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Competing skewness orderings are surveyed. It is argued that those based on natural skewness functional are preferable to those related to convex orderings.
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Stochastic Inequalities IMS Lecture Notes - Monograph Series Volume 22 (1993)
AMS 1991 subject classifications. Primary 62E10, 60E99. Key words and phrases. Asymmetry, peakedness, Lorenz order, star order, convex order, skewness functionals.
18 Barry C. Arnold and Richard A. Groeneυeld
2. A Budget of Skewness Orderings
To simplify discussion, we restrict attention to the class C of distributions with median 0 and finite first absolute moment. The moment condition is not crucial for some of the orderings but several orderings will only be well- defined when first moments exist and it is convenient to restrict our focus to such distributions. For any distribution FeC, we define its quantile (or inverse distribution) function F " 1 by
(2.1) ^(u) = sup{x : F(x) <u}, 0 < u < 1.
The zero median condition is equivalent to
(2.2) F-\±) = 0
and the first absolute moment is expressible as
(2.3) E(\X) = / 2 [F~\l - u) - F-\u)]du.
The class of median zero distributions with finite first moment can be iden- tified conveniently with the class of all non-decreasing functions defined on (0,1) satisfying (2.2) and (2.3). A symmetric distribution can be character- ized by the requirement that
(2.4) F-\l -u) + F~\u) = 0, Vue(0, ).
Skewness corresponds to the violation of condition (2.4). Positive values of (2.4) for some, most or all values of u will be associated with positive skewness or skewness to the right. Negative skewness is associated with negative values of (2.4). We will define skewness orderings denoted by a symbol < with a variety of subscripts and will in a cavalier fashion write them in terms of random variables or distribution functions i.e. X < Y <ί=^ FX < Fγ. It is generally conceded that measures of skewness and related skewness orderings should be scale invariant, i.e. for any positive constant c, X and cX exhibit the same degree of skewness. Accepting this viewpoint it is defensible to divide any random variable by its first absolute moment and effectively focus on skewness orderings defined on the class of random variables with
(2.5) J £ J φ = 0
and (2.6) E\X\ - 1.
20 Barry C. Arnold and Richard A. Groeneυeld
A star ordering (related to Oja's (1981) ordering) is defined by
(2.14) X < 2 Y iff | p M ί on (0,1) - φ
where the symbol f is to be read here and henceforth as non-decreasing. Progressively further weakening yields the following orders.
(2.15) X < 3 Y iff
and
In the above definitions fx and fy are the densities corresponding to Fx and Fy. An ordering based on stochastic ordering of the positive and negative parts of X and Y has been proposed:
(2.18) X < 6 Y iff X+ < (^) 5 t y + and Y~ < (^) s ί X~
Instead of stochastic ordering in (2.18) we might invoke Lorenz ordering. Thus we have
(2.19) X < 7 Y iff X+ < (^) L y+ and Y" < (^) L X~.
A convenient reference for the definition of <L is Arnold (1987). Second and higher order stochastic dominance could of course be used as a basis of a skewness order definition but we will not pursue that possibility. Finally we mention the David and Johnson (1956) skewness functional related to but distinct from sχ(u) defined in (2.8). It takes the form
Analogs to (2.9) and (2.10) are available
and
(
22)
+
Skewness and Kurtosis 23
5. Skewness Accentuating Transformations
We may reasonably seek to characterize all transformations g : IR —• IR which have the property that for any XeC 0 we have X less skew than g(X). It is evidently true that if we choose a function g such that #(0) = 0 and both g{x) and g(x)/x T on IR - {0} then X <τ g(X) for any Xe£ 0. Thus transformations of this kind accentuate skewness using the strongest (Van Zwet) skewness ordering. They thus accentuate skewness using any of the other orderings implied by the Van Zwet orderings. These transformations however do not necessarily accentuate skewness as measured by < 6.
6. Kurtosis
An analogous variety of kurtosis orderings exist. Most restrict attention to symmetric distributions. A reason for this is the difficulty of interpretation of the concept of kurtosis in the absence of symmetry. Setting aside such niceties for the moment, it is possible to provide kurtosis orderings analogous to several of the skewness orderings described in this paper. Some of these reduce to already known kurtosis orderings if symmetry is imposed. As in our skewness discussion we standardize all variables to have median 0 and first absolute moment equal to 1; i.e. we restrict attention to £ 0 To distinguish our kurtosis ordering from the corresponding parallel skewness orderings we place a superscript k above the inequality sign. Thus <ί> will be a kurtosis ordering analogous to the skewness ordering <2 Here is the list (as usual X = X+ - X~ where X+ > 0 and X" > 0).
X <\Y iff Fγl(Fχ+(x)) is convex on the
(6.1) is convex on the support of X~~.
X<k 2 Y iff Fyl (u)/F~l (tt) ΐ on (0,1)
(6.3) X<k 7 Y iff X+ <L Y+ and X~ <L Y
X<\Y iff £
24 Barry C. Arnold and Richard A. Groeneveld
In the case in which X and Y are symmetric random variables, this last ordering is equivalent to \X\ <L \Y. In the absence of symmetry, the Lorenz order of absolute values may be considered to be candidate variant kurtosis order. We may define
(6.5) X<k 8 Y iff \X<L\Y.
This ordering has an attractive simplicity. It certainly captures some of the idea of kurtosis when the random variables are symmetric. Interpretation in the asymmetric case is potentially more problematic. It is not difficult to construct an asymmetric example in which X <\Y but I ^ F and an example in which X <_ but X j£* Y. One advantage of the absolute Lorenz ordering (<g) is its potential for straightforward extension to higher dimensions. For m dimensional random vectors X and Y centered to have medians 0, we can define X <_ Y if and only if d(X,Q) < L d(Y, 0) where d is a metric in IRm^. More details on these kurtosis orderings and related summary measures of kurtosis will appear in a separate report.
REFERENCES
ARNOLD, B. C. (1987) Majoήzation and the Lorenz Order: A Bήef Introduction. Lecture Notes in Statistics 43, Springer-Verlag, Berlin. BALANDA, K. P. AND MACGILLIVRAY, H. L. (1988) Kurtosis: a critical review. Amer. Statist 42 111-119. DAVID, F. N. AND JOHNSON, N. L. (1956) Some tests of significance with ordered variables. J. Royal Stat. Soc. B18 1-20. MACGILLIVRAY, H. L. (1986) Skewness and asymmetry: measures and orderings. Ann. Statist. 14 994-1011. OJA, H. (1981) On location, scale, skewness and kurtosis of univariate distributions. Scand. J. Statist. 8 154-168. VAN ZWET, W. R. (1964) Convex Transformations of Random Variables. Mathe- matisch Centrum, Amsterdam.
DEPARTMENT OF STATISTICS DEPARTMENT OF STATISTICS UNIVERSITY OF CALIFORNIA, RIVERSIDE IOWA STATE UNIVERSITY RIVERSIDE, CA 92502 AMES, IA 50010