On kernel-based estimation of conditional Kendall's tau: finite-distance bounds and asymptotic behavior
aa r X i v : . [ m a t h . S T ] M a r On kernel-based estimation of conditional Kendall’s tau:finite-distance bounds and asymptotic behavior
Alexis Derumigny and Jean-David Fermanian
CREST-ENSAE,5, avenue Henry Le Chatelier91764 Palaiseau cedex, France.e-mail: [email protected] , [email protected] Abstract:
We study nonparametric estimators of conditional Kendall’s tau, a measure of concordancebetween two random variables given some covariates. We prove non-asymptotic pointwise and uniformbounds, that hold with high probabilities. We provide “direct proofs” of the consistency and the asymp-totic law of conditional Kendall’s tau. A simulation study evaluates the numerical performance of suchnonparametric estimators.
Keywords and phrases: conditional dependence measures, kernel smoothing, conditional Kendall’stau.
MSC 2010 subject classifications:
Primary 62H20; secondary 62G05, 62G08, 62G20..
1. Introduction
In the field of dependence modeling, it is common to work with dependence measures. Contrary to usual linearcorrelations, most of them have the advantage of being defined without any condition on moments, and ofbeing invariant to changes in the underlying marginal distributions. Such summaries of information are verypopular and can be explicitly written as functionals of the underlying copulas: Kendall’s tau, Spearman’s rho,Blomqvist’s coefficient... See Nelsen [1] for an introduction. In particular, for more than a century (Spearman(1904), Kendall (1938)), Kendall’s tau has become a popular dependence measure in [ − , . It quantifiesthe positive or negative dependence between two random variables X and X . Denoting by C , the uniqueunderlying copula of ( X , X ) that are assumed to be continuous, their Kendall’s tau can be directly definedas τ , := 4 Z [0 , C , ( u , u ) C , ( du , du ) − (1) = IP (cid:0) ( X , − X , )( X , − X , ) > (cid:1) − IP (cid:0) ( X , − X , )( X , − X , ) < (cid:1) , where ( X i, , X i, ) i =1 , are two independent versions of X := ( X , X ) . This measure is then interpreted asthe probability of observing a concordant pair minus the probability of observing a discordant pair . See [2]for an historical perspective on Kendall’s tau. Its inference is discussed in many textbooks (see [3] or [4],e.g.). Its links with copulas and other dependence measures can be found in [1] or [5].Similar dependence measure can be introduced in a conditional setup, when a p -dimensional covariate Z is available. When hundreds of papers refer to Kendall’s tau, only a few of them have considered conditionalKendall’s tau (as defined below) until now. The goal is now to model the dependence between the twocomponents X and X , given the vector of covariates Z . Logically, we can invoke the conditional copula imsart-generic ver. 2014/10/16 file: mainV2.tex date: March 8, 2019 . Derumigny and J.-D. Fermanian/On kernel-based estimation of conditional Kendall’s tau C , | Z = z of ( X , X ) given Z = z for any point z ∈ R p (see Patton [6, 7]), and the corresponding conditionalKendall’s tau would be simply defined as τ , | Z = z := 4 Z [0 , C , | Z = z ( u , u ) C , | Z = z ( du , du ) −
1= IP (cid:0) ( X , − X , )( X , − X , ) > (cid:12)(cid:12) Z = Z = z (cid:1) − IP (cid:0) ( X , − X , )( X , − X , ) < (cid:12)(cid:12) Z = Z = z (cid:1) , where ( X i, , X i, , Z i ) i =1 , are two independent versions of ( X , X , Z ) . As above, this is the probabilityof observing a concordant pair minus the probability of observing a discordant pair , conditionally on Z and Z being both equal to z . Note that, as conditional copulas themselves, conditional Kendall’s taus areinvariant w.r.t. increasing transformations of the conditional margins X and X , given Z . Of course, if Z is independent of ( X , X ) then, for every z ∈ R p , the conditional Kendall’s tau τ , | Z = z is equal to the(unconditional) Kendall’s tau τ , .Conditional Kendall’s tau, and more generally conditional dependence measures, are of interest per sebecause they allow to summarize the evolution of the dependence between X and X , when the covariate Z is changing. Surprisingly, their nonparametric estimates have been introduced in the literature only a fewyears ago ([8],[9],[10]) and their properties have not yet been fully studied in depth. Indeed, until now andto the best of our knowledge, the theoretical properties of nonparametric conditional Kendall’s tau estimateshave been obtained “in passing” in the literature, as a sub-product of the weak-convergence of conditionalcopula processes ([9]) or as intermediate quantities that will be “plugged-in” ([11]). Therefore, such propertieshave been stated under too demanding assumptions. In particular, some assumptions were related to theestimation of conditional margins, while this is not required because Kendall’s tau are based on ranks. In thispaper, we directly study nonparametric estimates ˆ τ , | z without relying on the theory/inference of copulas.Therefore, we will state their main usual statistical properties: exponential bounds in probability, consistency,asymptotic normality.Our τ , | Z = z has not to be confused with the so-called “conditional Kendall’s tau” in the case of truncateddata ([12], [13]), in the case of semi-competing risk models ([14], [15]), or for other partial information schemes( [16], [17], among others). Indeed, particularly in biostatistics or reliability, the inference of dependence mod-els under truncation/censoring can be led by considering some types of conditional Kendall’s tau, given somealgebraic relationships among the underlying random variables. This would induce conditioning by subsets.At the opposite, we will consider only pointwise conditioning events in this paper, under a nonparametricpoint-of-view. Nonetheless, such pointwise events can be found in the literature, but in some parametric orsemi-parametric particular frameworks, as for the identifiability of frailty distributions in bivariate propor-tional models ( [18], [19]). Other related papers are [20] or [21], that are dealing with extreme co-movements(bivariate extreme-value theory). There, the tail conditioning events of Kendall’s tau have probabilities thatgo to zero with the sample size.In Section 2, different kernel-based estimators of the conditional Kendall’s tau are discussed. In Section 3,the theoretical properties of the latter estimators are proved, first with finite-distance bounds and then underan asymptotic point-of-view. A short simulation study is provided in Section 4. Proofs are postponed intothe appendix. imsart-generic ver. 2014/10/16 file: mainV2.tex date: March 8, 2019 . Derumigny and J.-D. Fermanian/On kernel-based estimation of conditional Kendall’s tau
2. Definition of several kernel-based estimators of τ , | z Let ( X i, , X i, , Z i ) , i = 1 , . . . , n be an i.i.d. sample distributed as ( X , X , Z ) , and n ≥ . Assuming continuousunderlying distributions, there are several equivalent ways of defining the conditional Kendall’s tau: τ , | Z = z = 4 IP (cid:0) X , > X , , X , > X , (cid:12)(cid:12) Z = Z = z (cid:1) −
1= 1 − (cid:0) X , > X , , X , < X , (cid:12)(cid:12) Z = Z = z (cid:1) = IP (cid:0) ( X , − X , )( X , − X , ) > (cid:12)(cid:12) Z = Z = z (cid:1) − IP (cid:0) ( X , − X , )( X , − X , ) < (cid:12)(cid:12) Z = Z = z (cid:1) . Motivated by each of the latter expressions, we introduce several kernel-based estimators of τ , | Z = z : ˆ τ (1)1 , | Z = z := 4 n X i =1 n X j =1 w i,n ( z ) w j,n ( z ) (cid:8) X i, < X j, , X i, < X j, (cid:9) − , ˆ τ (2)1 , | Z = z := n X i =1 n X j =1 w i,n ( z ) w j,n ( z ) (cid:16) (cid:8) ( X i, − X j, ) . ( X i, − X j, ) > (cid:9) − (cid:8) ( X i, − X j, ) . ( X i, − X j, ) < (cid:9)(cid:17) , ˆ τ (3)1 , | Z = z := 1 − n X i =1 n X j =1 w i,n ( z ) w j,n ( z ) (cid:8) X i, < X j, , X i, > X j, (cid:9) , where denotes the indicator function, w i,n is a sequence of weights given by w i,n ( z ) = K h ( Z i − z ) P nj =1 K h ( Z j − z ) , (2)with K h ( · ) := h − p K ( · /h ) for some kernel K on R p , and h = h ( n ) denotes a usual bandwidth sequencethat tends to zero when n → ∞ . In this paper, we have chosen usual Nadaraya-Watson weights. Obviously,there are alternatives (local linear, Priestley-Chao, Gasser-Müller, etc., weight), that would lead to differenttheoretical results.The estimators ˆ τ (1)1 , | Z = z , ˆ τ (2)1 , | Z = z and ˆ τ (3)1 , | Z = z look similar, but they are nevertheless different, as shown inProposition 1. These differences are due to the fact that all the ˆ τ ( k )1 , | Z = z , k = 1 , , are affine transformationsof a double-indexed sum, on every pair ( i, j ) , including the diagonal terms where i = j . The treatment of thesediagonal terms is different for each of the three estimators defined above. Indeed, setting s n := P ni =1 w i,n ( z ) , it can be easily proved that ˆ τ (1)1 , | Z = z takes values in the interval [ − , − s n ] , ˆ τ (2)1 , | Z = z in [ − s n , − s n ] ,and ˆ τ (3)1 , | Z = z in [ − s n , . Moreover, there exists a direct relationship between these estimators, given bythe following proposition. Proposition 1.
Almost surely, ˆ τ (1)1 , | Z = z + s n = ˆ τ (2)1 , | Z = z = ˆ τ (3)1 , | Z = z − s n , where s n := P ni =1 w i,n ( z ) . This proposition is proved in A.1. As a consequence, we can easily rescale the previous estimators so thatthe new estimator will take values in the whole interval [ − , . This would yield ˜ τ , | Z = z := ˆ τ (1)1 , | Z = z − s n + s n − s n = ˆ τ (2)1 , | Z = z − s n = ˆ τ (3)1 , | Z = z − s n − s n − s n · imsart-generic ver. 2014/10/16 file: mainV2.tex date: March 8, 2019 . Derumigny and J.-D. Fermanian/On kernel-based estimation of conditional Kendall’s tau Note that none of the latter estimators depends on any estimation of conditional marginal distributions. Inother words, we only have to conveniently choose the weights w i,n to obtain an estimator of the conditionalKendall’s tau. This is coherent with the fact that conditional Kendall’s taus are invariant with respect toconditional marginal distributions. Moreover, note that, in the definition of our estimators, the inequalities arestrict (there are no terms corresponding to the cases i = j ). This is inline with the definition of (conditional)Kendall’s tau itself through concordant/discordant pairs of observations.The definition of ˆ τ (1)1 , | Z = z can be motivated as follows. For j = 1 , , let ˆ F j | Z ( ·| Z = z ) be an estimator ofthe conditional cdf of X j given Z = z . Then, a usual estimator of the conditional copula of X and X given Z = z is ˆ C , | Z ( u , u | Z = z ) := n X i =1 w i,n ( z ) (cid:8) ˆ F | Z ( X i, | Z = z ) ≤ u , ˆ F | Z ( X i, | Z = z ) ≤ u (cid:9) . See [9] or [10], e.g. The latter estimator of the conditional copula can be plugged into (1) to define an estimatorof the conditional Kendall’s tau itself: ˆ τ , | Z = z := 4 Z ˆ C , | Z ( u , u | Z = z ) ˆ C , | Z ( du , du | Z = z ) − (3) = 4 n X j =1 w j,n ( z ) ˆ C , | Z (cid:0) ˆ F | Z ( X j, | Z = z ) , ˆ F | Z ( X j, | Z = z ) (cid:12)(cid:12) Z = z (cid:1) − . Since the functions ˆ F j | Z ( ·| Z = z ) are non-decreasing, this reduces to ˆ τ , | Z = z = 4 n X i =1 n X j =1 w i,n ( z ) w j,n ( z ) (cid:8) X i, ≤ X j, , X i, ≤ X j, (cid:9) −
1= 4 n X i =1 n X j =1 w i,n ( z ) w j,n ( z ) (cid:8) X i, < X j, , X i, < X j, (cid:9) − o P (1) = ˆ τ (1)1 , | Z = z + o P (1) . Veraverbeke et al. [9], Subsection 3.2, introduced their estimator of τ , | z by (3). By the functional Delta-Method, they deduced its asymptotic normality as a sub-product of the weak convergence of the process √ nh (cid:0) ˆ C , | Z ( · , ·| z ) − C , | Z ( · , ·| z ) (cid:1) when Z is univariate. In our case, we will obtain more and stronger theoret-ical properties of ˆ τ (1)1 , | Z = z under weaker conditions by a more direct analysis based on ranks. In particular, wewill not require any regularity condition on the conditional marginal distributions, contrary to [9]. Indeed, inthe latter paper, it is required that F j | Z ( ·| Z = z ) has to be two times continuously differentiable (assumption ( ˜ R ) and its inverse has to be continuous (assumption ( R ). This is not satisfied for some simple univariatecdf as F j ( t ) = t ( t ∈ [0 , / ( t ∈ (1 , / t ( t ∈ (2 , / ( t > , for instance. Note that we couldjustify ˆ τ (3)1 , | Z = z in a similar way by considering conditional survival copulas.Let us define g , g , g by g ( X i , X j ) := 4 (cid:8) X i, < X j, , X i, < X j, (cid:9) − ,g ( X i , X j ) := (cid:8) ( X i, − X j, ) × ( X i, − X j, ) > (cid:9) − (cid:8) ( X i, − X j, ) × ( X i, − X j, ) < (cid:9) ,g ( X i , X j ) := 1 − (cid:8) X i, < X j, , X i, > X j, (cid:9) , where, for i = 1 , . . . , n , we set X i := ( X i, , X i, ) . Clearly, ˆ τ ( k )1 , | z is a smoothed estimator of E [ g k ( X , X ) | Z = Z = z ] , k = 1 , , . imsart-generic ver. 2014/10/16 file: mainV2.tex date: March 8, 2019 . Derumigny and J.-D. Fermanian/On kernel-based estimation of conditional Kendall’s tau Note that such dependence measures are of interest for the purpose of estimating (conditional or uncon-ditional) copula models too. Indeed, several popular parametric families of copulas have a simple one-to-onemapping between their parameter and the associated Kendall’s tau (or Spearman’s rho): Gaussian, Studentwith a fixed degree of freedom, Clayton, Gumbel and Frank copulas, etc. Then, assume for instance thatthe conditional copula C , | Z = z belongs is a Gaussian copula with a parameter ρ ( z ) . Then, by estimating itsconditional Kendall’s tau τ , | Z = z , we get an estimate of the corresponding parameter ρ ( z ) , and finally of theconditional copula itself. See [22], e.g.The choice of the bandwidth h could be done in a data-driven way, following the general conditional U-statistics framework detailed in Dony and Mason [23, Section 2]. Indeed, for any k ∈ { , , } and z ∈ Z ,denote by ˆ τ ( h, k ) − ( i,j ) , , | Z = z the estimator ˆ τ ( k )1 , | Z = z that is made with the smoothing parameter h and our dataset,when the i -th and j -th observations have been removed. As a consequence, the random function ˆ τ ( h, k ) − ( i,j ) , , | Z = · is independent of (cid:0) ( X i , Z i ) , ( X j , Z j ) (cid:1) . As usual with kernel methods, it would be tempting to propose h asthe minimizer of the cross-validation criterion CV DM ( h ) := 2 n ( n − n X i,j =1 (cid:16) g k ( X i , X j ) − ˆ τ ( h, k ) − ( i,j ) , , | Z =( Z i + Z j ) / (cid:17) K h ( Z i − Z j ) , for k = 1 , , or for ˜ τ , | Z = · . The latter criterion would be a “naively localized” version of the usual cross-validation method. Unfortunately, we observe that the function h CV DM ( h ) is most often decreasing inthe range of realistic bandwidth values. If we remove the weight K h ( Z i − Z j ) , then there is no reason why g k ( X i , X j ) should be equal to ˆ τ ( k ) − ( i,j ) , , | Z =( Z i + Z j ) / (on average), and we are not interested in the predictionof concordance/discordance pairs for which the Z i and Z j are far apart. Therefore, a modification of thiscriteria is necessary. We propose to separate the choice of h for the terms g k ( X i , X j ) − ˆ τ ( h, k ) − ( i,j ) , , | Z =( Z i + Z j ) / and the selection of the “convenient pairs” of observations ( i, j ) . This leads to the new criterion CV ˜ h ( h ) := 2 n ( n − n X i,j =1 (cid:16) g k ( X i , X j ) − ˆ τ ( h, k ) − ( i,j ) , , | Z =( Z i + Z j ) / (cid:17) ˜ K ˜ h ( Z i − Z j ) , (4)with a potentially different kernel ˜ K and a new fixed tuning parameter ˜ h . Even if more complex proceduresare possible, we suggest to simply choose ˜ K ( z ) := {| z | ∞ ≤ } and to calibrate ˜ h so that only a fraction of thepairs ( i, j ) has non-zero weights. In practice, set ˜ h as the empirical quantile of (cid:0) {| Z i − Z j | ∞ : 1 ≤ i < j = n } of order N pairs / ( n ( n − , where N pairs is the number of pairs we want to keep.
3. Theoretical results
Hereafter, we will consider the behavior of conditional Kendall’s tau estimates given Z = z belongs to somefixed open subset Z in R p . For the moment, let us state an instrumental result that is of interest per se. Let ˆ f Z ( z ) := n − P nj =1 K h ( Z j − z ) be the usual kernel estimator of the density f Z of the conditioning variable Z .Note that the estimators ˆ τ ( k )1 , | Z = z , k = 1 , . . . , are well-behaved only whenever ˆ f Z ( z ) > . Denote the jointdensity of ( X , Z ) by f X , Z . In our study, we need some usual conditions of regularity. imsart-generic ver. 2014/10/16 file: mainV2.tex date: March 8, 2019 . Derumigny and J.-D. Fermanian/On kernel-based estimation of conditional Kendall’s tau Assumption 3.1.
The kernel K is bounded, and set k K k ∞ =: C K . It is symmetrical and satisfies R K = 1 , R | K | < ∞ . This kernel is of order α for some integer α > : for all j = 1 , . . . , α − and every indices i , . . . , i j in { , . . . , p } , R K ( u ) u i . . . u i j d u = 0 . Moreover, E [ K h ( Z − z )] > for every z ∈ Z and h > . Set ˜ K ( · ) := K ( · ) / R K and k ˜ K k ∞ =: C ˜ K . Assumption 3.2. f Z is α -times continuously differentiable on Z and there exists a constant C K,α > s.t.,for all z ∈ Z , Z | K | ( u ) p X i ,...,i α =1 | u i . . . u i α | sup t ∈ [0 , (cid:12)(cid:12) ∂ α f Z ∂z i . . . ∂z i α ( z + th u ) (cid:12)(cid:12) d u ≤ C K,α . Moreover, C ˜ K, denotes a similar constant replacing K by ˜ K and α by two. Assumption 3.3.
There exist two positive constants f Z ,min and f Z ,max such that, for every z ∈ Z , f Z ,min ≤ f Z ( z ) ≤ f Z ,max . Proposition 2.
Under Assumptions 3.1-3.3 and if C K,α h α /α ! < f Z ,min , for any z ∈ Z , the estimator ˆ f Z ( z ) is strictly positive with a probability larger than − (cid:16) − nh p (cid:0) f Z ,min − C K,α h α /α ! (cid:1) / (cid:0) f Z ,max Z K + (2 / C K ( f Z ,min − C K,α h α /α !) (cid:1)(cid:17) . The latter proposition is proved in A.2. It guarantees that our estimators ˆ τ ( k )1 , | z , k = 1 , . . . , , are well-behaved with a probability close to one. The next regularity assumption is necessary to explicitly control thebias of ˆ τ , | Z = z . Assumption 3.4.
For every x ∈ R , z f X , Z ( x , z ) is differentiable on Z almost everywhere up to theorder α . For every ≤ k ≤ α and every ≤ i , . . . , i α ≤ p , let H k,~ι ( u , v , x , x , z ) := sup t ∈ [0 , (cid:12)(cid:12)(cid:12)(cid:12) ∂ k f X , Z ∂z i . . . ∂z i k (cid:16) x , z + th u (cid:17) ∂ α − k f X , Z ∂z i k +1 . . . ∂z i α (cid:16) x , z + th v (cid:17)(cid:12)(cid:12)(cid:12)(cid:12) , denoting ~ι = ( i , . . . , i α ) . Assume that H k,~ι ( u , v , x , x , z ) is integrable and there exists a finite constant C XZ ,α > such that, for every z ∈ Z and every h < , Z | K | ( u ) | K | ( v ) α X k =0 (cid:18) αk (cid:19) p X i ,...,i α =1 H k,~ι ( u , v , x , x , z ) | u i . . . u i k v i k +1 . . . v i α | d u d v d x d x is less than C XZ ,α . The next three propositions state pointwise and uniform exponential inequalities for the estimators ˆ τ ( k )1 , | Z = z ,when k = 1 , , . They are proved in A.3. We will denote c := c := 4 and c := 2 . Proposition 3 (Exponential bound with explicit constants) . Under Assumptions 3.1-3.4, for every t > such that C K,α h α /α ! + t ≤ f Z ,min / and every t ′ > , if C ˜ K, h < f z ( z ) , we have IP | ˆ τ ( k )1 , | Z = z − τ , | Z = z | > c k f z ( z ) (cid:16) C XZ ,α h α α ! + 3 f z ( z ) R K nh p + t ′ (cid:17) × (cid:18) f Z ( z ) f Z ,min (cid:16) C K,α h α α ! + t (cid:17)(cid:19)! ≤ (cid:16) − nh p t f Z ,max R K + (2 / C K t (cid:17) + 2 exp (cid:16) − ( n − h p t ′ f Z ,max ( R K ) + (8 / C K t ′ (cid:17) + 2 exp (cid:18) − nh p ( f z ( z ) − C ˜ K, h ) f Z ,max R ˜ K + 4 C ˜ K ( f z ( z ) − C ˜ K, h ) / (cid:19) , for any z ∈ Z and every k = 1 , , . imsart-generic ver. 2014/10/16 file: mainV2.tex date: March 8, 2019 . Derumigny and J.-D. Fermanian/On kernel-based estimation of conditional Kendall’s tau Alternatively, we can apply Theorem 1 in Major [24] instead of the Bernstein-type inequality that hasbeen used in the proof of Proposition 3.
Proposition 4 (Alternative exponential bound without explicit constants) . Under Assumptions 3.1-3.4, forevery t > such that C K,α h α /α ! + t ≤ f Z ,min / and every t ′ > s.t. t ′ ≤ h p ( R K ) f Z ,max /C K , there existsome universal constants C and α s.t. IP | ˆ τ ( k )1 , | Z = z − τ , | Z = z | > c k f z ( z ) (cid:16) C XZ ,α h α α ! + 3 f z ( z ) R K nh p + t ′ (cid:17) × (cid:18) f Z ( z ) f Z ,min (cid:16) C K,α h α α ! + t (cid:17)(cid:19)! ≤ (cid:16) − nh p t f Z ,max R K + (2 / C K t (cid:17) + 2 exp (cid:18) − nh p ( f z ( z ) − C ˜ K, h ) f Z ,max R ˜ K + 4 C ˜ K ( f z ( z ) − C ˜ K, h ) / (cid:19) + 2 exp (cid:16) nh p t R K ( R | K | ) f Z ,max + 8 C K R | K | f Z ,max t/ (cid:17) + C exp (cid:18) − α nh p t ′ f Z ,max ( R K ) (cid:19) , for any z ∈ Z and every k = 1 , , , if C ˜ K, h < f Z ( z ) and h p (cid:0) R | K | (cid:1) f z ,max < R K . Remark 5.
In Propositions 2, 3 and 4, when the support of K is included in [ − c, c ] p for some c > , f Z ,max can be replaced by a local bound sup ˜ z ∈ V ( z ,ǫ ) f Z (˜ z ) , denoting by V ( z , ǫ ) a closed ball of center z and any radius ǫ > , when h c < ǫ . As a corollary, the two latter result yield the weak consistency of ˆ τ ( k )1 , | Z = z for every z ∈ Z , when nh p → ∞ (choose the constants t and t ′ ∼ h p sufficiently small, in Proposition 4, e.g.).It is possible to obtain uniform bounds, by slightly strengthening our assumptions. Note that this nextresult will be true if n is sufficiently large, when Proposition 4 was true for every n . Assumption 3.5.
The kernel K is Lipschitz on ( Z , k · k ∞ ) , with a constant λ K and Z is a subset of anhypercube in R p whose volume is denoted by V . Moreover, K and K are regular in the sense of [25] or [26]. Proposition 6 (Uniform exponential bound) . Under the assumptions 3.1-3.5, there exist some constants L K and C K (resp. L ˜ K and C ˜ K ) that depend only on the VC characteristics of K (resp. ˜ K ), s.t., for every µ ∈ (0 , such that µf z ,min < C XZ ,α h α /α ! + b K R K f Z ,max /C K , if f Z ,max < ˜ C XZ , h / b ˜ K R ˜ K f Z ,max /C ˜ K , IP sup z ∈ Z | ˆ τ ( k )1 , | Z = z − τ , | Z = z | > c k f z ,min (1 − µ ) (cid:18) C XZ ,α h α α ! + 3 f z ,max R K nh p + t (cid:19)! ≤ L K exp (cid:0) − C f,K nh p (cid:0) µf z ,min − C XZ ,α h α α ! (cid:1) (cid:1) + C D exp (cid:18) − α nth p f Z ,max ( R K ) (cid:19) + L ˜ K exp (cid:0) − C f, ˜ K nh p ( f z ,max − ˜ C XZ , h ) / (cid:1) + 2 exp (cid:0) − A nh p t C − K A R K f z ,max ( R | K | ) (cid:1) + 2 exp( − A nh p t C K A ) , for n sufficiently large, k = 1 , , , and for every t > s.t. t ≤ h p ( R K ) f Z ,max /C K , − A C K A g Z K f z ,max ( Z | K | ) ln( h p Z K f z ,max ( Z | K | ) ) < n / h p/ t, and nh p t ≥ (cid:0) Z K (cid:1) f z ,max M ( p + β ) / log (cid:16) C K h p f z ,max R K (cid:17) , β = max (cid:0) , log D log n (cid:1) , D := ⌈V (cid:0) C K λ K h (cid:1) p ⌉ , for some universal constants C , α , M , A , A and a constant A g that depends on K and f z ,max . imsart-generic ver. 2014/10/16 file: mainV2.tex date: March 8, 2019 . Derumigny and J.-D. Fermanian/On kernel-based estimation of conditional Kendall’s tau We have denoted C f,K := log(1 + b K / (4 L K )) / ( L K b K f z ,max R K ) , for any arbitrarily chosen constant b K ≥ C K . Similarly, C f, ˜ K := log(1 + b ˜ K / (4 L ˜ K )) / ( L ˜ K b ˜ K f z ,max R ˜ K ) , b ˜ K ≥ C ˜ K . The previous exponential inequalities are not optimal to prove usual asymptotic results. Indeed, they directlyor indirectly rely on upper bounds of estimates, as in Hoeffding or Bernstein-type inequalities. In the case ofkernel estimates, this implies the necessary condition nh p → ∞ , at least. By a direct approach, it is possibleto state the consistency of ˆ τ ( k )1 , | Z = z , k = 1 , , , and then of ˜ τ , | Z = z , under the weaker condition nh p → ∞ . Proposition 7 (Consistency) . Under Assumption 3.1, if nh pn → ∞ , lim K ( t ) | t | p = 0 when | t | → ∞ , f Z and z τ , | Z = z are continuous on Z , then ˆ τ ( k )1 , | Z = z tends to τ , | Z = z in probability, when n → ∞ for any k = 1 , , . This property is proved in A.6. Moreover, Proposition 6 does not allow to state the strong uniform con-sistency of ˆ τ ( k )1 , | Z = z because the threshold t has to be of order h p at most. Here again, a direct approach ispossible, nonetheless. Proposition 8 (Uniform consistency) . Under Assumption 3.1, assume that nh pn / log n → ∞ , lim K ( t ) | t | p =0 when | t | → ∞ , K is Lipschitz, f Z and z τ , | Z = z are continuous on a bounded set Z , and there exists alower bound f Z , min s.t. f Z , min ≤ f Z ( z ) for any z ∈ Z . Then sup z ∈ Z (cid:12)(cid:12) ˆ τ ( k )1 , | Z = z − τ , | Z = z (cid:12)(cid:12) → almost surely,when n → ∞ for any k = 1 , , . This property is proved in A.7. To derive the asymptotic law of this estimator, we will assume:
Assumption 3.6. (i) nh pn → ∞ and nh p +2 αn → ; (ii) K ( · ) is compactly supported. Proposition 9 (Joint asymptotic normality at different points) . Let z ′ , . . . , z ′ n ′ be fixed points in a set Z ⊂ R p . Assume 3.1, 3.4, 3.6, that the z ′ i are distinct and that f Z and z f X , Z ( x , z ) are continuous on Z ,for every x . Then, as n → ∞ , ( nh pn ) / (cid:0) ˆ τ , | Z = z ′ i − τ , | Z = z ′ i (cid:1) i =1 ,...,n ′ D −→ N (0 , H ( k ) ) , k = 1 , , , where ˆ τ , | Z = z denotes any of the estimators ˆ τ ( k )1 , | Z = z , k = 1 , , or ˜ τ , | Z = z , and H is the n ′ × n ′ diagonalreal matrix defined by [ H ( k ) ] i,j = 4 R K { i = j } f Z ( z ′ i ) (cid:8) E [ g k ( X , X ) g k ( X , X ) | Z = Z = Z = z ′ i ] − τ , | Z = z ′ i (cid:9) , for every ≤ i, j ≤ n ′ , and ( X , Z ) , ( X , Z ) , ( X , Z ) are independent versions. This proposition is proved in A.8.
Remark 10.
The latter results will provide some simple tests of the constancy of the function z τ , | z ,and then of the constancy of the associated conditional copula itself. This would test the famous “simplifyingassumption” (“ H : C , | Z = z does not depend on the choice of z ”), a key assumption for vine modeling inparticular: see [27] or [28] for a discussion, [29] for a review and a presentation of formal tests for thishypothesis. imsart-generic ver. 2014/10/16 file: mainV2.tex date: March 8, 2019 . Derumigny and J.-D. Fermanian/On kernel-based estimation of conditional Kendall’s tau
4. Simulation study
In this simulation study, we draw i.i.d. random samples ( X i, , X i, , Z i ) , i = 1 , . . . , n , with univariate explana-tory variables ( p = 1 ). We consider two settings, that correspond to bounded and/or unbounded explanatoryvariables respectively:1. Z =]0 , and the law of Z is uniform on ]0 , . Conditionally on Z = z , X | Z = z and X | Z = z both follow a Gaussian distribution N ( z, . Their associated conditional copula is Gaussian and theirconditional Kendall’s tau is given by τ , | Z = z = 2 z − .2. Z = R and the law of Z is N (0 , . Conditionally on Z = z , X | Z = z and X | Z = z both follow aGaussian distribution N (Φ( z ) , , where Φ( · ) is the cdf of the Z . Their associated conditional copula isGaussian and their conditional Kendall’s tau is given by τ , | Z = z = 2Φ( z ) − .These simple frameworks allow us to compare the numerical properties of our different estimators indifferent parts of the space, in particular when Z is close to zero or one, i.e. when the conditional Kendall’stau is close to − or to . We compute the different estimators ˆ τ ( k )1 , | Z = z for k = 1 , , , and the symmetricallyrescaled version ˜ τ , | z . The bandwidth h is chosen as proportional to the usual “rule-of-thumb” for kerneldensity estimation, i.e. h = α h ˆ σ ( Z ) n − / with α h ∈ { . , . , , . , } and n ∈ { , , , } . Foreach setting, we consider three local measures of goodness-of-fit: for a given z and for any Kendall’s tauestimate (say ˆ τ , | Z = z ), let • the (local) bias: Bias ( z ) := E [ˆ τ , | Z = z ] − τ , | Z = z , • the (local) standard deviation: Sd ( z ) := E h(cid:0) ˆ τ , | Z = z − E [ˆ τ , | Z = z ] (cid:1) i / , • the (local) mean square-error: M SE ( z ) := E h(cid:0) ˆ τ , | Z = z − τ , | Z = z (cid:1) i .We also consider their integrated version w.r.t the usual Lebesgue measure on the whole support of z ,respectively denoted by IBias , ISd and
IM SE . Some results concerning these integrated measures are givenin Table 1 (resp. Table 2) for Setting (resp. Setting ), and for different choices of α h and n . For the sakeof effective calculations of these measures, all the theoretical previous expectations are replaced by theirempirical counterparts based on simulations.For every n , the best results seem to be obtained with α h = 1 . and the fourth (rescaled) estimator,particularly in terms of bias. This is not so surprising, because the estimators ˆ τ ( k ) , k = 1 , , , do not havethe right support at a finite distance. Note that this comparative advantage of ˜ τ in terms of bias decreases with n , as expected. In terms of integrated variance, all the considered estimators behave more or less similarly,particularly when n ≥ .To illustrate our results for Setting 1 (resp. Setting 2), the functions z Bias ( z ) , Sd ( z ) and M SE ( z ) have been plotted on Figures 1-2 (resp. Figures 3-4), both with our empirically optimal choice α h = 1 . . Wecan note that, considering the bias, the estimator ˜ τ behaves similarly as ˆ τ (1) when the true τ is close to − ,and similarly as ˆ τ (3) when the true Kendall’s tau is close to . But globally, the best pointwise estimatoris clearly obtained with the rescaled version ˜ τ , | Z = · , after a quick inspection of MSE levels, and even ifthe differences between our four estimators weaken for large sample sizes. The comparative advantage of ˜ τ , | z more clearly appears with Setting 2 than with Setting 1. Indeed, in the former case, the support of Z ’s distribution is the whole line. Then ˆ f Z does not suffer any more from the boundary bias phenomenon, imsart-generic ver. 2014/10/16 file: mainV2.tex date: March 8, 2019 . Derumigny and J.-D. Fermanian/On kernel-based estimation of conditional Kendall’s tau contrary to what happened with Setting 1. As a consequence, the biases induced by the definitions of ˆ τ ( k )1 , | z , k = 1 , , appear more strinkingly in Figure 3, for instance: when z is close to ( − (resp. ), the biases of ˆ τ (1)1 , | z (resp. ˆ τ (3)1 , | z ) and ˜ τ , | z are close, when the bias ˆ τ (3)1 , | z (resp. ˆ τ (1)1 , | z ) is a lot larger. Since the squaredbiases are here significantly larger than the variances in the tails, ˜ τ , | z provides the best estimator globallyconsidering ”both sides” together. But even in the center of Z ’s distribution, the latter estimator behavesvery well.In Setting 2 where there is no boundary problem, we also try to estimate the conditional Kendall’s tauusing our cross-validation criterion (4), with N pairs = 1000 . More precisely, denoting by h CV the minimizerof the cross-validation criterion, we try different choices h = α h × h CV with α h ∈ { . , . , , . , } . Theresults in terms of integrated bias, standard deviation and MSE are given in Table 3. We do not find anysubstantial improvements compared to the previous Table 2, where the bandwidth was chosen “roughly”. InTable 4, we compare the average h CV with the previous choice of h . The expectation of h CV is always higherthan the “rule-of-thumb” h ref , but the difference between both decreases when the sample size n increases.The standard deviation of h CV is quite high for low values of n , but decreases as a function of n . This maybe seen as quite surprising given the fact that the number of pairs N pairs used in the computation of thecriterion stays constant. Nevertheless, when the sample size increases, the selected pairs are better in thesense that the differences | Z i − Z j | can become smaller as more replications of Z i are available. References [1] R. Nelsen, An introduction to copulas, Springer Science & Business Media, 2007.[2] W. Kruskal, Ordinal measures of association, J. Amer. Statist. Ass. 53 (284) (1958) 814–861.[3] M. Hollander, D. Wolfe, Nonparametric Statistical Methods, Wiley, 1973.[4] E. Lehmann, Nonparametrics: Statistical Methods Based on Ranks., Holden-Day, 1975.[5] H. Joe, Multivariate models and multivariate dependence concepts, Chapman and Hall/CRC, 1997.[6] A. Patton, Estimation of multivariate models for time series of possibly different lengths, J. Appl.Econometrics 21 (2) (2006) 147–173.[7] A. Patton, Modelling asymmetric exchange rate dependence, Internat. Econom. Rev. 47 (2) (2006) 527–556.[8] I. Gijbels, N. Veraverbeke, M. Omelka, Conditional copulas, association measures and their applications,Comput. Statist. Data Anal. 55 (5) (2011) 1919–1932.[9] N. Veraverbeke, M. Omelka, I. Gijbels, Estimation of a conditional copula and association measures,Scand. J. Stat. 38 (4) (2011) 766–780.[10] J.-D. Fermanian, M. Wegkamp, Time-dependent copulas, J. Multivariate Anal. 110 (2012) 19–29.[11] J.-D. Fermanian, O. Lopez, Single-index copulas, J. Multivariate Anal. 165 (2018) 27–55.[12] W.-Y. Tsai, Testing the assumption of independence of truncation time and failure time, Biometrika77 (1) (1990) 169–177.[13] E. C. Martin, R. A. Betensky, Testing quasi-independence of failure and truncation times via conditionalkendall’s tau, Journal of the American Statistical Association 100 (470) (2005) 484–492.[14] L. Lakhal, L.-P. Rivest, B. Abdous, Estimating survival and association in a semicompeting risks model,Biometrics 64 (1) (2008) 180–188. imsart-generic ver. 2014/10/16 file: mainV2.tex date: March 8, 2019 . Derumigny and J.-D. Fermanian/On kernel-based estimation of conditional Kendall’s tau n = 100 n = 500 n = 1000 n = 2000 IBias ISd IMSE IBias ISd IMSE IBias ISd IMSE IBias ISd IMSE α h = . ˆ τ (1)1 , | Z = z -133 197 66.5 -34.5 84.9 9.86 -18.2 61.6 4.85 -10.9 46 2.65 ˆ τ (2)1 , | Z = z -12.9 187 43.7 -4.08 84.4 8.58 -0.9 61.5 4.49 -1.07 46 2.53 ˆ τ (3)1 , | Z = z
107 190 56.6 26.4 84.5 9.26 16.4 61.5 4.76 8.8 46 2.6 ˜ τ , | Z = z -0.91
213 48.2 -1.18 -0.149 α h = . ˆ τ (1)1 , | Z = z -88 150 35.8 -26.3 68 6.32 -13.9 50.7 3.33 -7.98 37.6 1.8 ˆ τ (2)1 , | Z = z -10.4 145 26.3 -5.97 67.9 5.6 -2.33 50.6 3.12 -1.39 37.5 1.74 ˆ τ (3)1 , | Z = z ˜ τ , | Z = z -2.06 157 26.7 -3.99 69.2 5.49 -1.21 51.2 3.05 -0.76 37.8 1.69 α h = ˆ τ (1)1 , | Z = z -67.8 123 24.5 -19.2 58.7 4.8 -11 43.1 2.52 -6.34 33 1.44 ˆ τ (2)1 , | Z = z -9.99 121 19 -3.95 58.6 4.39 -2.35 43.1 2.39 -1.39 33 1.4 ˆ τ (3)1 , | Z = z ˜ τ , | Z = z -3.48 128 18.1 -2.34 59.5 4.18 -1.46 43.4 2.29 -0.897 33.2 1.35 α h = . ˆ τ (1)1 , | Z = z -44.6 101 17.5 -15.9 50.4 4.12 -9.7 35.9 2.13 -5.52 27.6 1.28 ˆ τ (2)1 , | Z = z -5.81 100 14.9 -5.68 50.3 3.84 -3.84 35.9 2.02 -2.18 27.6 1.24 ˆ τ (3)1 , | Z = z
33 101 15.5 4.58 50.3 3.77 2.01 35.9 1.99 1.15 27.6 1.23 ˜ τ , | Z = z -1.09 104 -4.55 50.8 -3.19 36.1 -1.83 27.7 α h = ˆ τ (1)1 , | Z = z -37.8 91.4 17.3 -11.8 43.8 4.14 -7.2 31.2 2.35 -5.97 23.7 1.43 ˆ τ (2)1 , | Z = z -8.03 91.4 15.4 -3.93 43.8 3.94 -2.75 31.2 2.28 -3.44 23.7 1.39 ˆ τ (3)1 , | Z = z ˜ τ , | Z = z -4.5 94.2 13.5 -3.01 44.1 3.62 -2.24 31.3 2.12 -3.16 23.8 1.32 Table 1
Results of the simulation in Setting 1. All values have been multiplied by 1000. Bold values indicate optimal choices for thechosen measure of performance. − . − . − . . . . . Bias, alpha_h = 1.5 , n = 100 z B i a s a t po i n t z . . . Sd, alpha_h = 1.5 , n = 100 z S t anda r d de v i a t i on a t po i n t z . . . . MSE, alpha_h = 1.5 , n = 100 z M SE a t po i n t z Fig 1 . Local bias, standard deviation and MSE for the estimators ˆ τ (1) (red) , ˆ τ (2) (blue), ˆ τ (3) (green), ˜ τ (orange), with n = 100 and α h = 1 . in Setting 1. The dotted line on the first figure is the reference at 0. imsart-generic ver. 2014/10/16 file: mainV2.tex date: March 8, 2019 . Derumigny and J.-D. Fermanian/On kernel-based estimation of conditional Kendall’s tau − . − . − . . . . . Bias, alpha_h = 1.5 , n = 500 z B i a s a t po i n t z . . . . . . Sd, alpha_h = 1.5 , n = 500 z S t anda r d de v i a t i on a t po i n t z . . . . MSE, alpha_h = 1.5 , n = 500 z M SE a t po i n t z Fig 2 . Local bias, standard deviation and MSE for the estimators ˆ τ (1) (red) , ˆ τ (2) (blue), ˆ τ (3) (green), ˜ τ (orange), with n = 500 and α h = 1 . in Setting 1. The dotted line on the first figure is the reference at 0. n = 100 n = 500 n = 1000 n = 2000 IBias ISd IMSE IBias ISd IMSE IBias ISd IMSE IBias ISd IMSE α h = . ˆ τ (1)1 , | Z = z -207 227 180 -54.1 83.9 16.9 -29.6 55.3 5.81 -16.9 38.9 2.49 ˆ τ (2)1 , | Z = z
207 97 ˆ τ (3)1 , | Z = z
210 228 181 55.7 83.2 16.4 30.7 55.4 5.9 17.2 38.9 2.5 ˆ τ (4)1 , | Z = z
55 3.22 0.175 38.9 1.66 α h = . ˆ τ (1)1 , | Z = z -144 175 98.6 -33.3 60.6 7.5 -19.8 41.9 3.12 -10.6 30.5 1.42 ˆ τ (2)1 , | Z = z -2.33 163 56.2 1.73 59.4 5.56 -0.0619 41.7 2.51 0.665 30.4 1.24 ˆ τ (3)1 , | Z = z
140 176 99.2 36.8 60.7 7.73 19.7 42.1 3.12 11.9 30.5 1.45 ˆ τ (4)1 , | Z = z -3.15 170 30.3 1.69 60.2 3.85 -0.093 42.1 1.95 0.645 30.5 1.05 α h = ˆ τ (1)1 , | Z = z -99.8 143 57.7 -24.9 50.9 5.06 -13.5 36.6 2.28 -6.92 26.6 1.09 ˆ τ (2)1 , | Z = z ˆ τ (3)1 , | Z = z
102 139 54.4 26.7 51 5.13 15.8 36.6 2.33 9.83 26.6 1.11 ˆ τ (4)1 , | Z = z α h = . ˆ τ (1)1 , | Z = z -59.1 104 28.1 -14.7 42.3 3.87 -7.56 29.7 1.86 -4.17 21.8 0.932 ˆ τ (2)1 , | Z = z ˆ τ (3)1 , | Z = z ˆ τ (4)1 , | Z = z α h = ˆ τ (1)1 , | Z = z -37.2 88.2 23.9 -9.57 38.2 4.6 -3.75 26.2 2.34 -1.09 19.8 1.32 ˆ τ (2)1 , | Z = z ˆ τ (3)1 , | Z = z ˆ τ (4)1 , | Z = z Table 2
Results of the simulation in Setting 2. All values have been multiplied by 1000. Bold values indicate optimal choices for thechosen measure of performance. imsart-generic ver. 2014/10/16 file: mainV2.tex date: March 8, 2019 . Derumigny and J.-D. Fermanian/On kernel-based estimation of conditional Kendall’s tau −2 −1 0 1 2 − . − . . . . Bias, alpha_h = 1.5 , n = 100 z B i a s a t po i n t z −2 −1 0 1 2 . . . . Sd, alpha_h = 1.5 , n = 100 z S t anda r d de v i a t i on a t po i n t z −2 −1 0 1 2 . . . . . MSE, alpha_h = 1.5 , n = 100 z M SE a t po i n t z Fig 3 . Local bias, standard deviation and MSE for the estimators ˆ τ (1) (red) , ˆ τ (2) (blue), ˆ τ (3) (green), ˜ τ (orange), with n = 100 and α h = 1 . in Setting 2. The dotted line on the first figure is the reference at 0. −2 −1 0 1 2 − . − . . . . Bias, alpha_h = 1.5 , n = 500 z B i a s a t po i n t z −2 −1 0 1 2 . . . . . . . Sd, alpha_h = 1.5 , n = 500 z S t anda r d de v i a t i on a t po i n t z −2 −1 0 1 2 . . . . . . . MSE, alpha_h = 1.5 , n = 500 z M SE a t po i n t z Fig 4 . Local bias, standard deviation and MSE for the estimators ˆ τ (1) (red) , ˆ τ (2) (blue), ˆ τ (3) (green), ˜ τ (orange), with n = 500 and α h = 1 . in Setting 2. The dotted line on the first figure is the reference at 0. imsart-generic ver. 2014/10/16 file: mainV2.tex date: March 8, 2019 . Derumigny and J.-D. Fermanian/On kernel-based estimation of conditional Kendall’s tau n = 100 n = 500 n = 1000 n = 2000 IBias ISd IMSE IBias ISd IMSE IBias ISd IMSE IBias ISd IMSE α h = . ˆ τ (1)1 , | Z = z -111 154 66.2 -36.9 66.8 9.01 -22.4 48.2 4.06 -12.9 36.1 2.04 ˆ τ (2)1 , | Z = z
137 36.3 ˆ τ (3)1 , | Z = z
111 151 60.6 37.4 66.3 8.88 23.5 47.2 4.07 15.5 36.2 2.18 ˆ τ (4)1 , | Z = z α h = . ˆ τ (1)1 , | Z = z -67.4 117 35.7 -23.3 52.1 5.27 -13.9 37.8 2.4 -7.6 29 1.3 ˆ τ (2)1 , | Z = z ˆ τ (3)1 , | Z = z ˆ τ (4)1 , | Z = z α h = ˆ τ (1)1 , | Z = z -43 101 28 -15.8 45.7 4.44 -9.51 33.1 2.04 -4.68 25.1 1.07 ˆ τ (2)1 , | Z = z ˆ τ (3)1 , | Z = z ˆ τ (4)1 , | Z = z α h = . ˆ τ (1)1 , | Z = z -16.1 95.6 41.7 -6.36 43 6.35 -4.04 30.6 2.87 -1.11 22.1 1.34 ˆ τ (2)1 , | Z = z ˆ τ (3)1 , | Z = z
46 92.8 42.2 16.5 42.6 6.45 10.4 30.4 2.94 8.06 22.1 1.4 ˆ τ (4)1 , | Z = z Table 3
Results of the simulation in Setting 2 using h = α h × h CV where h CV has been chosen by cross-validation. All values havebeen multiplied by 1000. Bold values indicate optimal choices for the chosen measure of performance. n
100 500 1000 2000 E [ h CV ] Sd [ h CV ] h ref = n − / Table 4
Expectation and standard deviation of the bandwidth selected by cross-validation as a function of the sample size n , andcomparison with bandwidth h ref chosen by the rule-of-thumb. imsart-generic ver. 2014/10/16 file: mainV2.tex date: March 8, 2019 . Derumigny and J.-D. Fermanian/On kernel-based estimation of conditional Kendall’s tau [15] J.-J. Hsieh, W.-C. Huang, Nonparametric estimation and test of conditional kendall’s tau under semi-competing risks data and truncated data, Journal of Applied Statistics 42 (7) (2015) 1602–1616.[16] L. L. Chaieb, L.-P. Rivest, B. Abdous, Estimating survival under a dependent truncation, Biometrika93 (3) (2006) 655–669.[17] Y.-J. Kim, Estimation of conditional kendall’s tau for bivariate interval censored data, Communicationsfor Statistical Applications and Methods 22 (6) (2015) 599–604.[18] D. Oakes, Bivariate survival models induced by frailties, Journal of the American Statistical Association84 (406) (1989) 487–493.[19] A. K. Manatunga, D. Oakes, A measure of association for bivariate frailty distributions, Journal ofMultivariate Analysis 56 (1) (1996) 60–74.[20] A. V. Asimit, R. Gerrard, Y. Hou, L. Peng, Tail dependence measure for examining financial extremeco-movements, Journal of Econometrics 194 (2) (2016) 330–348.[21] A. Liu, Y. Hou, L. Peng, Interval estimation for a measure of tail dependence, Insurance: Mathematicsand Economics 64 (2015) 294–305.[22] A. Sabeti, M. Wei, R. V. Craiu, Additive models for conditional copulas, Stat 3 (1) (2014) 300–312.[23] J. Dony, D. Mason, Uniform in bandwidth consistency of conditional u-statistics, Bernoulli 14 (4) (2008)1108–1133.[24] P. Major, An estimate on the supremum of a nice class of stochastic integrals and u-statistics, ProbabilityTheory and Related Fields 134 (3) (2006) 489–537.[25] E. Giné, A. Guillou, Rates of strong uniform consistency for multivariate kernel density estimators,Annales de l’Institut Henri Poincare (B) Probability and Statistics 38 (6) (2002) 907–921.[26] U. Einmahl, D. Mason, Uniform in bandwidth consistency of kernel-type function estimators, Ann.Statist. 33 (3) (2005) 1380–1403.[27] E. F. Acar, C. Genest, J. Ne ˇ slehová, Beyond simplified pair-copula constructions, Journal of MultivariateAnalysis 110 (2012) 74–90.[28] I. Hobæk Haff, K. Aas, A. Frigessi, On the simplified pair-copula construction–simply useful or toosimplistic ?, J. Multivariate Anal. 101 (2010) 1296–1310.[29] A. Derumigny, J.-D. Fermanian, About tests of the “simplifying” assumption for conditional copulas,Depend. Model. 5 (1) (2017) 154–197.[30] R. Serfling, Approximation theorems of mathematical statistics, John Wiley & Sons, 1980.[31] A. Rinaldo, L. Wasserman, et al., Generalized density clustering, The Annals of Statistics 38 (5) (2010)2678–2722.[32] D. Bosq, J.-P. Lecoutre, Théorie de l’estimation fonctionnelle, Economica, 1987.[33] W. Stute, Conditional U-statistics, Ann. Probab. 19 (2) (1991) 812–825. Appendix A: Proofs
For convenience, we recall Berk’s (1970) inequality (see Theorem A in Serfling [30, p.201]). Note that, if m = 1 , this reduces to Bernstein’s inequality. Lemma 11.
Let m, n > , X , . . . , X n i.i.d. random vectors with values in a measurable space X and g : X m → [ a, b ] be a symmetric real bounded function. Set θ := E [ g ( X , . . . , X m )] and σ := V ar [ g ( X , . . . , X m )] . imsart-generic ver. 2014/10/16 file: mainV2.tex date: March 8, 2019 . Derumigny and J.-D. Fermanian/On kernel-based estimation of conditional Kendall’s tau Then, for any t > and n ≥ m , IP (cid:18) nm (cid:19) − X c g ( X i , . . . , X i m ) − θ ≥ t ! ≤ exp (cid:18) − [ n/m ] t σ + (2 / b − θ ) t (cid:19) , where P c denotes summation over all subgroups of m distinct integers ( i , . . . , i m ) of { , . . . n } . A.1. Proof of Proposition 1
Since there are no ties a.s., τ (1)1 , | Z = z = 4 n X i =1 n X j =1 w i,n ( z ) w j,n ( z ) (cid:16) (cid:8) X i, < X j, (cid:9) − (cid:8) X i, < X j, , X i, > X j, (cid:9)(cid:17) = 4 n X i =1 n X j =1 w i,n ( z ) w j,n ( z ) (cid:8) X i, < X j, (cid:9) + ˆ τ (3)1 , | Z = z − . But n X i =1 n X j =1 w i,n ( z ) w j,n ( z ) = n X i =1 n X j =1 w i,n ( z ) w j,n ( z ) (cid:16) (cid:8) X i, ≤ X j, (cid:9) + (cid:8) X i, > X j, (cid:9)(cid:17) = 2 n X i =1 n X j =1 w i,n ( z ) w j,n ( z ) (cid:8) X i, < X j, (cid:9) + n X i =1 w i,n , implying τ (1)1 , | Z = z = 2(1 − s n ) + ˆ τ (3)1 , | Z = z − , and then ˆ τ (1)1 , | Z = z = ˆ τ (3)1 , | Z = z − s n . Moreover, ˆ τ (2)1 , | Z = z = n X i =1 n X j =1 w i,n ( z ) w j,n ( z ) (cid:16) (cid:8) X i, > X j, , X i, > X j, (cid:9) + (cid:8) X i, < X j, , X i, < X j, (cid:9) − (cid:8) X i, > X j, , X i, < X j, (cid:9) − (cid:8) X i, < X j, , X i, > X j, (cid:9) = 2 n X i =1 n X j =1 w i,n ( z ) w j,n ( z ) (cid:16) (cid:8) X i, > X j, , X i, > X j, (cid:9) − (cid:8) X i, > X j, , X i, < X j, (cid:9)(cid:17) = 12 (cid:0) ˆ τ (1)1 , | Z = z + 1 (cid:1) + 12 (cid:0) ˆ τ (3)1 , | Z = z − (cid:1) = ˆ τ (1)1 , | Z = z + ˆ τ (3)1 , | Z = z τ (1)1 , | Z = z + s n = ˆ τ (3)1 , | Z = z − s n . (cid:3) A.2. Proof of Proposition 2
Lemma 12.
Under Assumptions 3.1, 3.2 and 3.3, we have for any t > , IP (cid:18)(cid:12)(cid:12) ˆ f Z ( z ) − f Z ( z ) (cid:12)(cid:12) ≥ C K,α h α α ! + t (cid:19) ≤ (cid:18) − nh p t f Z ,max R K + (2 / C K t (cid:19) . This Lemma is proved below. If, for some ǫ > , we have C K,α h α /α ! + t ≤ f Z ,min − ǫ , then ˆ f ( z ) ≥ ǫ > with a probability larger than − (cid:0) − nh p t / (2 f Z ,max R K + (2 / C K t ) (cid:1) . So, we should choose thelargest t as possible, which yields Proposition 2. imsart-generic ver. 2014/10/16 file: mainV2.tex date: March 8, 2019 . Derumigny and J.-D. Fermanian/On kernel-based estimation of conditional Kendall’s tau It remains to prove Lemma 12. Use the usual decomposition between a stochastic component and a bias: ˆ f Z ( z ) − f Z ( z ) = (cid:0) ˆ f Z ( z ) − E [ ˆ f Z ( z )] (cid:1) + (cid:0) E [ ˆ f Z ( z )] − f Z ( z ) (cid:1) . We first bound the bias from above. E [ ˆ f Z ( z )] − f Z ( z ) = Z R p K ( u ) (cid:16) f Z (cid:0) z + h u (cid:1) − f Z ( z ) (cid:17) d u . Set φ z , u ( t ) := f Z (cid:0) z + th u (cid:1) for t ∈ [0 , . This function has at least the same regularity as f Z , so it is α -differentiable. By a Taylor-Lagrange expansion, we get Z R p K ( u ) (cid:16) f Z (cid:0) z + h u (cid:1) − f Z ( z ) (cid:17) d u = Z R p K ( u ) (cid:18) α − X i =1 i ! φ ( i ) z , u (0) + 1 α ! φ ( α ) z , u ( t z , u ) (cid:19) d u , for some real number t z , u ∈ (0 , . By Assumption 3.1 and for every i < α , R R p K ( u ) φ ( i ) z , u (0) d u = 0 . Therefore, (cid:12)(cid:12)(cid:12) E [ ˆ f Z ( z )] − f Z ( z ) (cid:12)(cid:12)(cid:12) = (cid:12)(cid:12)(cid:12) Z R p K ( u ) 1 α ! φ ( α ) z , u ( t z , u ) d u (cid:12)(cid:12)(cid:12) = 1 α ! (cid:12)(cid:12)(cid:12) Z R p K ( u ) p X i ,...,i α =1 h α u i . . . u i α ∂ α f Z ∂z i . . . ∂z i α (cid:0) z + t z , u h u (cid:1) d u (cid:12)(cid:12)(cid:12) ≤ C K,α α ! h α . Second, the stochastic component may be written as ˆ f Z ( z ) − E [ ˆ f Z ( z )] = n − n X i =1 K h ( Z i − z ) − E h n − n X i =1 K h ( Z i − z ) i = n − n X i =1 (cid:0) g z ( Z i ) − E [ g ( Z i )] (cid:1) , where g ( Z i ) := K h ( Z i − z ) . Apply Lemma 11 with m = 1 and the latter g ( Z i ) . Here, we have b = − a = h − p C K , θ = E [ g ( Z )] ≥ and (cid:12)(cid:12) V ar [ g ( Z )] (cid:12)(cid:12) ≤ h − p f Z ,max R K , and we get IP (cid:12)(cid:12) n n X i =1 K h ( Z i − z ) − E [ K h ( Z i − z )] (cid:12)(cid:12) ≥ t ! ≤ (cid:18) − nt h − p f Z ,max R K + (2 / h − p C K t (cid:19) . (cid:3) A.3. Proof of Proposition 3
We show the result for k = 1 . The two other cases can be proven in the same way.Consider the decomposition ˆ τ , | Z = z − τ , | Z = z = 4 X ≤ i,j ≤ n w i,n ( z ) w j,n ( z ) (cid:8) X i < X j (cid:9) − (cid:0) X < X (cid:12)(cid:12) Z = Z = z (cid:1) = 4 n ˆ f Z ( z ) X ≤ i,j ≤ n K h ( Z i − z ) K h ( Z j − z ) (cid:16) (cid:8) X i < X j (cid:9) − IP (cid:0) X < X (cid:12)(cid:12) Z = Z = z (cid:1)(cid:17) =: 4ˆ f Z ( z ) X ≤ i,j ≤ n S i,j ( z ) . Therefore, for any positive numbers x and λ ( z ) , we have IP( | ˆ τ , | Z = z − τ , | Z = z | > x ) ≤ IP (cid:16) f Z ( z ) > λ ( z ) f Z ( z ) (cid:17) + IP (cid:16) λ ( z )) f Z ( z ) × | X ≤ i,j ≤ n S i,j ( z ) | > x (cid:17) ≤ IP (cid:16) | f Z ( z ) − f Z ( z ) | > λ ( z ) f Z ( z ) (cid:17) + IP (cid:16) λ ( z )) f Z ( z ) × | X ≤ i,j ≤ n S i,j ( z ) | > x (cid:17) . imsart-generic ver. 2014/10/16 file: mainV2.tex date: March 8, 2019 . Derumigny and J.-D. Fermanian/On kernel-based estimation of conditional Kendall’s tau For any t s.t. C K,α h α /α ! + t < f Z ,min / , set λ ( z ) = 16 f z ( z ) (cid:0) C K,α h α /α ! + t (cid:1) /f Z ,min . This yields IP (cid:16) | ˆ τ , | Z = z − τ , | Z = z | > x (cid:17) ≤ IP (cid:16) | f Z ( z ) − f Z ( z ) | > f Z ,min (cid:0) C K,α h α α ! + t (cid:1)(cid:17) + IP (cid:16) | X ≤ i,j ≤ n S i,j ( z ) | > f z ( z ) x λ ( z )) (cid:17) . By setting x = 4 f z ( z ) (cid:16) C XZ ,α h α α ! + 3 f z ( z ) R K nh p + t ′ (cid:17)(cid:18) f Z ( z ) f Z ,min (cid:16) C K,α h α α ! + t (cid:17)(cid:19) , and applying the next two lemmas 13 and 14, we get the result. (cid:3) Lemma 13.
Under Assumptions 3.1-3.3 and if C K,α h α /α ! + t < f Z ,min / for some t > , IP (cid:18) | f Z ( z ) − f Z ( z ) | > f Z ,min (cid:16) C K,α h α α ! + t (cid:17)(cid:19) ≤ (cid:18) − nh p t f Z ,max R K + (2 / C K t (cid:19) , and ˆ f Z ( z ) is strictly positive on these events.Proof : Applying the mean value inequality to the function x /x , we get the inequality (cid:12)(cid:12)(cid:12) / ˆ f Z ( z ) − /f Z ( z ) (cid:12)(cid:12)(cid:12) ≤ (cid:12)(cid:12) ˆ f Z ( z ) − f Z ( z ) (cid:12)(cid:12) /f ∗ Z , where f ∗ Z lies between ˆ f Z ( z ) and f Z ( z ) . Denote by E the event E := (cid:8) | ˆ f Z ( z ) − f Z ( z ) | ≤ C K,α h α /α ! + t (cid:9) . By Lemma 12, we obtain IP( E ) ≥ − (cid:16) − nh p t f Z ,max R K + (2 / C K t (cid:17) . (5)Therefore, on this event E , (cid:12)(cid:12) ˆ f Z ( z ) − f Z ( z ) (cid:12)(cid:12) ≤ f Z ,min / , so that f Z ,min / ≤ ˆ f Z ( z ) . We have also f Z ,min / ≤ f Z ( z ) and then f Z ,min / ≤ f ∗ Z . Combining the previous inequalities, we finally get (cid:12)(cid:12)(cid:12)(cid:12) f Z ( z ) − f Z ( z ) (cid:12)(cid:12)(cid:12)(cid:12) ≤ f Z ,min (cid:12)(cid:12) ˆ f Z ( z ) − f Z ( z ) (cid:12)(cid:12) ≤ f Z ,min (cid:16) C K,α h α α ! + t (cid:17) , on E . But since IP (cid:18) | f Z ( z ) − f Z ( z ) | > f Z ,min (cid:16) C K,α h α α ! + t (cid:17)(cid:19) ≤ IP( E c ) , we deduce the result. (cid:3) Lemma 14.
Under Assumptions 3.1-3.4, if C ˜ K, h < f z ( z ) , we have for any t > (cid:18)(cid:12)(cid:12)(cid:12) X ≤ i,j ≤ n S i,j ( z ) (cid:12)(cid:12)(cid:12) > C XZ ,α h α α ! + 3 f z ( z ) R K nh p + t (cid:19) ≤ (cid:18) − ( n − h p t f Z ,max ( R K ) + (8 / C K t (cid:19) + 2 exp (cid:18) − nh p ( f z ( z ) − C ˜ K, h ) f Z ,max R ˜ K + 4 C ˜ K ( f z ( z ) − C ˜ K, h ) / (cid:19) . Proof :
Note that P ≤ i,j ≤ n S i,j ( z ) = P ≤ i = j ≤ n (cid:0) S i,j ( z ) − E [ S i,j ( z )] (cid:1) + n ( n − E [ S , ( z )] + P ni =1 S i,i ( z ) . The “diagonal term” P ni =1 S i,i ( z ) = − IP (cid:0) X < X (cid:12)(cid:12) Z = Z = z (cid:1) P ni =1 K h ( Z i − z ) /n is negative andnegligible. It will be denoted by − ∆ n ( z ) < . Note that ˜ K ( · ) := K ( · ) / R K is a two-order kernel. Then, imsart-generic ver. 2014/10/16 file: mainV2.tex date: March 8, 2019 . Derumigny and J.-D. Fermanian/On kernel-based estimation of conditional Kendall’s tau ˜ f z ( z ) := P ni =1 ˜ K h ( Z i − z ) /n is a consistent estimator of f Z ( z ) . Therefore, due to Lemma 12 and with obviousnotations, we have for every ε > (cid:18)(cid:12)(cid:12) ˜ f Z ( z ) − f Z ( z ) (cid:12)(cid:12) ≥ C ˜ K, h ε (cid:19) ≤ (cid:18) − nh p ε f Z ,max R ˜ K + (2 / C ˜ K ε (cid:19) . This implies IP (cid:18) | R K n h p n X i =1 ˜ K h ( Z i − z ) − f Z ( z ) R K nh p | ≥ (cid:16) R K nh p (cid:17)(cid:16) C ˜ K, h ε (cid:17)(cid:19) ≤ (cid:18) − nh p ε f Z ,max R ˜ K + (2 / C ˜ K ε (cid:19) . By choosing ε s.t. C ˜ K, h / ε = f z ( z ) / , ∆ n will be smaller than f z ( z ) R K / (2 nh p ) with a probabilitythat is larger than − (cid:18) − nh p ε f Z ,max R ˜ K + (2 / C ˜ K ε (cid:19) . (6)Now, let us deal with the main term, that is decomposed as a stochastic component and a bias component.First, let us deal with the bias. Simple calculations provide, if i = j , E [ S i,j ( z )] = n − E (cid:20) K h ( Z i − z ) K h ( Z j − z ) (cid:16) (cid:8) X i < X j (cid:9) − IP (cid:0) X i < X j (cid:12)(cid:12) Z i = Z j = z (cid:1)(cid:17)(cid:21) = n − Z R p +2 K h ( z − z ) K h ( z − z ) (cid:16) (cid:8) x < x (cid:9) − IP (cid:0) X i < X j (cid:12)(cid:12) Z i = Z j = z (cid:1)(cid:17) × f X , Z ( x , z ) f X , Z ( x , z ) d x d z d x d z = n − Z R p +2 K ( u ) K ( v ) (cid:16) (cid:8) x < x (cid:9) − IP (cid:0) X i < X j (cid:12)(cid:12) Z i = Z j = z (cid:1)(cid:17) × (cid:18) f X , Z (cid:16) x , z + h u (cid:17) f X , Z (cid:16) x , z + h v (cid:17) − f X , Z ( x , z ) f X , Z ( x , z ) (cid:19) d x d u d x d v , because, for every z , Z R (cid:16) (cid:8) x < x (cid:9) − IP (cid:0) X < X (cid:12)(cid:12) Z = Z = z (cid:1)(cid:17) f X , Z ( x , z ) f X , Z ( x , z ) d x d x . Apply the Taylor-Lagrange formula to the function φ x , x , u , v ( t ) := f X , Z (cid:0) x , z + th u (cid:1) f X , Z (cid:0) x , z + th v (cid:1) . Withobvious notation, this yields E [ S i,j ( z )] = n − Z K ( u ) K ( v ) (cid:16) (cid:8) x < x (cid:9) − IP (cid:0) X i < X j (cid:12)(cid:12) Z i = Z j = z (cid:1)(cid:17) × (cid:18) α − X k =1 k ! φ ( k ) x , x , u , v (0) + 1 α ! φ ( α ) x , x , u , v ( t x , x , u , v ) (cid:19) d x d u d x d v = Z K ( u ) K ( v ) n α ! (cid:16) (cid:8) x < x (cid:9) − IP (cid:0) X i < X j (cid:12)(cid:12) Z i = Z j = z (cid:1)(cid:17) φ ( α ) x , x , u , v ( t x , x , u , v ) d x d u d x d v . Since φ ( α ) x , x , u , v ( t ) is equal to α X k =0 (cid:18) αk (cid:19) p X i ,...,i α =1 h α u i . . . u i k v i k +1 . . . v i α ∂ k f X , Z ∂z i . . . ∂z i k (cid:16) x , z + th u (cid:17) ∂ α − k f X , Z ∂z i k +1 . . . ∂z i α (cid:16) x , z + th v (cid:17) , imsart-generic ver. 2014/10/16 file: mainV2.tex date: March 8, 2019 . Derumigny and J.-D. Fermanian/On kernel-based estimation of conditional Kendall’s tau using Assumption 3.4, we get (cid:12)(cid:12) E [ S , ( z )] (cid:12)(cid:12) ≤ C XZ ,α h α / ( n α !) . (7)Second, the stochastic component will be bounded from above. Indeed, X ≤ i = j ≤ n ( S i,j ( z ) − E [ S i,j ( z )]) = 1 n X ≤ i = j ≤ n g z (cid:0) ( X i , Z i ) , ( X j , Z j ) (cid:1) , with the function g z defined by g z (cid:0) ( X i , Z i ) , ( X j , Z j ) (cid:1) := K h ( Z i − z ) K h ( Z j − z ) (cid:16) (cid:8) X i < X j (cid:9) − IP (cid:0) X i < X j (cid:12)(cid:12) Z i = Z j = z (cid:1)(cid:17) − E (cid:20) K h ( Z i − z ) K h ( Z j − z ) (cid:16) (cid:8) X i < X j (cid:9) − IP (cid:0) X i < X j (cid:12)(cid:12) Z i = Z j = z (cid:1)(cid:17)(cid:21) . The symmetrized version of g is ˜ g i,j = (cid:16) g z (cid:0) ( X i , Z i ) , ( X j , Z j ) (cid:1) + g z (cid:0) ( X j , Z j ) , ( X i , Z i ) (cid:1)(cid:17) / . We can nowapply Lemma 11 to the sum of the ˜ g i,j . With its notation, θ = E (cid:2) ˜ g i,j (cid:3) = 0 . Moreover, (cid:12)(cid:12)(cid:12) V ar h g z (cid:0) ( X i , Z i ) , ( X j , Z j ) (cid:1)i(cid:12)(cid:12)(cid:12) ≤ Z K h ( z − z ) K h ( z − z ) (cid:16) (cid:8) x < x (cid:9) − IP (cid:0) X i < X j (cid:12)(cid:12) Z i = Z j = z (cid:1)(cid:17) × f X , Z ( x , z ) f X , Z ( x , z ) d x d x d z d z ≤ Z K ( t ) K ( t ) h p f X , Z ( x , z − h t ) f X , Z ( x , z − h t ) d x d x d t d t ≤ h − p f Z ,max (cid:16) Z K (cid:17) , and the same upper bound applies for ˜ g i,j (invoke Cauchy-Schwarz inequality). Here, we choose b = − a =2 C K h − p . This yields IP (cid:16) n ( n − X ≤ i A.4. Proof of Proposition 4 With the notations of the proof of Proposition 3, we get the following lemma, that straightforwardly impliesthe result. imsart-generic ver. 2014/10/16 file: mainV2.tex date: March 8, 2019 . Derumigny and J.-D. Fermanian/On kernel-based estimation of conditional Kendall’s tau Lemma 15. Under the Assumptions and conditions of Proposition 4, we have IP (cid:18)(cid:12)(cid:12)(cid:12) X ≤ i,j ≤ n S i,j ( z ) (cid:12)(cid:12)(cid:12) > C XZ ,α h α α ! + 3 f z ( z ) R K nh p + t (cid:19) ≤ C exp (cid:18) − α nh p t f Z ,max ( R K ) (cid:19) + 2 exp (cid:18) − nh p ( f z ( z ) − C ˜ K, h ) f Z ,max R ˜ K + 4 C ˜ K ( f z ( z ) − C ˜ K, h ) / (cid:19) + 2 exp (cid:16) nh p t R K ( R | K | ) f Z ,max + 8 C K R | K | f Z ,max t/ (cid:17) . Proof : We lead exactly the same reasoning and the same notations as in Lemma 14, until (8). Now,with the same notations, introduce g i := E [˜ g i,j | X i , Z i ] and consider ξ i,j := ˜ g i,j − g i − g j . Then, ξ i,j isa degenerate (symmetrical) U-statistics because E [ ξ i,j | X i , Z i ] = E [ ξ i,j | X j , Z j ] = 0 , when i = j . Actually ξ i,j =: ξ z ( X i , Z i , X j , Z j ) for some function ξ z and set ℓ z : ( x , z , x , z ) h p C K ξ z (cid:0) ( x , z ) , ( x , z ) (cid:1) , (9)for a fixed z and a fixed h . This yields k ℓ z k ∞ ≤ . By usual changes of variables, we get Z ℓ z ( x , z , x , z ) f X , Z ( x , z ) f X , Z ( x , z ) d x d x d z d z ≤ h p ( R K f z ,max ) (4 C K ) + 6 h p R K f z ,max ( R | K | f z ,max ) (4 C K ) ≤ σ , with σ := h p C σ , C σ := Z K f z ,max / (2 C K ) , (10)because h p R K f z ,max ( R | K | f z ,max ) ≤ ( R K f z ,max ) . With the notations of [24], this implies D = 1 , m = 1 and L is arbitrarily small. Therefore, Theorem 2 in [24] yields IP (cid:16) n | X i = j ℓ z ( X i , Z i , X j , Z j ) | > x (cid:17) ≤ C exp (cid:18) − α xσ (cid:19) , (11)for some universal constants C and α when x ≤ nσ . By setting t/ C K x/ ( nh p ) and applyingLemma 11, this provides IP (cid:16) | X ≤ i = j ≤ n (cid:0) S i,j ( z ) − E [ S i,j ( z )] (cid:1) | ≥ t (cid:17) ≤ IP (cid:16) n | X ≤ i = j ≤ n ξ ij | ≥ t/ (cid:17) + IP (cid:16) | n n X i =1 g i | ≥ t/ (cid:17) ≤ C exp (cid:16) − α nth p f Z ,max ( R K ) (cid:17) + 2 exp (cid:16) nh p t R K ( R | K | ) f Z ,max + 8 / C K R | K | f Z ,max t (cid:17) , when t ≤ h p ( R K ) f Z ,max /C K . This inequality, (6) and (7) conclude the proof. (cid:3) A.5. Proof of Proposition 6 For k = 1 , we follow the paths of the proof of Proposition 4. Since ˆ τ , | Z = z − τ , | Z = z = 4 P ≤ i,j ≤ n S i,j ( z ) / ˆ f Z ( z ) ,we prove the result if we bound from above / ˆ f Z ( z ) and (cid:12)(cid:12) P ≤ i,j ≤ n S i,j ( z ) (cid:12)(cid:12) uniformly w.r.t. z ∈ Z . To be imsart-generic ver. 2014/10/16 file: mainV2.tex date: March 8, 2019 . Derumigny and J.-D. Fermanian/On kernel-based estimation of conditional Kendall’s tau specific, for any positive constant µ < , if | ˆ f Z ( z ) − f Z ( z ) | ≤ µf z ,min , then / ˆ f z ,max ( z ) ≤ f − z ,min (1 − µ ) − .We deduce IP( sup z ∈ Z | ˆ τ , | Z = z − τ , | Z = z | > x ) ≤ IP (cid:0) k ˆ f Z − f Z k ∞ > µf z ,min (cid:1) + IP( 4 f Z ,min (1 − µ ) sup z ∈ Z | X ≤ i,j ≤ n S i,j ( z ) | > x ) . First invoke the uniform exponential inequality, as stated in [31], Proposition 9: for every ε < b K R K f Z ,max /C K , IP (cid:0) k ˆ f Z − f Z k ∞ > ε + C XZ ,α h α α ! (cid:1) ≤ IP (cid:0) k ˆ f Z − E [ ˆ f Z ] k ∞ > ε (cid:1) ≤ L K exp (cid:0) − C f,K nh p ε (cid:1) , (12)for n sufficiently large. Then, apply Lemma 16, by setting ( x, ε ) so that x = 4 f z ,min (1 − µ ) (cid:16) C XZ ,α h α α ! + 3 f z ,max R K nh p + t (cid:17) and ε + C XZ ,α h α α ! = µf z ,min . (cid:3) Lemma 16. Under the assumptions and conditions of Proposition 6, we have IP (cid:18) sup z ∈ Z (cid:12)(cid:12)(cid:12) X ≤ i,j ≤ n S i,j ( z ) (cid:12)(cid:12)(cid:12) > C XZ ,α h α α ! + 3 f z ,max R K nh p + t (cid:19) ≤ C D exp (cid:18) − α nth p f Z ,max ( R K ) (cid:19) + L ˜ K exp (cid:0) − C f, ˜ K nh p ( f z ,max − ˜ C XZ , h ) / (cid:1) + 2 exp (cid:0) − A nh p t C − K A R K f z ,max ( R | K | ) (cid:1) + 2 exp (cid:0) − A nh p t C K A (cid:1) . Proof : We will use the arguments and notations of the proof of Lemmas 14 and 15. We still invoke thedecomposition P ≤ i,j ≤ n S i,j ( z ) = P ≤ i = j ≤ n (cid:0) S i,j ( z ) − E [ S i,j ( z )] (cid:1) + n ( n − E [ S , ( z )] + P ni =1 S i,i ( z ) . Firstlet us find a uniform bound for the “diagonal term” ∆ n ( z ) = P ni =1 S i,i ( z ) = R K ˜ f z ( z ) / ( nh p ) . As in (12), forevery ε < b ˜ K R ˜ K f Z ,max /C ˜ K , IP (cid:0) k ˜ f Z − f Z k ∞ > ε + ˜ C XZ , h (cid:1) ≤ L ˜ K exp (cid:0) − C f, ˜ K nh p ε (cid:1) , for n sufficiently large. This implies IP (cid:18) sup z ∈ Z | R K n h p n X i =1 ˜ K h ( Z i − z ) − f Z ( z ) R K nh p | ≥ (cid:16) R K nh p (cid:17)(cid:0) ε + ˜ C XZ , h (cid:1)(cid:19) ≤ L ˜ K exp (cid:0) − C f, ˜ K nh p ε (cid:1) . Choose ε s.t. ˜ C XZ , h / ε = f z ,max / . Then, sup z | ∆ n ( z ) | will be smaller than f z ,max R K / (2 nh p ) with a probability that is larger than − L ˜ K exp (cid:0) − C f, ˜ K nh p ε (cid:1) . (13)Moreover, it is easy to see that sup z ∈ Z (cid:12)(cid:12) E [ S , ( z )] (cid:12)(cid:12) ≤ C XZ ,α h α / ( n α !) . (14) imsart-generic ver. 2014/10/16 file: mainV2.tex date: March 8, 2019 . Derumigny and J.-D. Fermanian/On kernel-based estimation of conditional Kendall’s tau With the notations of Lemma 15’s proof, the stochastic component is driven by X ≤ i = j ≤ n ( S i,j ( z ) − E [ S i,j ( z )]) = 1 n X ≤ i = j ≤ n g z (cid:0) ( X i , Z i ) , ( X j , Z j ) (cid:1) = 1 n X ≤ i = j ≤ n ˜ g i,j = 1 n X ≤ i = j ≤ n ξ i,j + 2( n − n n X i =1 g i . Now apply Theorem 1 in [24], by recalling (9) and considering the family F := n ℓ z , z ∈ Z o , for a fixedbandwidth h . The constant σ has the same value as in (10). It is easy to check that the latter class of functionsis L dense (see [24]). Set ε ∈ (0 , . Since K is λ K -Lipschitz, every function ℓ z ∈ F can be approximated in L by a function ℓ z j ∈ F , for some j ∈ { , . . . , m } s.t. R | ℓ z − ℓ z i | dν ≤ ε , for any probability measure ν .Indeed, R | ℓ z − ℓ z i | dν ≤ λ K k z − z j k ∞ C K h − that is less than ε , if we cover Z by a grid of m points ( z j ) in Z s.t. k z − z j k ∞ ≤ εh/ (8 C K λ K ) := εδ . This can be done with m ≤ ε − p ⌈ Q pk =1 (cid:0) ( b k − a k ) /δ (cid:1) ⌉ = ε − p ⌈V δ − p ⌉ points. Then, with the notations of [24], L = p and D = V (8 C K λ K /h ) p . As above, this yields IP (cid:16) sup z ∈ Z n | X ≤ i = j ≤ n ξ Z ( X i , Z i , X j , Z j ) , ( X j , Z j ) (cid:1) | > t (cid:17) ≤ C D exp (cid:16) − α nh p t f Z ,max Z K (cid:17) , (15)when t ≤ h p ( R K ) f Z ,max /C K . It remains to bound IP(sup z ∈ Z | n − P ni =1 g i | > t/ . Consider the familyof functions F := { ( x , z ) ∈ R × Z 7→ h p C K E [ g z ( x , z , X , Z )] , z ∈ Z} . This family of functions is bounded is one and its variance is less than σ := h p R K f z ,max (cid:0) R | K | (cid:1) . ApplyPropositions 9 and 10 in [11] that is coming from [26]: for some universal constants A and A , some constant A g that depends on K and f z ,max (see Proposition 1 in [26]) and for every x > , IP (cid:16) sup z ∈ Z h p C K | n X i =1 E [ g z ( X i , Z i , X , Z ) | X i , Z i ] | > A (cid:0) x + A g n / σ ln(1 /σ ) (cid:1)(cid:17) ≤ (cid:16) exp (cid:0) − A x nσ (cid:1) + exp( − A x ) (cid:17) , or IP (cid:16) sup z ∈ Z n | n X i =1 g i | > A C K (cid:0) x − A g σn / h p ln( σ ) (cid:1)(cid:17) ≤ (cid:0) − A nh p x σ (cid:1) + 2 exp( − A nh p x ) . For any positive t s.t. A C K ( n − A g σ ln(1 /σ ) < n / h p t/ , note that we can find a real x > th p / (16 C K A ) .Then, we have IP (cid:16) sup z ∈ Z ( n − n | n X i =1 g i | > t (cid:17) ≤ (cid:0) − A nh p t C − K A R K f z ,max ( R | K | ) (cid:1) + 2 exp( − A nh p t C K A ) . (16)Therefore, for such t , we obtain from (16) and (15) that IP (cid:16) sup z ∈ Z | X ≤ i = j ≤ n (cid:0) S i,j ( z ) − E [ S i,j ( z )] (cid:1) | ≥ t (cid:17) ≤ C D exp (cid:18) − α nh p t R K ) f Z ,max (cid:19) + 2 exp (cid:0) − A nh p t C − K A R K f z ,max ( R | K | ) (cid:1) + 2 exp( − A nh p t C K A ) . for sufficiently large integers n . The latter inequality, (13) and (14) yield the exponential upper bound. (cid:3) imsart-generic ver. 2014/10/16 file: mainV2.tex date: March 8, 2019 . Derumigny and J.-D. Fermanian/On kernel-based estimation of conditional Kendall’s tau A.6. Proof of Proposition 7 Note that τ , | Z = z = E (cid:2) g k ( X , X ) (cid:12)(cid:12) Z = z , Z = z (cid:3) for every k = 1 , , , and that our estimators with theweights (2) can be written as ˆ τ ( k )1 , | Z = z := U n ( g k ) / { U n (1) + ǫ n } , where U n ( g ) := 1 n ( n − X ≤ i = j ≤ n g ( X i , X j ) K h ( z − Z i ) K h ( z − Z j ) E [ K h ( z − Z )] =: 1 n ( n − X ≤ i = j ≤ n g i,j , for any measurable bounded function g , with the residual diagonal term ǫ n := P ni =1 K h ( z − Z i ) / { n ( n − E [ K h ( z − Z )] } . By Bochner’s lemma (see Bosq and Lecoutre [32]), ǫ n is O P (( nh p ) − ) , and it will benegligible compared to U n (1) . Since the reasoning will be exactly the same for every estimator τ ( k )1 , | z , i.e. forevery function g k , k = 1 , , , we omit the sub-index k . Then, the functions g k will be simply denoted by g .The expectation of our U-statistics is E (cid:2) U n ( g ) (cid:3) := E (cid:2) g ( X , X ) K h ( z − Z ) K h ( z − Z ) (cid:3) / E [ K h ( z − Z )] = Z g ( x , x ) K ( t ) K ( t ) f X , Z ( x , z + h t ) f X , Z ( x , z + h t ) d x d x d t d t / E [ K h ( z − Z )] → f Z ( z ) Z g ( x , x ) f X , Z ( x , z ) f X , Z ( x , z ) d x d x = E (cid:2) g ( X , X ) (cid:12)(cid:12) Z = z , Z = z (cid:3) , applying Bochner’s lemma to z R g ( x , x ) f X | Z = z ( x ) f X | Z = z ( x ) d x d x = τ , | Z = z , that is a continuousfunction by assumption.Set θ n := E [ U n ( g )] , g ∗ ( x , x ) := ( g ( x , x ) + g ( x , x )) / and g ∗ i,j = ( g i,j + g j,i ) / for every ( i, j ) , i = j . Note that U n ( g ) = U n ( g ∗ ) . Since g ∗ is symmetrical, the Hájek projection ˆ U n ( g ∗ ) of U n ( g ∗ ) satisfies ˆ U n ( g ∗ ) := 2 P nj =1 E [ g ∗ ,j | X j , Z j ] /n − θ n . Note that E [ ˆ U n ( g ∗ )] = θ n = τ , | Z = z + o P (1) . Since V ar ( ˆ U n ( g ∗ ) =4 V ar ( E [ g ∗ ,j | X j , Z j ]) /n = O (( nh p ) − ) , then ˆ U n ( g ∗ ) = θ n + o P (1) = τ , | Z = z + o P (1) .Moreover, using the notation g i,j := g ∗ i,j − E [ g ∗ i,j | X j , Z j ] − E [ g ∗ i,j | X i , Z i ] + θ n for ≤ i = j ≤ n , we have U n ( g ∗ ) − ˆ U n ( g ∗ ) = P ≤ i = j ≤ n g i,j /n ( n − . By usual U-statistics calculations, it can be easily checked that V ar (cid:0) U n ( g ∗ ) − ˆ U n ( g ∗ ) (cid:1) = 1 n ( n − X ≤ i = j ≤ n X ≤ i = j ≤ n E [ g i ,j g i ,j ] = O (cid:0) n h p (cid:1) . Indeed, when all indices ( i , i , j , j ) are different, or when there is a single identity among them, E [ g i ,j g i ,j ] is zero. The first nonzero terms arise when there are two identities among the indices, i.e. i = i and j = j (or i = j and j = i ). In the latter case, we get an upper bound as O (( nh p ) − ) when f Z is continuousat z , by usual changes of variable techniques and Bochner’s Lemma. Then, U n ( g ∗ ) = ˆ U n ( g ∗ ) + o P (1) = τ , | Z = z + o P (1) . Note that U n (1) + ǫ n tends to one in probability (Bochner’s lemma). As a consequence, ˆ τ , | Z = z = U n ( g ∗ ) / ( U n (1) + ǫ n ) tends to τ , | Z = z / by the continuous mapping theorem. (cid:3) A.7. Proof of Proposition 8 Let us note that τ , | Z = z = E (cid:2) g k ( X , X ) (cid:12)(cid:12) Z = z , Z = z (cid:3) = Z g k ( x , x ) f X | Z = z ( x ) f X | Z = z ( x ) d x d x = φ k ( z ) f Z ( z ) , imsart-generic ver. 2014/10/16 file: mainV2.tex date: March 8, 2019 . Derumigny and J.-D. Fermanian/On kernel-based estimation of conditional Kendall’s tau where φ k ( z ) := R g k ( x , x ) f X , Z ( x , z ) f X , Z ( x , z ) d x d x . Also write ˆ τ ( k )1 , | Z = z = ˆ φ k ( z ) / ˆ f Z ( z ) , where ˆ φ k ( z ) := n − P ni,j =1 K h ( Z i − z ) K h ( Z j − z ) g k ( X i , X j ) and ˆ f Z ( z ) := n − P ni =1 K h ( Z i − z ) . Therefore, we have ˆ τ ( k )1 , | Z = z − τ , | Z = z = ˆ φ k ( z ) − φ k ( z )ˆ f Z ( z ) − τ , | Z = z ˆ f Z ( z ) − f Z ( z )ˆ f Z ( z ) × (cid:0) ˆ f Z ( z ) + f Z ( z ) (cid:1) . By usual uniform consistency results (see for example Bosq and Lecoutre [32]), sup z ∈ Z (cid:12)(cid:12) ˆ f Z ( z ) − f Z ( z ) (cid:12)(cid:12) → almost surely, as n → ∞ . We deduce that min z ∈ Z ˆ f Z ( z ) ≥ f Z , min / , and max z ∈ Z | ˆ f Z ( z ) + f Z ( z ) | ≤ z ∈ Z f Z ( z ) a.s.This means it is sufficient to prove the uniform strong consistency of ˆ φ k on Z , to obtain that sup z ∈ Z (cid:12)(cid:12) ˆ τ ( k )1 , | Z = z − τ ( k )1 , | Z = z (cid:12)(cid:12) tends to zero a.s.Note that, by Bochner’s Lemma, sup z ∈ Z (cid:12)(cid:12) E [ ˆ φ k ( z )] − φ k ( z ) (cid:12)(cid:12) → . Then, it remains to show that sup z ∈ Z (cid:12)(cid:12) ˆ φ k ( z ) − E [ ˆ φ k ( z )] (cid:12)(cid:12) → almost surely. Let ρ n > be such that we cover Z by the union of l n open balls B ( t l , ρ n ) ,where t , . . . , t l n ∈ R p and l n ∈ N ∗ . Then sup z ∈ Z (cid:12)(cid:12) ˆ φ k ( z ) − E [ ˆ φ k ( z )] (cid:12)(cid:12) ≤ sup l =1 ,...l n (cid:12)(cid:12) ˆ φ k ( t l ) − E [ ˆ φ k ( t l )] (cid:12)(cid:12) + A n , where A n := sup l =1 ,...l n sup z ∈ B ( t l ,ρ n ) (cid:12)(cid:12) ˆ φ k ( z ) − ˆ φ k ( t l ) − ( E [ ˆ φ k ( z )] − E [ ˆ φ k ( t l )]) (cid:12)(cid:12) . For any index l ∈ { , . . . , l n } and any z ∈ B ( t l , ρ n ) , a first-order expansion yields (cid:12)(cid:12) ˆ φ k ( z ) − ˆ φ k ( t l ) − ( E [ ˆ φ k ( z )] − E [ ˆ φ k ( t l )]) (cid:12)(cid:12) = (cid:12)(cid:12)(cid:12)(cid:12) n ( n − X ≤ i = j ≤ n g k ( X i , X j ) K h ( z − Z i ) K h ( z − Z j ) − n ( n − X ≤ i = j ≤ n g k ( X i , X j ) K h ( t l − Z i ) K h ( t l − Z j ) − (cid:16) E (cid:2) g k ( X , X ) K h ( z − Z ) K h ( z − Z ) (cid:3) − E (cid:2) g k ( X i , X j ) K h ( t l − Z i ) K h ( t l − Z j ) (cid:3)(cid:17)(cid:12)(cid:12)(cid:12)(cid:12) ≤ C Lip,K h p +1 | z − t l | (cid:16) E (cid:2) | g k ( X , X ) | (cid:3) + 1 n ( n − X ≤ i = j ≤ n | g k ( X i , X j ) | (cid:17) = O ( ρ n h p +1 ) = o (1) , for some constant C Lip,K and by choosing ρ n = o ( h p +1 n ) . Actually, we can cover Z in such a way that l n = O ( h − p (2 p +1) n ) . This is always possible because Z is a bounded set in R p . The previous upper bound isuniform w.r.t. l and z ∈ B ( t l , ρ n ) , proving A n = o (1) everywhere.Now, for every l = ≤ l n , apply Equation (8) for every z = t l . For any t > , this yields IP (cid:18) n ( n − (cid:12)(cid:12)(cid:12) X i = j g ( l ) (cid:0) ( X i , Z i ) , ( X j , Z j ) (cid:1) − E h g ( l ) (cid:0) ( X , Z ) , ( X , Z ) (cid:1)i(cid:12)(cid:12)(cid:12) > t (cid:19) ≤ exp (cid:16) − C nh pn t C + C t (cid:17) , for some positive constants C , C , C , by setting g ( l ) (cid:0) ( X i , Z i ) , ( X j , Z j ) (cid:1) := g k ( X i , X j ) K h ( t l − Z i ) K h ( t l − Z j ) . imsart-generic ver. 2014/10/16 file: mainV2.tex date: March 8, 2019 . Derumigny and J.-D. Fermanian/On kernel-based estimation of conditional Kendall’s tau Therefore, we deduce IP sup l =1 ,...l n (cid:12)(cid:12) ˆ φ k ( t l ) − E [ ˆ φ k ( t l )] (cid:12)(cid:12) ≥ t ! ≤ C h − p (2 p +1) n exp (cid:16) − C nh pn t C + C t (cid:17) , for some constant C . Finally, applying Borel-Cantelli lemma, sup z ∈ Z (cid:12)(cid:12) ˆ φ k ( z ) − E [ ˆ φ k ( z )] (cid:12)(cid:12) tends to zero a.s.,proving the result. (cid:3) A.8. Proof of Proposition 9 By Markov’s inequality, P ni =1 w i,n ( z ) = O P (( nh p ) − ) for any z , that tends to zero. Then, by Slutsky’stheorem, we get an asymptotic equivalence between the limiting laws of any ˆ τ ( k )1 , | z , k = 1 , , , and of theirlinearly transformed versions ˜ τ , | z . Thus, we will prove the asymptotic normality of ˆ τ ( k )1 , | z for some index k = 1 , , , simply denoted by ˆ τ , | z .Let g ∗ ( x , x ) := ( g k ( x , x ) + g k ( x , x )) / for some index k = 1 , , (that will be implicit in the proof).We now study the joint behavior of (ˆ τ , | Z = z ′ i − τ , | Z = z ′ i ) i =1 ,...,n ′ . We will extend Stute [33]’s approach, inthe case of multivariate conditioning variable z and studying the joint distribution of U-statistics at severalconditioning points. As in the proof of Proposition 7, the estimator with the weights given by (2) can berewritten as ˆ τ , | Z = z ′ i := U n,i ( g ∗ ) / ( U n,i (1) + ǫ n,i ) , where U n,i ( g ) := 1 n ( n − E [ K h ( z ′ i − Z )] n X j ,j =1 ,j = j g ( X j , X j ) K h ( z ′ i − Z j ) K h ( z ′ i − Z j ) , for any bounded measurable function g : R → R . Moreover, sup i =1 ,...,n ′ | ǫ n,i | = O P ( n − h − p ) . By a limitedexpansion of f X , Z w.r.t. its second argument, and under Assumption 3.4, we easily check that E (cid:2) U n,i ( g ) (cid:3) = τ , | Z = z ′ i + r n,i , where | r n,i | ≤ C h αn /f Z ( z ′ i ) , for some constant C that is independent of i .Now, we prove the joint asymptotic normality of (cid:0) U n,i ( g ) (cid:1) i =1 ,...,n ′ . The Hájek projection ˆ U n,i ( g ) of U n,i ( g ) satisfies ˆ U n,i ( g ) := 2 P nj =1 g n,i (cid:0) X j , Z j (cid:1) /n − θ n , where θ n := E (cid:2) U n,i ( g ) (cid:3) and g n,i ( x , z ) := K h ( z ′ i − z ) E (cid:2) g ( X , x ) K h ( z ′ i − Z ) (cid:3) / E [ K h ( z ′ i − Z )] . Lemma 17. Under the assumptions of Proposition 9, for any measurable bounded function g , ( nh p ) / (cid:16) ˆ U n,i ( g ) − E (cid:2) U n,i ( g ) (cid:3)(cid:17) i =1 ,...,n ′ D −→ N (0 , M ∞ ( g )) , as n → ∞ , where, for ≤ i, j ≤ n ′ , [ M ∞ ( g )] i,j := 4 R K { z ′ i = z ′ j } f Z ( z ′ i ) Z g (cid:0) x , x ) g (cid:0) x , x ) f X | Z = z ′ i ( x ) f X | Z = z ′ i ( x ) f X | Z = z ′ i ( x ) d x d x d x . This lemma is proved in A.9. Similarly as in the proof of Lemma 2.2 in Stute [33], for every i = 1 , . . . , n ′ and every bounded symmetrical measurable function g , we have ( nh p ) / V ar (cid:2) ˆ U n,i ( g ) − U n,i ( g ) (cid:3) = o (1) , whichimplies ( nh p ) / (cid:16) U n,i ( g ) − E (cid:2) U n,i ( g ) (cid:3)(cid:17) i =1 ,...,n ′ D −→ N (0 , M ∞ ( g )) , as n → ∞ . imsart-generic ver. 2014/10/16 file: mainV2.tex date: March 8, 2019 . Derumigny and J.-D. Fermanian/On kernel-based estimation of conditional Kendall’s tau Considering two measurable bounded functions g and g , we have U n,i ( c g + c g ) = c U n,i ( g ) + c U n,i ( g ) for every numbers c , c . By the Cramér-Wold device, we check that ( nh p ) / (cid:18)(cid:16) U n,i ( g ) − E (cid:2) U n,i ( g ) (cid:3)(cid:17) i =1 ,...,n ′ , (cid:16) U n,i ( g ) − E (cid:2) U n,i ( g ) (cid:3)(cid:17) i =1 ,...,n ′ (cid:19) D −→ N , " M ∞ ( g ) M ∞ ( g , g ) M ∞ ( g , g ) M ∞ ( g ) , as n → ∞ , where [ M ∞ ( g , g )] i,j := 4 R K { z ′ i = z ′ j } f Z ( z ′ i ) Z g (cid:0) x , x ) g (cid:0) x , x ) f X | Z = z ′ i ( x ) f X | Z = z ′ i ( x ) f X | Z = z ′ i ( x ) d x d x d x . Set ˜ τ , | Z = z ′ i := U n,i ( g ∗ ) / U n,i (1) . Since ( nh pn ) / (cid:0) ˆ τ , | Z = z ′ i − ˜ τ , | Z = z ′ i (cid:1) = O P (cid:0) ( nh pn ) / ǫ n,i (cid:1) is o P (1) , it issufficient to establish the asymptotic law of ( nh pn ) / (cid:0) ˜ τ , | Z = z ′ i − τ , | Z = z ′ i (cid:1) . Since E [ U n,i (1)] = 1+ o (( nh p ) − / ) and E [ U n,i ( g ∗ )] = τ , | Z = z ′ i + o (( nh pn ) − / ) , we get ( nh p ) / (cid:18)(cid:16) U n,i ( g ∗ ) − τ , | Z = z ′ i (cid:17) i =1 ,...,n ′ , (cid:16) U n,i (1) − (cid:17) i =1 ,...,n ′ (cid:19) D −→ N , " M ∞ ( g ∗ ) M ∞ ( g ∗ , M ∞ ( g ∗ , M ∞ (1) , as n → ∞ . Now apply the Delta-method with the function ρ ( x , y ) := x / y where x and y are real-valued vectors of size n ′ and the division has to be understood component-wise. The Jacobian of ρ is given by the n ′ × n ′ matrix J ρ ( x , y ) = h Diag (cid:0) y − , . . . y − n ′ (cid:1) , Diag (cid:0) − x y − , · · · − x n ′ y − n ′ (cid:1)i , where, for any vector v of size n ′ , Diag ( v ) is the diagonal matrix whose diagonal elements are the v i , with i = 1 , . . . , n ′ . We deduce ( nh p ) / (cid:0) ˜ τ , | Z = z ′ i − τ , | Z = z ′ i (cid:1) i =1 ,...,n ′ D −→ N (0 , H ) , as n → ∞ , setting H := J ρ ( ~τ , e ) " M ∞ ( g ∗ ) M ∞ ( g ∗ , M ∞ ( g ∗ , M ∞ (1) J ρ ( ~τ , e ) T , where ~τ = (cid:0) τ , | Z = z ′ i (cid:1) i =1 ,...,n ′ and e is the vector of size n ′ whose all components are equal to . Thus, we have J ρ ( ~τ , e ) = h Id n ′ , − Diag ( ~τ ) i , denoting by Id n ′ the identity matrix of size n ′ and by Diag ( ~τ ) the diagonalmatrix of size n ′ whose diagonal elements are the τ , | z ′ i , for i = 1 , . . . , n ′ . To be specific, we get H = M ∞ ( g ∗ ) − Diag ( ~τ ) M ∞ ( g ∗ , − M ∞ ( g ∗ , Diag ( ~τ ) + Diag ( ~τ ) M ∞ (1) Diag ( ~τ ) . For i, j in { , . . . , n ′ } and using the symmetry of the function g ∗ , we obtain [ M ∞ ( g ∗ )] i,j = 4 Z K { z ′ i = z ′ j } E [ g ∗ ( X , X ) g ∗ ( X , X ) | Z = Z = Z = z ′ i ] /f Z ( z ′ i ) , [ Diag ( ~τ ) M ∞ ( g ∗ , i,j = 4 τ , | Z = z ′ i Z K { z ′ i = z ′ j } E [ g ∗ ( X , X ) | Z = Z = z ′ i ] /f Z ( z ′ i )= 4 Z K { z ′ i = z ′ j } τ , | Z = z ′ i /f Z ( z ′ i ) = [ M ∞ ( g ∗ , Diag ( ~τ )] i,j = [ Diag ( ~τ ) M ∞ (1) Diag ( ~τ )] i,j . As a consequence, we obtain [ H ] i,j = 4 R K { z ′ i = z ′ j } f Z ( z ′ i ) (cid:16) E [ g ∗ ( X , X ) g ∗ ( X , X ) | Z = Z = Z = z ′ i ] − τ , | Z = z ′ i (cid:17) . (cid:3) imsart-generic ver. 2014/10/16 file: mainV2.tex date: March 8, 2019 . Derumigny and J.-D. Fermanian/On kernel-based estimation of conditional Kendall’s tau A.9. Proof of Lemma 17 Let us first evaluate the variance-covariance matrix M n,n ′ := [ Cov ( ˆ U n,i , ˆ U n,j )] ≤ i,j ≤ n ′ . Note that E (cid:2) g n,i ( X j , Z j ) (cid:3) = E (cid:2) ˆ U n,i (cid:3) = E (cid:2) U n,i ( g ) (cid:3) , and that (cid:16) ( nh p ) / (cid:0) ˆ U n,i − E [ U n,i ( g )] (cid:1)(cid:17) i =1 ,...,n ′ = 2 h p/ n / n X j =1 (cid:0) g n,i ( X j , Z j ) − E [ U n,i ( g )] (cid:1) i =1 ,...,n ′ , that is a sum of independent vectors. Thus, Cov ( ˆ U n,i , ˆ U n,j ) = 4 n − Cov (cid:0) g n,i (cid:0) X , Z (cid:1) , g n,j (cid:0) X , Z (cid:1)(cid:17) , for every i, j in { , . . . , n ′ } , and E (cid:2) g n,i ( X , Z ) g n,j ( X , Z ) (cid:3) = Z K h ( z ′ i − z ) K h ( z ′ j − z ) E (cid:2) g ( X , x ) K h ( z ′ i − Z ) (cid:3) E (cid:2) g ( X , x ) K h ( z ′ j − Z ) (cid:3) E [ K h ( z ′ i − Z )] E [ K h ( z ′ j − Z )] f X , Z ( x , z ) d x d z ∼ h p f Z ( z ′ i ) f Z ( z ′ j ) Z g (cid:0) x , x ) g (cid:0) x , x ) K h ( z ′ i − z ) K h ( z ′ j − z ) K h ( z ′ i − w ) K h ( z ′ j − w ) × f X , Z ( x , z ) f X , Z ( x , w ) f X , Z ( x , w ) d x d z d x d w d x d w ∼ h p f Z ( z ′ i ) f Z ( z ′ j ) Z g (cid:0) x , x ) g (cid:0) x , x ) K ( u ) K ( u ) K ( u ) K ( z ′ j − z ′ i h + u ) f X , Z ( x , z ′ i − h u ) × f X , Z ( x , z ′ i − h u ) f X , Z ( x , z ′ j − h u ) d x d u d x d u d x d u . If i = j and K is compactly supported, the latter term is zero when n is sufficiently large, and Cov ( ˆ U n,i , ˆ U n,j ) = − n − E [ U n,i ] E [ U n,j ] ∼ − n − τ , | Z = z ′ i τ , | Z = z ′ j = o (cid:0) ( nh p ) − (cid:1) . Otherwise, i = j and, as E h g n,i (cid:0) X , Z (cid:1)i = O (1) , we have V ar (cid:16)(cid:0) g n,i ( X , Z ) (cid:1) (cid:17) ∼ h p f Z ( z ′ i ) Z g (cid:0) x , x ) g (cid:0) x , x ) K ( u ) K ( u ) K ( u ) f X , Z ( x , z ′ i − h u ) × f X , Z ( x , z ′ i − h u ) f X , Z ( x , z ′ i − h u ) d x d u d x d u d x d u ∼ R K h p f Z ( z ′ i ) Z g (cid:0) x , x ) g (cid:0) x , x ) f X | Z = z ′ i ( x ) f X | Z = z ′ i ( x ) f X | Z = z ′ i ( x ) d x d x d x , by Bochner’s lemma. We have proved that, for every i, j ∈ { , . . . , n ′ } , nh p [ M n,n ′ ] i,j → R K { z ′ i = z ′ j } f Z ( z ′ i ) Z g (cid:0) x , x ) g (cid:0) x , x ) f X | Z = z ′ i ( x ) f X | Z = z ′ i ( x ) f X | Z = z ′ i ( x ) d x d x d x , as n → ∞ . Therefore, nh p M n,n ′ tends to M ∞ .We now verify Lyapunov’s condition with third-order moments, so that the usual multivariate central limittheorem would apply. It is then sufficient to show that (cid:16) h p/ n / (cid:17) n X j =1 E h(cid:12)(cid:12) g n,i ( X j , Z j ) − E [ U n,i ( g )] (cid:12)(cid:12) i = o (1) . (17)For any j = 1 , . . . , n , we have E h(cid:12)(cid:12) g n,i ( X j , Z j ) − E [ U n,i ( g )] (cid:12)(cid:12) i ∼ Z (cid:12)(cid:12)(cid:12) f Z ( z ′ i ) Z g ( x , x ) K h ( z ′ i − z ) K h ( z ′ i − z ) f X , Z ( x , z ) d x d z − E (cid:2) U n,i ( g ) (cid:3)(cid:12)(cid:12)(cid:12) f X , Z ( x , z ) d x d z . imsart-generic ver. 2014/10/16 file: mainV2.tex date: March 8, 2019 . Derumigny and J.-D. Fermanian/On kernel-based estimation of conditional Kendall’s tau By the change of variable z = z ′ i − h t and z = z ′ i − h t , we get E h(cid:12)(cid:12) g n,i ( X j , Z j ) − E [ U n,i ( g )] (cid:12)(cid:12) i ∼ h − p Z (cid:12)(cid:12)(cid:12) f Z ( z ′ i ) Z g ( x , x ) K ( t ) K ( t ) f X , Z ( x , z ′ i − h t ) d x d t − h p E (cid:2) U n,i ( g ) (cid:3)(cid:12)(cid:12)(cid:12) f X , Z ( x , z ′ i − h t ) d x d t = O ( h − p ) , because of Bochner’s lemma, under our assumptions. Therefore, we have obtained (cid:16) h p/ n / (cid:17) n X j =1 E h(cid:12)(cid:12) g n,i ( X j , Z j ) − E [ U n,i ( g )] (cid:12)(cid:12) i = O ( h p/ n − / nh − p ) = O (( nh p ) − / ) = o (1) . Therefore, we have checked Lyapunov’s condition and the result follows. (cid:3)(cid:3)