in (39), we have A n ( t ) = ( S − n − I d ) 1 √ n n (cid:88) j =1 (cid:0) X j CS + ( t, X j ) − tψ ( t ) (cid:1) − S − n X n √ n n (cid:88) j =1 (cid:0) CS + ( t, X j ) − ψ ( t ) (cid:1) − ψ ( t ) (cid:0) S − n − I d (cid:1) √ nX n + (cid:16) √ n ( S − n − I d ) + 12 √ n n (cid:88) j =1 (cid:0) X j X (cid:124) j − I d (cid:1)(cid:17) tψ ( t )= A n, ( t ) − A n, ( t ) − A n, ( t ) + A n, ( t ) , say, and thus it remains to prove that each of (cid:107) A n,k (cid:107) H , k ∈ { , , , } , is o P (1) . Letting (cid:107) · (cid:107) denote the spectralnorm, it follows that (cid:107) A n, (cid:107) H ≤ (cid:13)(cid:13) √ n ( S − n − I d ) (cid:13)(cid:13) (cid:13)(cid:13)(cid:13) n n (cid:88) j =1 (cid:0) X j CS + ( t, X j ) − tψ ( t ) (cid:1)(cid:13)(cid:13)(cid:13) H . Here, the first factor on the right hand side is O P (1) , and the second converges to zero almost surely because of thestrong law of large numbers in H . As for (cid:107) A n, (cid:107) H , it holds that (cid:107) A n, (cid:107) H ≤ (cid:13)(cid:13) S − n (cid:13)(cid:13) (cid:13)(cid:13) √ nX n (cid:13)(cid:13) (cid:13)(cid:13)(cid:13) n n (cid:88) j =1 (cid:0) CS + ( t, X j ) − ψ ( t ) (cid:1)(cid:13)(cid:13)(cid:13) L . Here, each of the first two factors on the right hand side are O P (1) , and the last one converges to zero almost surelybecause of the strong law of large numbers in L . The term (cid:107) A n, (cid:107) H is bounded from above by (cid:107) A n, (cid:107) H ≤ (cid:107)√ n ( S − n − I d ) (cid:107) (cid:107) X n (cid:107) (cid:90) exp( −(cid:107) t (cid:107) ) w a ( t ) d t. Hence (cid:107) A n, (cid:107) H = o P (1) since (cid:107) X n (cid:107) = o P (1) . Finally, we have (cid:107) A n, (cid:107) H ≤ (cid:13)(cid:13)(cid:13) √ n ( S − n − I d ) + 12 √ n n (cid:88) j =1 ( X j X (cid:124) j − I d ) (cid:13)(cid:13)(cid:13) (cid:90) (cid:107) t (cid:107) exp( −(cid:107) t (cid:107) ) w a ( t ) d t. From display (2.13) of [32], the factor preceding the integral is o P (1) , and thus (cid:107) A n, (cid:107) H = o P (1) . The proof ofTheorem 3.1 is completed. A.3 Proof of Theorem 4.8
Proof.
Since the proof is analogous to that given in [14], it will only be sketched. The first observation is that thequantities Ψ (cid:96),n ( t ) , (cid:96) ∈ { , , , } , defined in (29), (30) have the following almost sure limits: Ψ ,n ( t ) a . s . −→ ψ + X ( t ) , Ψ ,n ( t ) a . s . −→ ∇ ψ + X ( t ) , Ψ ,n ( t ) a . s . −→ −∇ ψ − X ( t ) , Ψ ,n ( t ) a . s . −→ E [ XX (cid:124) CS − ( t, X )] . Here, the convergence of Ψ ,n ( t ) is assertion a) of Lemma 6.6 of [14], and the remaining claims follow mutatismutandis the reasoning given in the proof of Lemma 6.6. of [14]. From (27) and (28), we have L n ( s, t ) = (cid:88) i,j =1 L i,jn ( s, t ) , (44)19esting normality in any dimension by Fourier methods in a multivariate Stein equationwhere L i,jn ( s, t ) = L j,in ( t, s ) (cid:124) and – putting I ± n,j := Y n,j Y (cid:124) n,j ± I d – L , n ( s, t ) = 1 n n (cid:88) j =1 Y n,j CS + ( s, Y n,j ) Y (cid:124) n,j CS + ( t, Y n,j ) , L , n ( s, t ) = − n n (cid:88) j =1 Y n,j CS + ( s, Y n,j ) Y (cid:124) n,j Ψ ,n ( t ) ,L , n ( s, t ) = − n n (cid:88) j =1 Y n,j CS + ( s, Y n,j ) t (cid:124) Y n,j Ψ ,n ( t ) (cid:124) , L , n ( s, t ) = − n n (cid:88) j =1 Y n,j CS + ( s, Y n,j )Ψ ,n ( t ) (cid:124) I + n,j ,L , n ( s, t ) = − n n (cid:88) j =1 Y n,j CS + ( s, Y n,j ) t (cid:124) I − n,j Ψ ,n ( t ) , L , n ( s, t ) = 1 n n (cid:88) j =1 Y n,j Ψ ,n ( s ) Y (cid:124) n,j Ψ ,n ( t ) ,L , n ( s, t ) = 1 n n (cid:88) j =1 Y n,j Ψ ,n ( s ) t (cid:124) Y n,j Ψ ,n ( t ) (cid:124) , L , n ( s, t ) = 12 n n (cid:88) j =1 Y n,j Ψ ,n ( s )Ψ ,n ( t ) (cid:124) I + n,j ,L , n ( s, t ) = 12 n n (cid:88) j =1 Y n,j Ψ ,n ( s ) t (cid:124) I − n,j Ψ ,n ( t ) , L , n ( s, t ) = 1 n n (cid:88) j =1 s (cid:124) Y n,j Ψ ,n ( s ) t (cid:124) Y n,j Ψ ,n ( t ) (cid:124) ,L , n ( s, t ) = 12 n n (cid:88) j =1 s (cid:124) Y n,j Ψ ,n ( s )Ψ ,n ( t ) (cid:124) I + n,j , L , n ( s, t ) = 12 n n (cid:88) j =1 s (cid:124) Y n,j Ψ ,n ( s ) t (cid:124) I − n,j Ψ ,n ( t ) ,L , n ( s, t ) = 14 n n (cid:88) j =1 I + n,j Ψ ,n ( s )Ψ ,n ( t ) (cid:124) I + n,j , L , n ( s, t ) = 14 n n (cid:88) j =1 I + n,j Ψ ,n ( s ) t (cid:124) I − n,j Ψ ,n ( t ) ,L , n ( s, t ) = 14 n n (cid:88) j =1 Ψ ,n ( s )I − n,j st (cid:124) I − n,j Ψ ,n ( t ) . From (44), it follows that (cid:98) σ n,a = (cid:80) i,j =1 (cid:98) σ i,jn,a , where (cid:98) σ i,jn,a = 4 (cid:90) (cid:90) z n ( s ) (cid:124) L i,jn ( s, t ) z n ( t ) w a ( s ) w a ( t ) d s d t. (45)Notice that (cid:98) σ i,jn,a = (cid:98) σ j,in,a . In view of (22) and (21), we have L ( s, t ) = (cid:80) i,j =1 L i,j ( s, t ) , where L i,j ( s, t ) = E [ w i ( s, X ) w j ( t, X ) (cid:124) ] , and w ( t, X ) = X CS + ( t, X ) , w ( t, X ) = − Xψ + X ( t ) , w ( t, X ) = − t (cid:124) X ∇ ψ + X ( t ) ,w ( t, X ) = 12 ( XX (cid:124) + I d ) ∇ ψ − X ( t ) , w ( t, X ) = − E [ XX (cid:124) CS − ( t, X )]( XX (cid:124) − I d ) t. Therefore, σ a = (cid:80) i,j =1 σ i,ja , where σ i,ja = 4 (cid:90) (cid:90) z ( s ) (cid:124) L i,j ( s, t ) z ( t ) w a ( s ) w a ( t ) d s d t and, by symmetry, L i,j ( s, t ) = L j,i ( t, s ) (cid:124) and hence σ i,ja = σ j,ia . We thus have to prove (cid:98) σ i,jn,a P −→ σ i,ja for each choiceof i, j ∈ { , . . . , } . To this end, we proceed in two steps. The first one is to replace L i,jn ( s, t ) in (45) with L i,jn, ( s, t ) .Here, L i,jn, ( s, t ) originates from L i,jn ( s, t ) by throughout replacing Y n,j with X j , and this replacement also affects thequantities Ψ (cid:96),n ( t ) , (cid:96) ∈ { , . . . , } . Moreover, we replace z n ( t ) with z n, ( t ) = n − (cid:80) nj =1 X j CS + ( t, X j ) − tψ ( t ) .Putting (cid:98) σ i,jn, ,a = 4 (cid:90) (cid:90) z n, ( s ) (cid:124) L i,jn, ( s, t ) z n, ( t ) w a ( s ) w a ( t ) d s d t, it follows from Fubini’s theorem that (cid:98) σ i,jn, ,a P −→ σ i,ja . The second, much more technical step is to prove (cid:98) σ i,jn,a − (cid:98) σ i,jn, ,a = o P (1) . To this end, notice that z n ( s ) (cid:124) L i,jn ( s, t ) z n ( t ) − z n, ( s ) (cid:124) L i,jn, ( s, t ) z n, ( t ) = z n ( s ) (cid:124) (cid:0) L i,jn ( s, t ) − L i,jn, ( s, t ) (cid:1) z n ( t )+ (cid:0) z n ( s ) − z n, ( s ) (cid:1) (cid:124) L i,jn, ( s, t ) z n ( t )+ z n, ( s ) (cid:124) L i,jn, ( s, t ) (cid:0) z n ( t ) − z n, ( t ) (cid:1) , (46)20esting normality in any dimension by Fourier methods in a multivariate Stein equationwhere (cid:12)(cid:12)(cid:0) z n ( s ) − z n, ( s ) (cid:1) (cid:124) L i,jn, ( s, t ) z n ( t ) (cid:12)(cid:12) ≤ (cid:13)(cid:13) z n ( s ) − z n, ( s ) (cid:13)(cid:13)(cid:13)(cid:13) L i,jn, ( s, t ) (cid:13)(cid:13) (cid:13)(cid:13) z n ( t ) (cid:13)(cid:13) , (cid:12)(cid:12) z n, ( s ) (cid:124) L i,jn, ( s, t ) (cid:0) z n ( t ) − z n, ( t ) (cid:1)(cid:12)(cid:12) ≤ (cid:13)(cid:13) z n, ( s ) (cid:13)(cid:13)(cid:13)(cid:13) L i,jn, ( s, t ) (cid:13)(cid:13) (cid:13)(cid:13) z n ( t ) − z n, ( t ) (cid:13)(cid:13) . We have (cid:107) z n, ( t ) (cid:107) ≤ n − (cid:80) nj =1 (cid:107) X j (cid:107) + (cid:107) t (cid:107) ψ ( t ) , and a Taylor expansion yields (cid:107) z n ( t ) (cid:107) ≤ n n (cid:88) j =1 (cid:0) (cid:107) X j (cid:107) + (cid:107) X j (cid:107)(cid:107) t (cid:107)(cid:107) ∆ n,j (cid:107) + (cid:107) ∆ n,j (cid:107) + (cid:107) t (cid:107)(cid:107) ∆ n,j (cid:107) (cid:1) + (cid:107) t (cid:107) ψ ( t ) , (cid:107) z n ( t ) − z n, ( t ) (cid:107) ≤ n n (cid:88) j =1 (cid:107) ∆ n,j (cid:107) + 2 (cid:107) t (cid:107) n n (cid:88) j =1 (cid:107) ∆ n,j (cid:107)(cid:107) X j (cid:107) . Notice that each of the terms (cid:107) L i,jn, ( s, t ) (cid:107) are bounded from above by terms of the type k (cid:107) s (cid:107) (cid:96) (cid:107) t (cid:107) m , multiplied withfinitely many products of the type n − (cid:80) nj =1 (cid:107) X j (cid:107) β , with k ≤ , (cid:96), m ∈ { , } , and β ∈ { , , , } . In view of thecondition E (cid:107) X (cid:107) < ∞ and the fact that n − (cid:80) nj =1 (cid:107) ∆ n,j (cid:107) k (cid:107) X k (cid:107) (cid:96) a . s . −→ (see Proposition A.2 of [14]), it followsthat (cid:90) (cid:90) (cid:12)(cid:12)(cid:0) z n ( s ) − z n, ( s ) (cid:1) (cid:124) L i,jn, ( s, t ) z n ( t ) (cid:12)(cid:12) w a ( s ) w a ( t ) d s d t P −→ , (cid:90) (cid:90) (cid:12)(cid:12) z n, ( s ) (cid:124) L i,jn, ( s, t ) (cid:0) z n ( t ) − z n, ( t ) (cid:1)(cid:12)(cid:12) w a ( s ) w a ( t ) d s d t P −→ . As a consequence, we only have to consider the first term on the right hand side of (46). To this end, notice that (cid:12)(cid:12) z n ( s ) (cid:124) (cid:0) L i,jn ( s, t ) − L i,jn, ( s, t ) (cid:1) z n ( t ) (cid:12)(cid:12) ≤ (cid:13)(cid:13) z n ( s ) (cid:13)(cid:13)(cid:13)(cid:13) L i,jn ( s, t ) − L i,jn, ( s, t ) (cid:13)(cid:13) (cid:13)(cid:13) z n ( t ) (cid:13)(cid:13) . To find an upper bound for (cid:107) L i,jn ( s, t ) − L i,jn, ( s, t ) (cid:107) , we have to consider each case i, j ∈ { , . . . , } such that i ≤ j separately. We will elaborate on the case i = j = 1 ; the other cases are treated similarly. We have (cid:13)(cid:13) L , n ( s, t ) − L , n, ( s, t ) (cid:13)(cid:13) = (cid:13)(cid:13)(cid:13) n n (cid:88) j =1 (cid:0) Y n,j Y (cid:124) n,j CS + ( s, Y n,j ) CS + ( t, Y n,j ) − X j X (cid:124) j CS + ( s, X j ) CS + ( t, X j ) (cid:1)(cid:13)(cid:13)(cid:13) , and a Taylor expansion yields (cid:107) L , n ( s, t ) − L , n, ( s, t ) (cid:107) ≤ n n (cid:88) j =1 (cid:107) X j (cid:107) (cid:0) (cid:107) t (cid:107)(cid:107) ∆ n,j (cid:107) + (cid:107) s (cid:107)(cid:107) ∆ n,j (cid:107) (cid:1) + 4 n n (cid:88) j =1 (cid:107) X j (cid:107) (cid:0) (cid:107) s (cid:107)(cid:107) t (cid:107)(cid:107) ∆ n,j (cid:107) (cid:1) + 8 n n (cid:88) j =1 (cid:107) X j (cid:107)(cid:107) ∆ n,j (cid:107) (cid:0) (cid:107) s (cid:107)(cid:107) ∆ n,j (cid:107) (cid:1)(cid:0) (cid:107) t (cid:107)(cid:107) ∆ n,j (cid:107) (cid:1) + 4 n n (cid:88) j =1 (cid:107) ∆ n,j (cid:107) (cid:0) (cid:107) s (cid:107)(cid:107) ∆ n,j (cid:107) (cid:1)(cid:0) (cid:107) t (cid:107)(cid:107) ∆ n,j (cid:107) (cid:1) . From Proposition A.2 of [14], it follows that (cid:107) L , n ( s, t ) − L , n, ( s, t ) (cid:107) . s . −→ .To prove (32), we need the integrals L ,a ( x ) := (cid:90) tψ ( t ) CS + ( t, x ) w a ( t ) d t = (2 π ) d (2 a + 1) d +1 x exp (cid:16) − (cid:107) x (cid:107) a + 2 (cid:17) ,L ,a ( x ) := (cid:90) tt (cid:124) xψ ( t ) CS − ( t, x ) w a ( t ) d t = (2 π ) d (2 a + 1) d +2 (cid:0) (2 a + 1) x − (cid:107) x (cid:107) x (cid:1) exp (cid:16) − (cid:107) x (cid:107) a + 2 (cid:17) ,I ,a ( x, y ) := (cid:90) CS + ( t, x ) CS + ( t, y ) w a ( t ) d t = (cid:16) πa (cid:17) d exp (cid:16) − (cid:107) x − y (cid:107) a (cid:17) ,I ,a ( x, y ) := (cid:90) t CS + ( t, x ) CS − ( t, y ) w a ( t ) d t = (cid:16) πa (cid:17) d ( x − y )2 a exp (cid:16) − (cid:107) x − y (cid:107) a (cid:17) . P i,j ,a := Y (cid:124) n,i Y n,j I ,a ( Y n,i , Y n,j ) − L ,a ( Y n,j ) (cid:124) Y n,j ,P i,j,k ,a := Y (cid:124) n,i Y n,j I ,a ( Y n,i , Y n,k ) − L ,a ( Y n,k ) (cid:124) Y n,j ,P i,j,k ,a := Y (cid:124) n,i Y n,k Y (cid:124) n,j I ,a ( Y n,i , Y n,k ) − Y (cid:124) n,j L ,a ( Y n,k ) ,P i,j,k ,a := Y (cid:124) n,i ( Y n,j Y (cid:124) n,j + I d ) Y n,k I ,a ( Y n,i , Y n,k ) − Y (cid:124) n,k ( Y n,j Y (cid:124) n,j + I d ) L ,a ( Y n,k ) ,P i,j,k ,a := Y (cid:124) n,i Y n,k Y (cid:124) n,k ( Y n,j Y (cid:124) n,j − I d ) I ,a ( Y n,i , Y n,k ) − Y (cid:124) n,k ( Y n,j Y (cid:124) n,j − I d ) L ,a ( Y n,k ) , straightforward calculations give (cid:98) σ , n,a = 4 n n (cid:88) i,j,k =1 P i,j ,a P k,j ,a , (cid:98) σ , n,a = − n n (cid:88) i,j,k,(cid:96) =1 P i,j ,a P (cid:96),j,k ,a , (47) (cid:98) σ , n,a = − n n (cid:88) i,j,k,(cid:96) =1 P i,j ,a P (cid:96),j,k ,a , (cid:98) σ , n,a = − n n (cid:88) i,j,k,(cid:96) =1 P i,j ,a P (cid:96),j,k ,a , (cid:98) σ , n,a = − n n (cid:88) i,j,k,(cid:96) =1 P i,j ,a P (cid:96),j,k ,a , (cid:98) σ , n,a = 4 n n (cid:88) i,j,k,(cid:96),m =1 P i,j,k ,a P m,j,(cid:96) ,a , (cid:98) σ , n,a = 4 n n (cid:88) i,j,k,(cid:96),m =1 P i,j,k ,a P m,j,(cid:96) ,a , (cid:98) σ , n,a = 2 n n (cid:88) i,j,k,l,m =1 P i,j,k ,a P m,j,(cid:96) ,a , (cid:98) σ , n,a = 2 n n (cid:88) i,j,k,l,m =1 P i,j,k ,a P m,j,(cid:96) ,a , (cid:98) σ , n,a = 4 n n (cid:88) i,j,k,(cid:96),m =1 P i,j,k ,a P m,j,(cid:96) ,a , (cid:98) σ , n,a = 2 n n (cid:88) i,j,k,(cid:96),m =1 P i,j,k ,a P m,j,(cid:96) ,a , (cid:98) σ , n,a = 2 n n (cid:88) i,j,k,(cid:96),m =1 P i,j,k ,a P m,j,(cid:96) ,a , (cid:98) σ , n,a = 1 n n (cid:88) i,j,k,(cid:96),m =1 P i,j,k ,a P m,j,(cid:96) ,a , (cid:98) σ , n,a = 1 n n (cid:88) i,j,k,(cid:96),m =1 P i,j,k ,a P m,j,(cid:96) ,a , (cid:98) σ , n,a = 1 n n (cid:88) i,j,k,(cid:96),m =1 P i,j,k ,a P m,j,(cid:96) ,a . d n \ a ( −√ , √ d (0 , / √ d (0 , √ /π ) d I n,a, . for ∆ a ( replications, nominal level 0.95)23esting normality in any dimension by Fourier methods in a multivariate Stein equation d n/a ∞ ∞ . -quantiles for a d/ π − d/ T n,a under H (100000 replications) n CvM AD SW BHEP HV T . T T T T T ∞ N (0 ,
20 5 5 5 5 5 5 5 5 5 5 550 5 5 5 5 5 5 5 5 5 5 5100 5 5 5 5 5 5 5 5 5 5 5NMix1 20 20 23 25 26 25 27
28 28 28
27 2750 45 50 56 55 52 58 60
61 61
60 59100 75 81 85 84 82 87 88
89 89
88 88t (0 ,
20 30 33 34 33
36 36 36
35 35 34 3550 57 61 64 61 63
65 63 59 56 52100 83 85
86 84
88 88
86 80 76 64t (0 ,
20 15 17 19 18
20 20 20 20 20 2050 27 30 35 31
36 36 35 34 33 32100 43 48
50 56 55 56 53 49 45 40t (0 ,
20 8 9 10 9
11 11 11 11 11 1150 11 12 15 13
15 16 16 16 16 16100 14 16 23 17
21 22 22 21 20 20 χ (5)
20 34 38
42 35 42 43 43 42 41 4050 73 80
83 74 86 86 87 86 85 83100 97 99 *
99 97 99 99 *
99 99 99 χ (15)
20 14 15 17 17 16 18
19 19 19 19
45 45 45
77 77 (0 ,
20 10 11 11 11
13 13 13 13 13 1350 14 16 20 17
20 20 20 19 19 19100 21 24 31 25
30 30 28 26 24 23U ( −√ , √
20 14 17
12 0 10 4 2 1 1 150 44 58
55 0 55 33 5 1 0 0100 84 95 *
94 0 96 90 48 2 1 0P
V II (5)
20 15 17 19 18
20 20 20 20 20 2150 27 30 35 31
36 36 35 34 33 32100 43 48
50 56 55 56 53 49 45 41P
V II (10)
20 8 9 10 9
11 11 11 11 11 1150 11 12 16 12
15 16 16 16 16 16100 14 16 23 17
21 22 22 20 20 20Table 6: Empirical power ( d = 1 , α = 0 . , replications)24esting normality in any dimension by Fourier methods in a multivariate Stein equation n BHEP HZ HV EN T . T T T T T ∞ N (0 , I )
20 5 5 5 5 5 5 5 5 5 550 5 5 5 5 5 5 5 5 5 5100 5 5 5 5 5 5 5 5 5 5NMix1 20 39 34 32 37 38
41 41
40 39 3850 83 74 68 82 85 88
88 88 86100 99 96 97 99 99 99 * * * *
NMix2 20 20 17
20 23 24 25 25 25 2550 38 30
39 45 48 49 48 47 44100 60 47
61 68 72 72 70 66 55t (0 , I )
20 47 45 54 49 49 51
53 53 53
84 82 84 83 83 81 78100
97 97 97 97
98 98
97 95 90t (0 , I )
20 25 22
26 27 29 30 31 31 3150 49 42
50 49 53 55 54 54 52100 75 67
76 71 76 77 75 72 66t (0 , I )
20 11 10
12 12 14 14 15 15
50 17 14
18 19 22 24 25 25 25100 27 20
28 26 31 33 34 33 33 ( χ (5))
20 48 44 38 46 46 48
48 47 4650 93 87 80 92 93 94
95 95
94 93100 * * * * * * * * * ( χ (15))
20 18 16 17 17 17 19
20 20 20
54 52100 78 62 71 77 78 84 88
88 88 ( χ (20))
20 15 13 14 14 14 15
16 16 16 16
50 34 27 31 33 33 38 41
43 43
77 77
Γ(5 ,
20 26 23 23 24 24 27
27 27 2650 64 53 53 61 62 68 71
71 69100 93 84 87 93 94 96 97
97 97
Γ(4 ,
20 32 28 27 30 30 33
33 33 3250 75 64 61 73 73 79
81 81
80 79100 98 92 93 97 98
99 99 99 99 99
Logistic (0 ,
20 11 10
12 13 14 15
50 18 15
20 20 23 24 25 25 25100 29 23
31 29 34 35 34 33 31U ( −√ , √
20 12
98 98
V II (5)
20 20 18
21 22 24 26 26 26 2750 39 32
40 41 45 47 46 46 45100 63 53
64 62 67 68 66 62 58P
V II (10)
20 10 8
10 11 11 12
13 13 13
50 13 11
14 15 18 19 20 20 20100 19 14
21 20 24 26 27 27 26P
V II (20)
20 7 6
50 7 7 d = 2 , α = 0 . , replications)25esting normality in any dimension by Fourier methods in a multivariate Stein equation n BHEP HZ HV EN T . T T T T T ∞ N (0 , I )
20 5 5 5 5 5 5 5 5 5 550 5 5 5 5 5 5 5 5 5 5100 5 5 5 5 5 5 5 5 5 5NMix1 20 39 35 33 41 40 43
43 41 4050 89 81 66 91 91 94
95 95
93 92100 *
98 95 * * * * * * *
NMix2 20 28 24
33 34 38 40 41 41 4150 59 49
66 65 72 75 75 75 73100 85 74
88 87 92 93 94 92 87t (0 , I )
20 56 53 65 62 58 63 65
65 6550 93 90
94 94
89 93 93 93 92 91100 * * *
98 99 * *
99 99 98t (0 , I )
20 29 26
35 32 37 39
50 62 54
67 57 67 70 70 70 69100 90 83
91 80 88 90 89 88 84t (0 , I )
20 12 11
15 14 17 18 19 19
50 22 17
26 22 28 32 34 35 35100 37 28
42 30 40 46 48 48 47 ( χ (5))
20 48 43 38 49 46 50
50 49 4850 95 89 82 96 94
97 97 97 97 * * * * * * * * * ( χ (15))
20 17 15 17 18 16 18
19 19 19 19
50 45 34 38 48 44 51 56
57 56100 82 64 69 84 81 88 92
93 93 ( χ (20))
20 13 12 14 14 13 14
15 15 1550 34 25 30 36 31 39 43
44 44100 67 48 56 70 65 75 81
83 83 Γ(5 ,
20 25 22 23 25 23 26
27 27 2650 65 53 53 68 64 71
76 76
75 74100 96 86 87 97 96 98
99 99 99 99
Γ(4 ,
20 30 27 27 32 29 32
33 33 3250 77 65 62 79 76 82 85
85 83100 99 94 93 99 99 * * * * *
Logistic (0 ,
20 11 10
13 13 15 16
17 17 17
50 18 14
22 19 24 27 29 29 29100 31 23
36 27 35 39 39 39 38U ( −√ , √
20 11
98 98
V II (5)
20 20 17
24 23 27 29
30 30 30
50 41 34
47 42 50 54 55 54 53100 69 57
73 63 72 76 75 73 69P
V II (10)
20 9 8
11 11 12 13
14 14 14
50 13 10
16 14 18 21 23 23 23100 20 14
24 18 24 29 31 31 31P
V II (20)
20 6 6
10 8 10 12 13 14 14Table 8: Empirical power ( d = 3 , α = 0 . , replications)26esting normality in any dimension by Fourier methods in a multivariate Stein equation n BHEP HZ HV EN T . T T T T T ∞ N (0 , I )
20 5 5 5 5 5 5 5 5 5 550 5 5 5 5 5 5 5 5 5 5100 5 5 5 5 5 5 5 5 5 5NMix1 20 25 22 31 32 27 33
34 34 3350 85 74 50 94 87 94
92 90 86100 *
98 77 * * * * * * *
NMix2 20 32 27
48 40 51 56 58 59 5950 76 67
89 79 89 93 94 94 94100 96 92 *
99 96 99 99 * * * t (0 , I )
20 62 59 79 76 67 76 79
81 81 * *
99 99 99100 * * * * * * * * * * t (0 , I )
20 31 28 54 47 37 48 52 54 54
50 77 71
88 68 82 88
89 89 89
100 98 96
99 99
88 96 98
98 98t (0 , I )
20 12 11
20 15 21 24 25
26 26
50 28 23
44 26 39 48 52 54 53100 54 44
69 36 54 67 72 73 72 ( χ (5))
20 39 35 36
39 46
48 48
47 4550 94 87 80
94 97
98 98 98 * * * * * * * * * ( χ (15))
20 13 12 15 16 13 15
17 17 17 17
50 38 29 35 52 37 49 56
58 58
95 95 ( χ (20))
20 11 9 12 13 11 12 13
13 1350 28 22 28 39 27 36 42
44 43100 61 43 51 77 60 74 83
86 86 Γ(5 ,
20 18 16 21 24 18 22 24
24 2450 59 47 20 74 58 71 78
78 76100 95 85 83 99 95 99 99 * * Γ(4 ,
20 23 20 25 29 23 28
30 30 30
87 85100 99 94 91 * * * * * * Logistic (0 ,
20 9 8
13 11 14 16
17 17 17
50 15 13
26 17 24 30 33
34 34
100 29 22
42 23 34 43 47 47 47U ( −√ , √
20 9
95 0 75 49 20 5 0 0 0P
V II (5)
20 16 14
25 20 27 30
32 3250 39 32
56 39 54 62 65 65 65100 71 60
83 59 77 84 86 85 83P
V II (10)
20 8 7
11 9 11 13
14 14 14
50 11 9
19 12 18 23 26 27 27100 18 13
28 16 24 32 37 38 38P
V II (20)
20 6 5
50 7 6
11 8 10 13 15 16 16Table 9: Empirical power ( d = 5 , α = 0 . , replications) a replications)27 e s ti ngno r m a lit y i n a nyd i m e n s i onby F ou r i e r m e t hod s i n a m u lti v a r i a t e S t e i n e qu a ti on κ ( a ) = 65536 π / (4 a + 8 a + 3) (2 a + 4 a + 1) / √ a (2 a + 3) / (2 a + 1) / ( a + 1) / (cid:16)(cid:16)(cid:16) √ a + 3 √ a + 165536 (cid:16) a + 3276839107037 a + 900300839107037 a + 26214413035679 a + 4394716179239107037 a + 804230854413035679 a + 6815602342439107037 a + 10307527499239107037 a + 10083401608039107037 a + 8459430257639107037 a + 6069150804439107037 a + 3707418959639107037 a + 638900434813035679 a + 277318582413035679 a + 300007533539107037 a + 88627556939107037 a + 7015004913035679 a + a + 182203513035679 a + 17975713035679 a + 8599509 a + 32513035679 √ a + 9047497595239107037 a + 34195456039107037 a + 6557286439107037 a + 431605504039107037 a + 1118006508839107037 a (cid:17) + (cid:0) a + 48 a + 72 a + 84 a + 45 (cid:1) (2 a + 1) (cid:0) a + 4 a + 1 (cid:1) ( a + 1) / (cid:17)(cid:112) a + 8 a + 3 − √ a + 3 √ a + 14194304 (cid:16) a + 1048576004557280077 a + 41943044557280077 a + 95006228484557280077 a + 12509511684557280077 a + 137726770780164557280077 a + 84135992023044557280077 a + 193604464629764557280077 a + 244911942307844557280077 a + 3150868184512651040011 a + 170924492624004557280077 a + 37870617543521519093359 a + 64469777596044557280077 a + 31036486353924557280077 a + 12571615594904557280077 a + 1412883115121519093359 a + 13030129762506364453 a + 32242833256262717 a + a + 117347893059 a + 435833572236 a + 636893059 a + 98150011304 √ a + 2605134412544506364453 a + 717252853761519093359 a + 7386431488651040011 a + 6454746890241519093359 a + 14627513466881519093359 a (cid:17)(cid:17)(cid:112) a + 4 a + 1 − √ a + 3 √ a + 1 √ (cid:16) a + 543162368021160654407 a + 43620761621160654407 a + 24655167488021160654407 a + 1677721621160654407 a + 4318245683221160654407 a + 13365461447065621160654407 a + 8969714778112021160654407 a + 17225220586700821160654407 a + 18644014650060821160654407 a + 173905711889922351183823 a + 126218373632002351183823 a + 876956342720261242647 a + 3808701297952021160654407 a + 1740120190462421160654407 a + 671946094870421160654407 a + 7233127312967053551469 a + 1926359674047053551469 a + 1537776695261242647 a + a + 33641318261242647 a + 3084786261242647 a + 179523261242647 a + 4995261242647 √ a + 101279560632321113718653 a + 139213668352783727941 a + 107806746214421160654407 a + 28485989171202351183823 a + 5184583481753621160654407 a (cid:17)(cid:17) e s ti ngno r m a lit y i n a nyd i m e n s i onby F ou r i e r m e t hod s i n a m u lti v a r i a t e S t e i n e qu a ti on κ ( a ) = 4238729565 π √ a (4 a + 8 a + 3) / √ a + 3 a + 2( a + 3 / ( a + 2 a + 1 / ( a + 5 / a + 5 / ( a + 2) ( a + 1) ( a + 3 / a + 1 / (cid:18) − √ a + 8 a + 3 √ a + 64 a + 84 a + 40 a + 5 √ a + 3 a + 2189423662804146875 (cid:18) a + 14212479451218934325182662246446529006931033345 a + 12580967273506896100823101933795402223 a + 582697931332256810829296435246529006931033345 a + 1575030841745889017557747259215509668977011115 a + 13861129700142534438263187653646529006931033345 a + a + 2794299890386033933772046721615509668977011115 a + 70449087227773480486348369817646529006931033345 a + 44639066041418351072429 a + 56792221859153182725169889659003705 a + 1616889961418351072429 a + 145382366443231507323965931523101933795402223 a + 68478220343365680467658355521033977931800741 a + 54299950068233390994227246529006931033345 a + 2669642365607623083950085169889659003705 a + 3665864337153003723161615509668977011115 a + 8649623898353227032410069401646529006931033345 a + 6816457006309691477749731019337954022230 a + 313914401212712825651246529006931033345 a + 438481418351072429 √ a + 20656910835466566401431956684815509668977011115 a + 1276573406441989829589834137646529006931033345 a + 206979347198605284565731019337954022230 a + 758502645909287149466768957445169889659003705 a + 41506997715629950040330081158446529006931033345 a + 2596132102476118558088180531246529006931033345 a + 32611772182894725530319303722 a + 54975581388846529006931033345 a + 43283381268239299379246529006931033345 a + 61676766235818589172095244761646529006931033345 a + 1362185223804593222334335238409305801386206669 a + 3473843670149914665209036846529006931033345 a + 17432876021284068073047704700815509668977011115 a + 168846695533956300815509668977011115 a + 49715829083619402422314746529006931033345 a + 31276439860669737200014073036846529006931033345 a + 120987510741401646529006931033345 a + 7128396562640048461357675945615509668977011115 a + 4909991378105224070076583116846529006931033345 a + 3823338534973964083737395215509668977011115 a + 100359730557832239231033977931800741 a + 319714513733402084822220815509668977011115 a + 38099521418351072429 a + 2596991489225523246529006931033345 a + 41961808233206994849525789900846529006931033345 a + 1802431059372836702144858 a + 52696221277640962806807232409646529006931033345 a + 3683363953049646529006931033345 a + 9467780993801174376590957911166 a + 521454215655265638295478955583 a + 246382529614299548385437286446529006931033345 a + 2099786311112841274762507331019337954022230 a + 77042495835667672680366081033977931800741 a + 96273213709186088777757491246529006931033345 a + 2479084089124740626531961615246529006931033345 a + 868575931910092824285497246529006931033345 a e s ti ngno r m a lit y i n a nyd i m e n s i onby F ou r i e r m e t hod s i n a m u lti v a r i a t e S t e i n e qu a ti on + 1208957773329706612398750980646529006931033345 a + 819390424757043209305801386206669 a + 145459816889470238887469931019337954022230 a + 208859765438314437141286219305801386206669 a + 63523003822972843136892527393058013862066690 a + 88868334735738098170934774646529006931033345 a + 229336483421457278332057780646529006931033345 a + 547100879380497752743746657246529006931033345 a (cid:17) + (cid:16) / a ) ( a + 3 / ( a + 2 a + 1 / ( a + 5 / a + 5 / ( a + 1) ( a + 3 / a + 1 / √ a + 4 a + 16214043258290038234375 (cid:16) a + 8 a + 28 a + 56 a + 76 a + 80 a + 63 a + 30 a + 10516 (cid:17) + √ (cid:112) a + 3 a + 2 √ a + 1 (cid:16) a + 89350321631502512566506962309611522071347752763346078125 a + 8395518981191541073286241841198002456307625 a + 3712297682887162628273801705553922071347752763346078125 a + 913849879634598027664546373083136690449250921115359375 a + 81640671439342759539138843571650562071347752763346078125 a + a + 1633851323087451063964056414027776690449250921115359375 a + 431898627571736122657069717132410882071347752763346078125 a + 6029264458520205190699109 a + 33389951396266533831311362071347752763346078125 a + 211741357520205190699109 a + 911256225356037430357120383975424138089850184223071875 a + 6131265682608049365461479693877248690449250921115359375 a + 11734861729999873853970448384690449250921115359375 a + 51826136668193132779021533184690449250921115359375 a + 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