RELATIONSHIPS BETWEEN MOMENTS OF TWO RELATED SETS OF ORDER STATISTICS AND SOME EXTENSIONS

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1 A. Ist. Statist. Math. Vol. 45, No. 2, (1993) RELATIONSHIPS BETWEEN MOMENTS OF TWO RELATED SETS OF ORDER STATISTICS AND SOME EXTENSIONS N. BALAKRISHNAN.1, Z. GOVINDARAJULU 2 AND K. BALASUBRAMANIAN 3 1Departmet of Mathematics ad Statistics, McMaster Uiversity, 1280 Mai Street West, Hamilto, Otario, Caada L8S gk1 2Departmet of Statistics, Uiversity of Ketucky, Lexigto, KY , U.S.A. 3 Idia Statistical Istitute, SJS Sasawal Marg, New Delhi , Idia (Received September 2, 1991; revised April 21, 1992) Abstract. Govidarajulu expressed the momets of order statistics from a symmetric distributio i terms of those from its folded form. He derived these relatios aalytically by dividig the rage of itegratio suitably ito parts. I this paper, we establish these relatios through probabilistic argumets which readily exted to the idepedet ad o-idetically distributed case. Results for radom variables havig arbitrary multivariate distributios are also derived. Key words ad phrases: Double expoetial distributio, expoetial distributio, order statistics, outliers, permaets, recurrece relatios. 1. Itroductio If XI: ~_ X2: ~_... ~ X: are the order statistics obtaied from a radom sample of size from a radom variable X symmetric about 0 ad Yl: < Y2:, _< "'" _< Y: are the correspodig order statistics from Y = IXI, Govidarajulu (1963) expressed the momets of X~:~ i terms of the momets of Y<:~,, 1 _< r' _< ' _<. He derived these relatios by dividig the rage of itegratio ad maipulatig the itegrals algebraically. Govidarajulu (1966) applied these results to compute the meas, variaces ad covariaces of order statistics from a double expoetial distributio by makig use of the kow explicit expressios of these quatities from a stadard expoetial distributio. Recetly, Balakrisha (1988a) exteded the results of Govidarajulu to a sigle-outlier model which were used by Balakrisha ad Ambagaspitiya (1988) to study the robustess features of liear estimators for the parameters of a double expoetial distributio. Balakrisha's results for the sigle-outlier model has * The first author would like to thak the Natural Scieces ad Egieerig Research Coucil of Caada for fudig this research. 243

2 244 N. BALAKRISHNAN ET AL. bee exteded by him (Balakrisha (1989)) to the case whe X~'s are idepedet, symmetric but o-idetical by applyig the methods used by Govidarajulu (1963) ad Balakrisha (1988b). I this paper, we first of all derive Govidarajulu's result by a probabilistic argumet. This method leds itself easily to exted the results to the case whe Xi's are margially symmetric but ot ecessarily idepedet. Further extesio of the result whe Xi's joitly have a arbitrary multivariate distributio is also preseted. 2. Probabilistic proof for the i.i.d, case ad a extesio Let us deote E(X~:) by p~:~, E(Xk:) by #(k) for k _> 2, E(X~:Xs:) by #~,s:~ for 1 < r < s <, ad similarly E(Yr:) by ~:~, E(Y~ ) by y(k)~: for k _> 2, ad E(Y~:~Y~:) by v~,s:~ for 1 < r < s <. The, through divisio of the rage of itegratio ad direct algebraic maipulatio of the resultig itegrals, Govidarajulu (1963) established the followig two relatios betwee the two sets of momets of order statistics: Forl<r<adk=l,2,..., (2.1) r: i %-i:-i + (-1) k i "i-v+l:i ; ad for 1 _< r < s <, k i=o i=r (2.2) #~,s:~ = 2 -~ i=0 i=s s-1 } We shall provide below a probabilistic proof for the relatio i (2.1) while the relatio i (2.2) may be proved easily usig similar argumets. Suppose X~: > 0. The, the umber of X's < 0 ca at most be r - 1; let us suppose that umber is i (0 < i < r - 1) with the remaiig - i X's the costitutig a radom sample from Y. It is readily see i this case that the coditioal distributio of X~: give that i (< r) of the X's are egative is the same as the ucoditioal distributio of Y~-i:-i. Suppose X~: _< 0. The, the umber of X's _< 0 will at least be r; let us suppose that umber is i (r < i < ). By otig the that these i X's costitute a radom sample from -Y, it is readily see i this case that the coditioal distributio of X~:~ give that i (> r) of the X's are egative is the same as the ucoditioal distributio of -Y/-~+l:i. The relatio i (2.1) the follows immediately. The aforemetioed probabilistic argumet eables us to exted Govidarajulu's relatios i (2.1) ad (2.2) to the case whe the radom variable X, with cumulative distributio fuctio F(x), is ot ecessarily symmetric. I this case, the coditioal distributio of X give X > 0 ad the coditioal i=r

3 RELATIONS AMONG TWO SETS OF ORDER STATISTICS 245 distributio of X give X < 0 will play the role of the distributios of Y ad -Y, respectively. Now, by deotig the momets of order statistics from the former as before by us:,~, u(k)s:m ad L's,t:,~ ad the momets of order statistics from the latter by L,~:,~,- ~(k)~:,~ ad L'~.t:,~,- we have the followig geeralized relatios: For 1 < r < ad k = 1, 2,..., r--i (2.3), (k) X TM ii.~(k) x-" 1-, _(k) t~r: ~ ~ ~ r--i:--i ~- L llil/r:i i=0 i=r ad for 1 < r < s _<, (2.4) #r,sm r--i s--1 = E l-lilzr-i,s-i:-i + E IIiF/r,s:i + E YIiF]r:il]s-i:-i, i=o i=s i=r where Hi is the probability that exactly i X's < 0 give by () (F(O))i( 1 _ F(O))_ % IIi= i i = 0,1,...,. 3. Probabilistic proof for the i.i.d, case ad a extesio Suppose Xi, i -- 1, 2,...,, are idepedet radom variables with cdf Fi(x) ad pdf fi(x), i = 1,2,...,, each symmetric about 0. Let Y/ = IXil, i = 1,2,...,, ad Xx: < X2: _< "" _< X: ad YI:~ -< Y2: _< "'" < Y~:~ be the correspodig order statistics. Let us deote E(Xr:~) by #r:, E(X~:) by ~:' (k) for k > 2, ad E(Xr:Xs:) by #~,~:~ for 1 i r < s i. Further, let Y~[1¼'t_2i... td deote the r-th order statistic from - i Y variables obtaied by deletig Yh, Yl2, -, Yh. [l~,l~,...,@(k) from Y1, Y2,..., Y~, with the correspodig k-th momet deoted by "~:~-i. [ll,12,...,l~] ad the product momet by,~,~:~_i. With these otatios, Balakrisha (1989) has geeralized the relatios i (2.1) ad (2.2) by usig permaet represetatio of desity of order statistics give by Vaugha ad Veables (1972) ad direct algebraic maipulatio similar to those of Govidarajulu (1963). The geeralized relatios are as follows: For l<r<adk=l,2,..., (3.1) r~,r:".(k) =-2-/~ E ~,[ll,...,l,!(k) r--~:--z } +(-1)k E E 1]i--r--l:i [l~...,,~_,](k) i=r l<ll<...<l~_~< ad for l_<r<s_<, [ i-~o l<_ll<...<li~_ ~,[h... ld r--i,s--i:--i

4 246 N. BALAKRISHNAN ET AL. l][ll,...,l-i] -~-E E i--s+l,i--r+l:i i=s l<_ll<...<l~-i< s--1 } -- E ~ l][ll... l~i. [li+l... l] s--i:--i ~i--r4-1:i i=r l~ll<...<li~_ where {/1,12,...,li} A {li+l,...,l~} = 0. We shall provide here a probabilistic proof for the relatio i (3.1) while the relatio i (3.2) may be similarly proved. Suppose X~:~ > 0. The, the umber ofx's < 0 is at most r-l; let us suppose Xh, Xz2,..., Xl~ are the oly X's < 0. It is the readily see that the coditioal distributio of X~:~ give that Xh, Xl~,..., X h are egative is the same as the ucoditioal distributio of y[l_~;~,_z~. Suppose X~:~ _< 0. The, the umber of X's < 0 is at least r. By usig a similar argumet, it is the see that the coditioal distributio of X~:~ give that Xl~_~+~,..., Xt~ are egative is the [11... l~_~] same as the ucoditioal distributio of -~i-~+1:i. The relatio i (3.1) the readily follows. The probabilistic argumet provided above makes it possible to exted Balakrisha's relatios i (3.1) ad (3.2) to the case whe the radom variables Xi's are ot ecessarily symmetric. Let ~'s:-m [h... l~](k) ad Z]s,t:_ [11... l,~] m deote the sigle ad the product momets of order statistics from the coditioal distributio of - m radom variables obtaied by deletig Xh,..., Xlm from X1, X2,..., X~, _[11... l~](k) give that all these - m variables are positive. Similarly, let ~:~-m ad -[h... l~] ~,t:~-,~ deote the correspodig momets of order statistics from the coditioal distributio give that all the - m variables are egative. We may the prove the followig geeralized relatios aalogous to those i (2.3) ad (2.4): Forl<r<adk--1,2,..., (3.3) ~r: r-1 i=0 l<lz<...<li<_ [ll,...,id(k) + E E II(h... l~)"~:~, i=r l~_li+l<...<l~< ad forl_<r<s_<, (3.4) #~,~:~ r--1 =E E ll(/1,...,li) 1]r-i,s-i:-i Cll... i=o l~ll<...<li~ z:[/i+l,..-,/~] + E H(/1... li ) ~r,s:i i=s l~l~+l<...~l< _. 1J-(/1,...,li ) l]s--i:-- il]r:i i=r l~_ll<...<li~_

5 RELATIONS AMONG TWO SETS OF ORDER STATISTICS 247 where (3.5) II(ll... ll) = Pll " " " Pliql~+l " " " ql~, with Pi = P(Xi ~_ O) = 1 - qi. Remark. It is easy to see that the relatios i (3.3) ad (3.4) simply reduce to those i (3.1) ad (3.2) for the special case whe all the X's are symmetric about 0 i which case II(z~... t~) -- 2-~V{/1,..-,l~} C {1,2,...,}Vi = O, 1,...,. 4. Results for the i.i.d, case It is importat to metio here that while provig the relatios i (3.3) ad (3.4) the oly place where the idepedece of the radom variables X1, X2,..., X~ is used is i the expressio for II(ll... z~) i (3.5) i the form of a product of margial probabilities. Therefore, if we redefie II(ll... z~) as (4.1) II(ll... l~) = P{Xzl < 0,...,Xl~ < 0, Xl~+l > 0,...,Xl > 0}, the the relatios i (3.3) ad (3.4) cotiue to hold eve for the i.i.d, case, viz., whe Xi's joitly have a arbitrary cotiuous multivariate distributio. Ackowledgemets The authors would like to thak a Associate Editor ad a referee for suggestig some chages which led to a improvemet i the presetatio of this paper. REFERENCES Balakrisha, N. (1988a). Recurrece relatios amog momets of order statistics from two related outlier models, Biometrical J., 30, Balakrisha, N. (1988b). Recurrece relatios for order statistics from idepedet ad o-idetically distributed radom variables, A. Ist. Statist. Math., 40, Balakrisha, N. (1989). Recurrece relatios amog momets of order statistics from two related sets of idepedet ad o-idetically distributed radom variables, A. Ist. Statist. Math., 41, Balakrisha, N. ad Ambagaspitiya, R. S. (1988). Relatioships amog momets of order statistics i samples from two related outlier models ad some applicatios, Comm. Statist. Theory Methods, 17, Govidarajulu, Z. (1963). Relatioships amog momets of order statistics i samples from two related populatios, Techometrics, 5, Govidarajulu, Z. (1966). Best liear estimates uder symmetric cesorig of the parameters of a double expoetial populatio, J. Amer. Statist. Assoc., 61, Vaugha, R. J. ad Veables, W. N. (1972). Permaet expressios for order statistics desities, J. Roy. Statist. Soc. Ser. B, 34,

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