Convex transformations: A new approach to skewness and kurtosis )

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1 MATHEMATISCHE SECTIE Convex tranformation: A new approah to kewne and kurtoi ) by W. R. von Zwet ) UDC Samenvatting In dit artikel worden een tweetal orde-relatie voor waarhijnlijkheidverdelingen voorgete/d, die - beter dan de klaieke maten gebaeerd op derde en vierde momenten - aangeven wanneer een verdeling een grotere heefheid of kurtoi bezit dan een andere verdeling. Voort wordt een aantal karakterieringen en toepaingen van deze orde-relatie behande/d. Bewijzen worden in dit artikel niet gegeven; deze zijn te vinden in een meer uitgebreide, aan dit onderwerp gewijde tudie [5]. 1. Introdution Every tatitiian will have at leat an intuitive idea of what i meant by the onept of,kewne" and,.kurtoi" of a probability ditribution and he will be aware of the fat that thee hould play an important role in appliation. He will alo probably feel vaguely diatified with the exiting meaure for thee onept, i.e. the tandardized third and fourth entral moment, and indeed there are at leat two perfetly good reaon for thi uneay feeling. The firt one i that, aording to thee meaure, any pair of probability ditribution that poe finite fourth moment may be ompared a to kewne and kurtoi, wherea one feel that pair of uh ditribution exit that are quite inomparable in thee repet. The eond reaon i that, to the author' knowledge, very few intereting appliation of any generality exit. It i fairly obviou that both diadvantage are loely related: the reaon for the apparent lak of appliation i preiely the fat that omparion of probability ditribution on the bai of thee meaure i o often meaningle. From the above it will be lear that at the root of the trouble lie the fat that thee meaure impoe a imple ordering - i.e. an ordering where every pair of element are omparable - on too large a la of probability ditribution. Rather than retriting ourelve to onidering maller lae of ditribution we hall try and find a more atifatory approah by onidering partial ordering - i.e. ordering where not every pair of ditribution are neearily omparable - to replae the laial meaure. *) RapportS 328 van de afdeling Mathematihe Statitiek van het Mathematih Centrum te Amterdam; Lezing gehouden op de Statitihe Dag **) Sou-hef voor mathematihe tatitiek van het Mathematih Centrum te Amterdam. Statitia Neerlandia 18 (1964) nr S. van de Geer and M. Wegkamp (ed.), Seleted Work of Willem van Zwet, Seleted Work in Probability and Statiti, DOI / _1, Springer Siene+Buine Media, LLC

2 It will be hown in thi paper that two partial order relation exit that eem to over our intuitive idea about kewne and kurtoi. Thee order relation will not only be een to imply the ordering aording to the laial meaure, but alo to be o muh tronger than the laial ordering a to permit meaningful appliation. No proof will be given in thi paper; they may be found in [5], where a more extenive tudy of the ubjet i made. Part of the material preented here and in (5] wa previouly diued in [4]. 2. Notation Let ~ be a non-degenerate real-valued random variable 1) and let I be the mallet interval for whih P (~el) = 1. We define the ditribution funtion Fof~ by F(x) = l;p(~ < x) +!P(~ <x) and the expetation and entral moment of ~ by ~~ = f x df(x), I a 2 (~) = p 2 (~) = J (x - 8~)2 df (x), and I. J.lk (~) = J (x- ~~)k df(x), k = 3,4,..., I where the right-hand ide denote STIELTJES integral. We hall ay that thee expetation exit only if they are finite. The ditribution given by F i aid to be ymmetrial about x 0 e I if F(x 0 - x) + F(x 0 + x) = I for all real x. Let ~l:n < ~ 2 :n <... < ~n:n denote an ordered ample of ize n from the ditribution F; ~i:n i alled the i- th order tatiti of a ample of ize n from F. In the greater part of thi paper we hall onfine our attention to the la!f of ditribution funtion F atifying (a) F i twie ontinuouly differentiable on I; (b) F' (x) > 0 on I; () There exit integer i and n, 1 < i < n, uh that ~~i:n exit. For Fe.17 the invere funtion G i uniquely defined on (0,1) by 2) GF(x) = x for x e I. 1) We denote random variable by underlining their ymbol. 2) We hall uually not ue braket to denote ompoite funtion and write GF and GF(x) rather than G(F(.)) and G(F(x)). Statitia Neerlandia 18 (1964) nr

3 We hall alo be onerned with the ubla!i ( ~ of ymmetri ditribution in~. When we onider imultaneouly two random variable, ~ and ~, with ditribution funtion F and F*, we hall adopt imilar onvention and notation with regard to~ and F*, and write : 1*, ~ i : n and G*. A real-valued funtion p defined on I i aid to be onvex on I if for all x 1, x 2 e I and 0 <.A <. I rp (.Axt + (1 -.A) X2) <..Ap (xl) + (1 -.A) rp (x2), i.e. the graph of p lie below any hord. We note that thi definition implie ontinuity of rp on I, exept perhap at it endpoint, if thee exit. A real-valued funtion p on I i aid to be anti-~ymmetrial and onave-onvex on I about x 0 e I, if for all x 0 - x e I, x 6 + x e I, p (x 0 - x) + p (x 0 + x) = 2p (x 0 ), and if rp i onave for x < x 0 and onvex for x > x 0, x e I; x 0 will be alled a entral point of p. 3. Convex and onave-onvex tranformation Suppoe that p i non-dereaing and onvex on I and onider the random variable ~ and rp (~). Apart from an overall linear hange of ale uh a tranformation of the random variable ~ to the random variable p (~) effet a ontration of the lower part of the ale of meaurement and an extenion of the upper part. A, moreover, thi deformation inreae toward both end of the ale, the tranformation from ~ to rp {~) produe what one intuitively feel to be an inreaed kewne to the right. The following theorem hold : Theorem 3.1 If p i a non-dereaing onvex funtion on /, whih i not ontant on I, and if p,2k+i (~) and p, 2 k+i (rp (~))exit, then 1'2k+l (~).,;::: 1'2k+I,_(rp_(~_)) l' k 2 ~, lor = I, '... a2k+l (~) a2k+l (p (~)) It i intuitively equally appealing that a non-dereaing, antiymmetri and onave-onvex tranformation of a ymmetrially ditributed random variable hould lead to an inreaed kurtoi of the ditribution. We have: Theorem 3.2 Let p be a non-dereaing, antiymmetrial, onave-onvex funtion on /, Statitia Neerlandia 18 (1964) nr

4 whih i not ontant on /, and let the ditribution given by F be ymmetrial about x 0, where x 0 denote a entral point of q;. Then, if tlq;2 k (~)exit, l'u (~).;;;:: Pu (q; (~)) 2 " ( )..._, 2 k ( ( -,for k = 2,3,... a ~ a q; ~)) 4. Two weakoo0rder relation In the remaining part of thi paper we hall onfine our attention to ditribution funtion Fe g;; part of the reult, however, remain valid without thi retrition. Returning to the theorem of etion 3 we remark that they obviouly ontinue to hold if one replae q; (~) by any other random variable with the ame ditribution, i.e. they hold for any ~ with ditribution funtion F* ~ati~fying F*q; (x) = P (~* < f[j" (x)) = P (q; (~) < q; (x)) = P (~ < x) = F (x), or q: (x) = G*F(x) on I. We therefore define the following order relation on " and 9' repetively: Definition 4.1 If F,'F* e g;, then F < F* (or equivalently F* > F) if and only if G* F i onvex on I. Definition 4.2 If F, F* e 9', then F < F* (or equivalently F* > F) if and only if G* F i onvex for x > x 0, x e I, where x 0 denote the point of ymmetry of F. We hall ay in thi ae that F -preede or -preede F*, or that F* -follow or -follow F, and that the two are -omparable or -omparable. We hall alo peak of -ordering, -ordering, -omparion, -omparion, et., where the letter and tand for onvex and ymmetrial. Aording to the above the meaning of thee definition i lear: F < F* if and only if a random variable with ditribution F may be tranformed into one with ditribution F* by an inreaing and onvex tranformation; for ymmetrial ditribution, F < F* if and only if thi an be done by an inreaing, antiymmetrial, onave-onvex tranformation. From the theorem of the preeding etion the impliation are alo obviou: we have every right to ay that F < F* implie that F* ha greater kewne to the right than F, wherea for ymmetrial ditribution F < F* implie that F* ha greater kurtoi than F. Statitia Neerlandia 18 (1964) nr

5 Sine it i eaily een that both order relation are reflexive (F -< F) and tranitive (F-< F*, F* -< F** implie F-< F**) they are weak ordering. If one define an equivalene relation - by Definition 4.3 If F, F*e!F, then F- F* if and only iff (x) = F* (ax + h) for ome ontant a > 0 and h, it i alo eay to how that F - F* if and only if F < F* and F* < F; for F, F* e fj' one find that F - F* if and only iff< F* and F < F. Hene by paing to the olletion!f and fj' of equivalene lae one may define partial - - ordering (F-< F*, F* -< F implie F = F*) on!f and fj' by ordering equivalene lae aording to the - and -ordering of their repreentative. In tatitial parlane the above aert that - and -ordering are both independent of loation and ale parameter. The lae!f and 9' are the lae of type of law belonging to 91' and f/. We may onequently retrit our attention to - and -omparion of tandardized ditribution funtion. Here we give only two example of - and -ordering. The gamma ditribution may be hown to be -following one another with dereaing value of the parameter, wherea the ymmetri beta ditribution -follow one another with inreaing value of the parameter. Further example may be found in [5]. 5. Charaterization theorem In thi etion we give two theorem that provide a number of haraterization of the order relation < and < in term of inequalitie for expeted value and odd moment of order tatiti. Theorem 5.1 Let R be a dene ubet of (0,1). Then for F, F*e 91' the following tatement are equivalent: (l) F < F*; (2) F(tf~i:n) < F* (ts'~*i:n) for all n = 1,2,... and i = 1,2,..., n, for whih ts'~a:n and tl~*i : n exit; f.'2k+l (~i ; n) l-'2k+l (~* i;n) (3) 2k+l ) < 2k+l for an k = 1,2,... 'n = 1,2,... ' and (J (~i:n (J (~ i:n) i = 1,2,..., n, for whih p 2 k+l (~i:n) and p 2 k+1 (~*i:n) exit; Statitia Neerlandia 18 (1964) nr

6 i (4) If i and n tend to infinitv in uh a way that lim- = r, r e R, then n i (5) If i and n tend to infinity in uh a way that lim - = r, r E R, then for at n leat one value of k = 1,2,.... limy'- (P2k+1 (~* i: n) _ P2k+l (~i ; n)) > O. (}2 k+l (x*. ) a2 k+l (x. ) - 1;n -1 :n Two remark hould be made about thi theorem. The firt one i that for a given ditribution F onvexity would eem to be a rather heavy requirement to prove the inequalitie of theorem 3.1. The equivalene of tatement (I) and (5) of theorem 5.1 how, however, that if thee inequalitie are to hold even for a ingle value of k and for the la of ditribution of large ample order tatiti from a given ditribution, then onvexity i neeary a well a uffiient. The eond remark i that the equivalene of tatement (2) and ( 4) and of (3) and (5) enable u to derive mall ample inequalitie from their large ample ounterpart. Theorem 5.2 Let R be a dene ubet of(!, 1). Then for F, F*t f/ the following tatement are equivalent: (1) F < F*; (2) F(t!~i : n> < F*(t!~*i : n) for all n = 1,2,... and n + 2 -~ < i < n, for whih.(1! (!;)~ i:n exit; i (3) If i and n tend to infinity in uh a way that lim - = r, r e R, then n. i (4) If i and n tend to infinity in uh a way that lim - = r,! < r < 1, then for all k = 1,2,... n I. _ 1-(,U2k+t (z*i:n) P2k+l (~i:n))... O trn v n - -- ~ a2 k+l (x*.. ) a2 k+l (x.. ) ' - t,n - t.n (5) Statement (4) i valid for all r e Rand at leat one value of k = 1,2,... Statitia Neerlandia 18 (1964) nr

7 We note that mall ample inequalitie onerning odd moment are laking; the orreponding large ample reult i given in tatement (4). Fori < n + 1, 2 0 < r <! and R dene in (O,t}, the inequalitie of theorem 5.2 are of oure revered. For large lae of ditribution Fe f/' mall ample inequalitie between i-r:j. F(S:!"i n)andquantitieofthetype --- may be obtained by -ompari-. n + l-2o on with a la of ditribution funtion for whih the invere funtion G are inomplete beta funtion. For thee reult we refer to [5], where one may alo find till another haraterization of the order relation < and < in term of a meaure of kewne baed on the median. 6. Appliation Although it ha been made lear that the relation< and < may be taken to indiate inreaing kewne and kurtoi, we till have to demontrate that thee relation meet with more ue in appliation than the laial meaure baed on third and fourth moment. To thi end three example of omparion of ditribution will be onidered where kewne or kurtoi obviouly play an important role. The firt example i taken from a paper by J. L. HODGES jr. and E. L. LEH MANN [2]. They diu the relative aymptoti effiieny ew:n (F) of WILCOXON' two ample tet Wto the normal ore tet N, for the ae where the underlying ditribution i of type F. Numerial evidene lead them to uppoe that ew:n will inreae a the tail of the underlying ditribution grow heavier. Appliation of the relation < to a formula for ew:n (F) given in [2] immediately yield the deired reult: Under ertain regularity ondition, F, F*ef/' and F < F* implie ew:n (F) < < w:n (F*). The eond example onern a paper by H. HoTELLING [3] where the behaviour of STUDENT' tet under non-tandard ondition i tudied. Let :!" 1, :!" 2,, :!"n be a random ample from a ditribution Fe~. for whih either p, = 8:!" exit, or Fe f/'; in the latter ae we define p, by F(p,) = t Furthermore Jet ~-p ~ ~n=--y"n, where ~ = - n;=t n 1 n L :!"i and 2 = -- L (:!"i-.f) 2 n-1;=t Statitia Neerlandia 18 (1964) nr

8 The probability that ln will exeed a ontant value t will be denoted by P (In > t I F) and we define p t F Rn (F) = lim Cn > t I \ P <tn > I I f/>} where f/> denote the normal ditribution funtion. Suppoe that, auming the underlying ditribution to be normal, one arrie out STUDENT' right-ided tet for the hypothei p, < p, 0, wherea in fat F i not normal at ail. Then obviouly Rn (F) denote the limit of the ratio of the atual ize and the aumed ize of the tet a both thee ize tend to zero. It may therefore erve to provide a rough idea of what to expet when the aumption of normality i violated. For n = 3 numerial value found by HoTELLING for ome ymmetrial ditribution eem to indiate - paradoxiaiiy enough at firt ight- that Rn (F) dereae a the tail of F beome heavier. Making ue of an expreion for Rn (F) given in (3] one eaily how thi idea to be orret for -ordered ymmetri ditribution, wherea a imilar reult may be proved for -ordered ditribution. In fat we have: lf F, F*e ' and if either~~. tt~ exit and F < F*, or F, F*e f/ and F < F*, then Rn (F) > Rn (F*) for n = 2,3,.... Finally we diu the relative effiieny of ample median to ample mean in etimating the point of ymmetry of a ymmetri ditribution. Let ~ 1 ;f 2,, ~n denote a random ample from a ditribution Fe f/ with finite variane rr (~). and uppoe one wihe to etimate tf~. Two unbiaed etimate that are generally ued are the ample median and the ample mean ~n+l -2-:n 1 n.&n =- L ~t n 1=1 where we have uppoed n to be odd. The hoie between them hould depend on the ratio of their (mall ample) effiienie elf(~n;l:n) al(~) r (F) = = -~-----;:- The following reult i eaily obtained: n eff(gn) nal(~n;l :n) For ditribution F, F*e f/ having finite variane, F < F* implie rn (F)< <: r n (F*) for n = 1, 3, 5,.... Statitia Neerlandia 18 (1964) nr

9 Thi reult upport the tatement by G. W. BROWN and J. W. TUKEY [1] that,it i probable that the relative effiienie of mean and median are greatly affeted by the length of the tail". Referene [I) G. W. BROWN, J. W. TUKEY, Some ditribution of ample mean, Ann. Math. Statit.l7 (1946), [2] J. L. HoDGES jr., E. L. LEHMANN, Comparion of normal ore and Wiloxon tet, Pro. 4th Berkeley Symp. (1961) Vol. 1, [3] H. HoT LLJNo, The behavior of ome tandard tatitial tet under nontandard ondition, Pro. 4th Berkeley Symp. (1961) Vol. 1, [4] W. R. VAN Zw T, Two weak-order relation for ditribution funtion, Report S 303 (VP 17) Dept. of Math. Stat. (1962), Mathematih Centrum, Amterdam. {S] W. R. VAN Zw T, Convex tranformation of random variable, Math. Centre Trat 7 (1964), Mathematih Centrum, Amterdam. Statitia Neerlandia 18 (1964) nr