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Stochastic Inequalities IMS Lecture Notes - Monograph Series Volume 22 (1993)
SKEWNESS AND KURTOSIS ORDERINGS:
AN INTRODUCTION
By BARRY C. ARNOLD and RICHARD A. GROENEVELD
University of California, Riverside and Iowa State University
Competing skewness orderings are surveyed. It is argued that those
based on natural skewness functional are preferable to those related to
convex orderings. Analogous kurtosis orderings are also discussed. Here
the role of convex and Lorenz orderings appears more natural.
1. Introduction
What is skewness? An analogous question regarding inequality led Dal-
ton eventually down the majorization path via the enunciation of clearly
agreed upon inequality reducing transformations. Can a similar analysis be
performed with skewness? In a sense the answer is easy; skewness is asym-
metry, plain and simple. It is of course easy to recognize symmetric distri-
butions but not so easy to decide whether one non symmetric distribution is
more unsymmetric than another. Robin Hood (i.e. rich to poor) transfers are
at the heart of the accepted inequality orderings. It is natural to search for
analogous basic operations which will increase skewness. The present paper
surveys suggested skewness orderings (although not in the detail provided
by MacGillivray (1986)) but puts its major focus on promoting a particular
group of skewness orderings. Clear parallels may be discerned between some
of these orderings and the Lorenz inequality ordering generated by Robin
Hood operations.
What is Kurtosis? This is a bit harder. To quote our dictionary (Web-
ster's of course) it is the state or quality of peakedness or flatness of the
graphic representation of a statistical distribution. Again a plethora of com-
peting orderings have been proposed (see Balanda and MacGillivray (1988)
for a recent survey). Again we champion a particular ordering related to
Lorenz ordering.
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.
Jo
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
^(u) is I on (0,i)and f on (1,1)
(2.16) X < 4 Y iff Fy\u)fY(0) < Fϊ\u)fx(0) on (0,1) - φ
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)
Λ,M. r g:i;ri
+
ff.»<< \ 2
Skewness and Kurtosis 21
An advantage of the David-Johnson functional is that they are well
defined without any assumption regarding the existence of E\X. All three
functions axe uniformly bounded in absolute value by 1. Skewness orderings
<5, <ί/ and <χ are defined in the natural way using these functionals. Note
that <5 is equivalent to <s.
We remark finally without further comment on the possibility of intro-
ducing a non-negative weight function ψ(v) inside the integral in definitions
(2.9), (2.10), (2.21) and (2.22).
3. Inter—Relationships Among the Orderings
Thirteen skewness orderings were introduced in Section 2. How are they
related? As MacGillivray (1986) noted, convex ordering is very strong and
implies many reasonable definitions of skewness. Specifically from her paper
we have the relations
It is not difficult to verify that we also have
I — ^ λ
and
(3.4) < 2 = ^ < 7
The key observations justifying (3.4) are that X <2 Y implies X +^ <* Y+
and Y~~ <* X" and star-ordering implies Lorenz ordering (see for example
Arnold (1987, p. 78)).
The convex ordering <i is much stronger than necessary. It is our con-
tention that the candidate orderings most worthy of consideration are <s
(equivalently <$), <6 and <s. Additionally it is felt that the concept of
skewness does not necessarily involve comparison of positive and negative
parts of random variables. On this basis < 6 is not as appealing. We are
left with <s and <$ together with the more forgiving integrated versions
provided by <!/,<£, <λ and <χ.
At this juncture, if we were forced to select a single skewness ordering
to recommend, it might well be <u. Note that vχ(u) < vγ(u) V ue(0, ) is
equivalent to
E(X\X < F^iu)) + E(X\X > Fχλ (l - u))
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
support of X+ and Fyl(Fx-(x))
(6.1) is convex on the support of X~~.
X<k 2 Y iff Fyl (u)/F~l (tt) ΐ on (0,1)
(6.2) and Fγi(u)/Fχl(u) 1 on (0,1)
(6.3) X<k 7 Y iff X+ <L Y+ and X~ <L Y
X<\Y iff £
Fγ\ + v)- Fy\ - v)]dv
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
