On moment indeterminacy of the Benini income distribution
OON MOMENT INDETERMINACY OF THE BENINI INCOME DISTRIBUTION
Christian Kleiber
The Benini distribution is a lognormal-like distribution generalizingthe Pareto distribution. Like the Pareto and the lognormal distributionsit was originally proposed for modeling economic size distributions, no-tably the size distribution of personal income. This paper explores aprobabilistic property of the Benini distribution, showing that it is notdetermined by the sequence of its moments although all the momentsare finite. It also provides explicit examples of distributions possessingthe same set of moments. Related distributions are briefly explored.
Keywords:
Benini distribution, characterization of distributions, in-come distribution, moment problem, statistical distributions, Stieltjesclass.
AMS 2010 Mathematics Subject Classification:
Primary 60E05,Secondary 62E10, 44A60.
JEL Classification:
C46, C02. INTRODUCTION
In the late 19th century, the eminent Italian economist Vilfredo Pareto observed thatempirical income distributions are well described by a straight line on a doubly logarith-mic plot (Pareto, 1895, 1896, 1897). Specifically, with F = 1 − F denoting the survivalfunction of an income distribution with c.d.f. F , Pareto observed that, to a good degree ofapproximation,ln F ( x ) = a − a ln x. (1.1)The distribution implied by this equation is called the Pareto distribution.Not much later, the Italian statistician and demographer Rodolfo Benini found that asecond-order polynomialln F ( x ) = a − a ln x − a (ln x ) (1.2)sometimes provides a markedly better fit (Benini, 1905, 1906). The distribution implied bythis equation is called the Benini distribution.The present paper is concerned with a probabilistic property of the Benini distribution,namely whether it is possible to characterize this distribution in terms of its moments. The Version: April 29, 2013. Correspondence to: Christian Kleiber, Faculty of Business and Economics,Universit¨at Basel, Peter Merian-Weg 6, 4002 Basel, Switzerland. [email protected] a r X i v : . [ s t a t . O T ] J un C. KLEIBER moment problem asks, for a given distribution F with finite moments µ k ≡ E [ X k ] = (cid:82) ∞−∞ x k d F ( x ) of all orders k = 1 , , . . . , whether or not F is uniquely determined by thesequence of its moments. See, for example, Shohat and Tamarkin (1950) for analytical orStoyanov (2013, Sec. 11) for probabilistic aspects of the moment problem. If a distributionis uniquely determined by the sequence of its moments it is called moment-determinate,otherwise it is called moment-indeterminate. Cases where the support of the distribution isthe positive half-axis R + = [0 , ∞ ) or an unbounded subset thereof are called Stieltjes-typemoment problems. The Benini distribution thus poses a Stieltjes-type moment problem. Itis shown below that the Benini moment problem is indeterminate. Drawing on a classicalexample going back to Stieltjes (1894/1895) explicit examples of distributions possessingthe same set of moments are constructed. Certain generalizations of the Benini distributionare briefly explored, all of which are moment-indeterminate.2. THE BENINI DISTRIBUTION
Pareto’s observation (1.1) leads to a distribution of the form F ( x ) = 1 − (cid:16) xσ (cid:17) − α , x ≥ σ > , where α >
0. Benini’s observation (1.2) leads to a distribution of the form F ( x ) = 1 − exp (cid:26) − α ln xσ − β (cid:16) ln xσ (cid:17) (cid:27) , x ≥ σ > , (2.1)where α, β ≥
0, with ( α, β ) (cid:54) = (0 , β = 0 gives the Pareto distribution.For parsimony, Benini (1905) often worked with the special case where α = 0, i.e. with F ( x ) = 1 − exp (cid:26) − β (cid:16) ln xσ (cid:17) (cid:27) (2.2)= 1 − (cid:16) xσ (cid:17) − β (ln x − ln σ ) , x ≥ σ > . Here σ > β > β, σ ). For the purposes of the present paper the scale parameter σ is immaterial. Theobject under study is, therefore, the Ben( β, ≡ Ben( β ) distribution with F ( x ) = 1 − exp {− β (ln x ) } , x ≥ . (2.3)It may be worth noting that the Benini distributions are stochastically ordered w.r.t. β .Specifically, it follows directly from (2.3) that F ( x ; β ) ≤ F ( x ; β ) for all x ≥ ⇐⇒ β ≤ β , (2.4) OMENT INDETERMINACY OF THE BENINI DISTRIBUTION F ( x ; β ) is larger than F ( x ; β ) under this condition in the sense of the usual stochas-tic order, often called first-order stochastic dominance in economics.Noting further that the c.d.f. of a Weibull distribution is F ( x ) = 1 − exp( − x a ), x > a >
0, it follows that eq. (2.3) describes a log-Weibull distribution with a = 2. The Weibulldistribution with a = 2 is also known (up to scale) as the Rayleigh distribution, especiallyin physics, and so the Benini distribution may be seen as the log-Rayleigh distribution. Itmay also be seen as a log-chi distribution with two degrees of freedom (again up to scale);i.e., the logarithm of a Benini random variable follows the distribution of the square rootof a chi-square random variable with two degrees of freedom.The density implied by (2.3) is f ( x ) = 2 β ln xx exp (cid:8) − β (ln x ) (cid:9) , x ≥ , (2.5)and hence is similar to the density of the more familiar lognormal distribution. The log-normal distribution is perhaps the most widely known example of a distribution that isnot determined by its moments, although all its moments are finite (Heyde, 1963). Thesimilarity of the lognormal and the Benini densities now suggests that the Benini distribu-tion might also possess this somewhat pathological property. The remainder of the presentpaper explores this issue.Figure 1 depicts some two-parameter Benini densities, showing that distributions withsmaller values of β are associated with heavier tails, as indicated by (2.4).From a modeling point of view, the significance of the Benini distribution lies in thefact that it generalizes the Pareto distribution while itself being ‘lognormal-like’. It thusenables to discriminate between these two widely used distributions, at least approximately.Further details on the Benini distribution, including an independent rediscovery in actuarialscience motivated by failure rate considerations (Shpilberg, 1977), may be found in Kleiberand Kotz (2003, Ch. 7.1). The appendix of Kleiber and Kotz (2003) also provides a briefbiography of Rodolfo Benini.3. THE BENINI DISTRIBUTION AND THE MOMENT PROBLEM
The following proposition provides two basic properties of the Benini distribution thatare relevant in the context of the moment problem.
Proposition (a) The moments µ k , k ∈ N , of the Benini distribution Ben( β ) aregiven by µ k ≡ E [ X k ] = 1 + k (2 β ) − (1 / e k / (8 β ) D − (cid:18) − k √ β (cid:19) (3.1)= 1 + k √ π √ β e k / (4 β ) (cid:26) (cid:18) k √ β (cid:19)(cid:27) . (3.2) C. KLEIBER f ( x ) Figure 1 .— Some Benini densities; β = 2 , , . Here, D − is a parabolic cylinder function and erf denotes the error function.(b) The moment generating function (m.g.f.) of the Benini distribution does not exist.Proof. (a) We have µ k ≡ E [ X k ] = k (cid:90) ∞ x k − F ( x ) d x = 1 + k (cid:90) ∞ x k − exp {− β (ln x ) } d x = 1 + k (cid:90) ∞ e kx − βx d x = 1 + k (2 β ) − (1 / e k / (8 β ) D − (cid:18) − k √ β (cid:19) , using Gradshteyn and Ryzhik (2007), no. 3.462, eq. 1. This proves (3.1). The alternativerepresentation (3.2) is established via the relation (Olver et al., 2010, § OMENT INDETERMINACY OF THE BENINI DISTRIBUTION D − ( x ) = (cid:114) π e x / erfc (cid:18) x √ (cid:19) , where erfc( · ) is the complementary error function, together with erfc( x ) = 1 − erf( x ) anderf( − x ) = − erf( x ).(b) The defining integral is E [ e tX ] = (cid:90) ∞ e tx β ln xx exp (cid:8) − β (ln x ) (cid:9) d x =: (cid:90) ∞ h ( x ) d x. Now the leading term inln h ( x ) = tx + ln(2 β ln x ) − ln x − β (ln x ) is the linear term, hence E [ e tX ] = ∞ for all t > (cid:3) The representation (3.2) can also be obtained using
Mathematica (Wolfram Research,Inc., 2013), version 9.0.1.0.As an illustration, Table I provides the first four moments of selected Benini distributions,namely those from Figure 1. These moments are rather large, especially for small values of β . TABLE ILower-order moments of Benini distributions. E [ X ] E [ X ] E [ X ] E [ X ] β = 2 1.98 4.48 11.81 37.20 β = 1 2.73 9.88 50.59 387.19 β = 0 . Proposition 1 showed that the Benini distribution has moments of all orders, but nom.g.f. Distributions possessing these properties are candidates for moment indeterminacy,although these facts alone are not conclusive. Unfortunately, no tractable necessary andsufficient condition for moment indeterminacy is currently known.For exploring determinacy, the Carleman criterion (e.g. Stoyanov, 2013, Sec. 11) some-times provides an answer. In a Stieltjes-type problem, the condition C S := ∞ (cid:88) k =1 µ − k k = ∞ C. KLEIBER implies that the underlying distribution is characterized by its moments.However, Proposition 1 indicates that the moments of the Benini distribution grow ratherrapidly. In view of erf( x ) ≥
0, for x ≥
0, it follows from (3.2) that E [ X k ] ≥ k √ π √ β e k / (4 β ) . Using the ratio test this further implies that C S = ∞ (cid:88) k =1 µ − k k ≤ ∞ (cid:88) k =1 (cid:18) √ βk √ π (cid:19) k e − k/ (8 β ) < ∞ . (3.3)So the Carleman condition cannot establish determinacy here.This suggests to explore indeterminacy instead. Indeed, Theorem 2 shows that all Beninidistributions are moment-indeterminate. Two proofs are given, one utilizing a converseto the Carleman criterion due to Pakes (2001) and the other utilizing the Krein criterion(Stoyanov, 2000, 2013). Theorem The Benini distribution Ben ( β ) is moment-indeterminate for any β > .Proof 1. Pakes (2001, Th. 3) showed that if there exists x ≥ < f ( x ) < ∞ for x > x , the condition C S < ∞ together with the convexity of the function ψ ( x ) := − ln f ( e x ) on the interval (ln x , ∞ ) implies moment indeterminacy. C S < ∞ was shown in(3.3). For the Benini distribution, the function ψ ( x ) = − ln f ( e x ) = − ln(2 βx ) + x + βx is easily seen to be convex on the interval (0 , ∞ ) in view of β > (cid:3) Proof 2.
In the case of a Stieltjes-type moment problem, the Krein criterion requires, fora strictly positive density f and some c >
0, that the logarithmic integral K S [ f ] = (cid:90) ∞ c − ln f ( x )1 + x d x (3.4)is finite. For the Benini distribution this integral is, choosing c = e , K S [ f ] = − (cid:90) ∞ e ln(2 β ln x ) − ln x − β (ln x ) x d x. This quantity is finite for all β > (cid:3) OMENT INDETERMINACY OF THE BENINI DISTRIBUTION
A STIELTJES CLASS FOR THE BENINI DISTRIBUTION
The methods used in the proof of Theorem 2 only establish existence of further distri-butions possessing the same set of moments as the Benini distribution. It is known fromBerg and Christensen (1981) that if a distribution is moment-indeterminate, then thereexist infinitely many continuous and also infinitely many discrete distributions possessingthe same moments. It is, therefore, of interest to find explicit examples of such objects.A Stieltjes class (Stoyanov, 2004) corresponding to a moment-indeterminate distributionwith density f is a set S ( f, p ) = { f ε ( x ) | f ε ( x ) := f ( x )[1 + ε p ( x )] , x ∈ supp( f ) } , where p is a perturbation function satisfying E [ X k p ( X )] = 0 for all k = 0 , , , . . . . If − ≤ p ( x ) ≤ ε ∈ [ − , S ( f, p ) is called a two-sided Stieltjes class. Counterexamplesto moment determinacy in the literature are typically of this type. It is also possible to haveone-sided Stieltjes classes, for which p only needs to be bounded from below, and ε ≥ Theorem The distributions with densities f ε , ≤ ε ≤ , f ε ( x ) = f ( x ) (cid:26) ε x exp {− ( x − / + β (ln x ) } sin { ( x − / } C β ln x (cid:27) , x ≥ , all have the same moments as the Benini distribution Ben ( β ) with density f . Here C > is a normalizing constant defined in the proof.Proof. Consider the (unscaled) perturbation˜ p ( x ) = x exp {− ( x − / + β (ln x ) } sin { ( x − / } β ln x , x ≥ . This perturbation has the following properties:(P1). lim x → + ˜ p ( x ) = ∞ .(P2). Basic properties of the sine function imply that ˜ p ( x ) ≥ , , ∞ ), the function ˜ p is continuous, with ˜ p (2) < ∞ and lim x →∞ ˜ p ( x )= 0. Hence ˜ p ( x ) is bounded there.Let C = sup x ∈ [2 , ∞ ) | ˜ p ( x ) | and set p ( x ) = ˜ p ( x ) /C . It follows from (P1)–(P3) that p isunbounded from above and bounded from below, specifically − ≤ p ≤ ∞ . By construction, C. KLEIBER f ε ≥
0. The moments of the corresponding random variable X ε with density f ε , 0 ≤ ε ≤ E [ X kε ] = (cid:90) ∞ x k f ε ( x ) d x = (cid:90) ∞ x k f ( x ) { ε p ( x ) } d x = (cid:90) ∞ x k f ( x ) d x + εC (cid:90) ∞ x k exp {− ( x − / } sin { ( x − / } d x =: E [ X k ] + J. It remains to show that J = 0. Now (cid:90) ∞ x k exp {− ( x − / } sin { ( x − / } d x = (cid:90) ∞ ( x + 1) k exp {− x / } sin { x / } d x = k (cid:88) j =0 (cid:18) kj (cid:19) (cid:90) ∞ x k − j exp {− x / } sin { x / } d x = 0in view of (cid:90) ∞ x n exp {− x / } sin { x / } d x = 0 (4.1)for all n ∈ N . In particular, (cid:82) ∞ f ε ( x ) d x = 1. (cid:3) Note that Theorem 3 provides a further proof of the moment indeterminacy of the Beninidistribution.Apart from the shifted argument, the perturbation employed here draws on the pioneeringwork of Stieltjes (1894/1895). In modern terminology, Stieltjes showed that the relation(4.1) leads to a family of distributions whose moments coincide with those of a certaingeneralized gamma distribution, implying that the latter is moment-indeterminate.Stieltjes (1894/1895) has a further, and more widely known, example of a distributionthat is not determined by its moments, the lognormal distribution. The counterexample heprovides for that distribution employs the perturbation p ( x ) = sin(2 π ln x ) , x > , (4.2)which was further developed by Heyde (1963). It can also lead to a Stieltjes class for theBenini distribution. However, note that in view of the exponential term common to both OMENT INDETERMINACY OF THE BENINI DISTRIBUTION β , otherwise the resulting ratio diverges for x → ∞ . Methods outlined byStoyanov and Tolmatz (2005) may help to construct Stieltjes classes based on (4.2) andthe lognormal density that cover the entire range of the shape parameter β , at the price ofsomewhat greater analytical complexity.5. RELATED DISTRIBUTIONS
It is natural to augment Pareto’s equation (1.1) by higher-order terms going beyond thesecond-order term proposed by Benini (1905). Not surprisingly, curves of the formln F ( x ) = a − a ln x − a (ln x ) − . . . − a k (ln x ) k (5.1)soon began to appear in the subsequent Italian-language literature on economic statistics;see, e.g., Bresciani Turroni (1914) and Mortara (1917) for some early contributions. Some-what later, the Austrian statistician Winkler (1950) independently also experimented withpolynomials in ln x . Specifically, he fitted a quadratic—i.e., the three-parameter Beninidistribution (2.1)—to the U.S. income distribution of 1919.Dropping a scale parameter, i.e. setting a = 0, eq. (5.1) gives the c.d.f. F ( x ) = 1 − exp (cid:40) − k (cid:88) j =1 a j (ln x ) j (cid:41) , x ≥ , (5.2)where a , . . . , a k ≥
0, with corresponding density f ( x ) = exp (cid:40) − k (cid:88) j =1 a j (ln x ) j (cid:41) (cid:40) k (cid:88) j =1 ja j (ln x ) j − (cid:41) x , x ≥ . (5.3)Using the Krein criterion it is not difficult to see that these generalized Benini distri-butions are moment-indeterminate, provided ( a , . . . , a k ) (cid:54) = (0 , . . . ,
0) as otherwise not allmoments exist.A further generalization of the Benini distribution proceeds along different lines. In sec-tion 2 it was noted that the Benini distribution may be seen as the log-Rayleigh distribution,up to scale. It is then natural to consider the log-Weibull family, with c.d.f. F ( x ) = 1 − exp {− (ln x ) a } , x ≥ , where a >
0, and corresponding density f ( x ) = a (ln x ) a − x exp {− (ln x ) a } , x ≥ . C. KLEIBER
Indeed, Benini (1905, p. 231) briefly discusses this model and reports that, for his data,when a = 2 .
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