characterization through distributional properties of dual generalized order statistics

Journal of the Egyptian Mathematical Society (2012) 20, 211–214 Egyptian Mathematical Society Journal of the Egyptian Mathematical Society www.etms-eg.org www.elsevier.com/locate/joems ORIGINAL ARTICLE Characterization through distributional properties of dual generalized order statistics A.H Khan a b a,* , Imtiyaz A Shah a, M Ahsanullah b Department of Statistics and Operations Research, Aligarh Muslim University, Aligarh 202 002, India Department of Management Sciences, Rider University, Lawrenceville, NJ 08648-3099, USA Received 25 June 2011; revised 30 June 2012 Available online December 2012 KEYWORDS Order statistics; Lower record statistics; Upper record statistics; Generalized order statistics; Dual generalized order statistics; Contraction; Dilation; Characterization of distributions; Generalized exponential; Generalized Pareto; Generalized power function; Gumbel; Weibull; Inverse Weibull; Exponential; Power function; Pareto; Distributions Abstract Distributional properties of two non-adjacent dual generalized order statistics have been used to characterize distributions Further, one sided contraction and dilation for the dual generalized order statistics are discussed and then the results are deduced for generalized order statistics, order statistics, lower record statistics, upper record statistics and adjacent dual generalized order statistics ª 2012 Egyptian Mathematical Society Production and hosting by Elsevier B.V All rights reserved Introduction * Corresponding author E-mail address: ahamidkhan@rediffmail.com (A.H Khan) Peer review under responsibility of Egyptian Mathematical Society Production and hosting by Elsevier Kamps [6] introduced the concept of generalized order statistics (gos) as follows: Let X1, X2, , Xn be a sequence of independent and identically distributed (iid) random variables (rv) with the absolutely continuous distribution function (df) F(x) and the probability density function (pdf) f(x), x (a, b) Let P n N; ~ ¼ ðm1 ; m2 ; ; mnÀ1 Þ RnÀ1 ; Mr ¼ nÀ1 n P 2; k > 0; m j¼r mj , 1110-256X ª 2012 Egyptian Mathematical Society Production and hosting by Elsevier B.V All rights reserved http://dx.doi.org/10.1016/j.joems.2012.10.002 212 A.H Khan et al such that cr = k + (n À r) + Mr > for all r {1, 2, , n À 1} If m1 = m2 = Á Á Á = mnÀ1 = m, then X(r, n, m, k) is called the rth m-gos and its pdf is given as: " # mỵ1 r1 cr1 cr 1 ẵFxị ẵFxị fxị; fXr;n;m;kị xị ẳ r 1ị! mỵ1 1:1ị a < x < b; Qr where cr = k + (n À r)(m + 1) and crÀ1 ¼ i¼1 ci Based on the generalized order statistics (gos), Burkschat et al [4] introduced the concept of the dual generalized order statistics (dgos) where the pdf of the rth m-dgos X*(r, n, m, k) is given as " # mỵ1 r1 cr1 cr 1 ẵFxị ẵFxị fX r;n;m;kị xị ẳ fxị; r 1ị! mỵ1 a < x < b; Remark 1.1 It may be seen that if Y is a measureable function of X with the relation Y = h(X), then Y*(r, n, m, k) = h(X*(r, n, m, k)) and Y(r, n, m, k) = h(X(r, n, m, k)), if h is increasing function (e.g., Yr:n = h(Xr:n) and YL(r) = h(XL(r)), where Xr:n and XL(r) are the rth order statistic and lower record, respectively) Moreover, Y(r, n, m, k) = h(X*(r, n, m, k)) and Y*(r, n, m, k) = h(X(r, n, m, k)), if h is decreasing function (e.g., YnÀr+1:n = h(Xr:n) and YU(r) = h(XL(r)), where YU(r) is the rth upper record) Remark 1.2 The following elementary facts will be needed in the next section: eÀay (i) if Y = log X $ Gum(a) (i.e., F Y yị ẳ e ; < y < Àa 1; a > 0), then X $ in W(a) (i.e., F X xị ẳ ex ; < x < 1; a > 0) (ii) if -log X$ Gum(a), then X $ Wei(a), (i.e., F X xị ẳ a À eÀx ; < x < 1; a > 0) ay (iii) if Y = log X $ exp(a) (i.e., F Y yị ẳ e ; < y < 1; a > 0), then X $ Par(a), (i.e., FX(x) = À xÀa, < x < 1, a > 0) (iv) if -log X $ exp(a),then X $ pow(a), (i.e., FX(x) = xa, < x < 1, a > 0) (v) if Y = logX $ genexp(a) (i.e.,F Y yị ẳ ẵ1 m ỵ 1ị Theorem 2.1 Let X*(s, n, m, k) be the sth m-dgos from a sample of size n drawn from a continuous population with the pdf f(x) and the df F(x), then for r < s n, d X r ỵ j; n; m; kị ẳ X s; n; m; kị ỵ Ysrj:s1 ; then X $ gen1 Par(a) (i.e., F X ðxÞ ẳ ẵ1 m ỵ 1ịxa mỵ1 ; m ỵ 1Þa < x < 1; a > 0) (vi) if -log X $ genexp(a), then X $ genpow(a) (i.e., 1 F X xị ẳ ẵ1 m þ 1Þxa Šmþ1 ; < x < ðm þ 1ịa ; a > 0) j ẳ 0; 1; 2:1ị where YsÀrÀj:sÀ1 is the (s À r À j)th order statistic from a sample of size (s À 1) drawn from exp(a) distribution and is independent of X* (s, n, m, k) if and only if X1 $ genexp(a) and d X ¼ Y denotes that X and Y have the same df Proof To prove the necessary part, let the moment generating function (mgf) of X*(r, n, m, k) be MXÃðrÞ ðtÞ, then d which is obtained just by replacing Fxị ẳ Fxị by F(x) Ahsanullah [1] has characterized uniform distribution under random contraction for adjacent dgos Khan and Shah [7] have characterized distributions using distributional properties of non-adjacent lower records, upper records and order statistics In this paper, distributional properties of the dgos have been used to characterize a general form of distributions for non- adjacent dgos under random translation, dilation and contraction, thus generalizing the results of Ahsanullah [1] Further, results in terms of lower records, upper records and order statistics are deduced One may also refer to Alzaid and Ahsanullah [2], Beutner and Kamps [3], Wesolowski and Ahsanullah [8] and Castan˜o-Martı´nez et al [5] for the related results eay mỵ1 ; 1a logm ỵ 1Þ < y < 1; a > 0), Characterizing results X r; n; m; kị ẳ X s; n; m; kị ỵ Y, MXrị tị ẳ MXsị tị MY ðtÞ implies that Since for the genexp(a) distribution, we have MXrị tị ẳ cr Cr at ịCmỵ1 ị CrÀ1 : t c t r ðr À 1Þ! m ỵ 1ịra Cr a ỵ mỵ1 ị Therefore, MY tị ẳ MXrị tị MXsị tị ẳ Csị Cr À at Þ À Á: CðrÞ C s À at But this is the mgf of YsÀr:sÀ1, which is the (s À r)th order statistic from a sample of size (s À 1) drawn from exp(a) To prove the sufficiency part, we have for s P r + 1, fXà r;n;m;kị xị ẳ Z x fX s;n;m;kị yị fYsr:s1 x yịdy Z x as 1ị! ẳ ẵeaxyị r ẵ1 r 1ị!s r 1ị! 0 À eÀaðxÀyÞ ŠsÀrÀ1  fXà ðs;n;m;kÞ ðyÞdy: ð2:2Þ Differentiate both the sides of (2.2) w.r.t x, to get  d fX r;n;m;kị xị ẳ ar fX rỵ1;n;m;kị xị À fXà ðr;n;m;kÞ ðxÞ dx  à or, fXà ðr;n;m;kÞ xị ẳ ar FX rỵ1;n;m;kị xị FX r;n;m;kị xị Now, since (Ahsanullah [1]) Fxị fX rỵ1;n;m;kị xị: FX rỵ1;n;m;kị xị FX r;n;m;kị xị ẳ crỵ1 fxị Therefore, we have mỵ1ịẵFxịm fxị ẵ1Fxịịmỵ1 Fxị ẳ ẵ1 m ỵ 1ịeax mỵ1 ẳ a, which implies that Hence the proof h Remark 2.1 Let Xr:n be the rth order statistic from a sample of size n drawn from a continuous population with the pdf f(x) and the df F(x), then for r < s n, d Xsj:n ẳ Xr:n ỵ Xsrj:nr ; j ẳ 0; 1; 2:3ị where Xsrj:nr is independent of Xr:n if and only if X1 $ exp(a) Characterization through distributional properties of dual generalized order statistics This can be established by noting that order statistic appear in the generalized order statistics (gos) model as well as in dual generalized order statistics (dgos) model Therefore at m = 0, (2.1) may be written as d Xnrjỵ1:n ẳ Xnsỵ1:n ỵ XsÀrÀj:sÀ1 ; j ¼ 0; 1; r < s n; which implies d Xsj:n ẳ Xr:n ỵ XsÀrÀj:nÀr ; j ¼ 0; 1; 213 Remark 2.6 In case of ordinary order statistics, i.e., at m = 0, we have d XsÀj:n ¼ Xr:n Á XsÀrÀj:nÀr ; j ¼ 0; 1; r < s n; where XsÀrÀj:nÀr is independent of Xr:n if and only if X1 $ Par(a), as obtained by Castan˜o-Martı´nez et al [5] and Khan and Shah Imtiyaz [7] Remark 2.7 As m fi À 1, we get r < s n; obtained by replacing (n À s + 1) by r and (n À r + 1) by s as given by Khan and Shah [7] Remark 2.2 Alzaid and Ahsanullah [2] have proved that d Xr:n ¼ XrÀ1:n ỵ V d XLrỵjị ẳ XLsị Ysrj:s1 ; j ¼ 0; 1; r < s; where YsÀrÀj:sÀ1 is the (s À r À j)th order statistic from a sample of size (s À 1) drawn from the Par(a) distribution and is independent of XL(s), the sth lower records if and only if X1 $ inW(a) where V $ exp(n À r + 1) if and only if X1 $ exp (1) Remark 2.3 Castan˜o-Martı´nez et al [5] have shown that Corollary 2.2 Let X(s, n, m,k) be the sth m-gos from a sample of size n drawn from a continuous population with the pdf f(x) and the df F(x), then for r < s n, d Xs:n ẳ Xr:n ỵ V d d where V ¼ À log W with W $ Be(n À s + 1,s À r) if and only if X1 $ exp(1) Remark 2.4 As m fi À 1, genexp(a) tends to the Gum(a) and X*(r, n, m, k) to XL(r), the rth lower records Therefore, we have d XLrỵjị ẳ XLsị ỵ Ysrj:s1 ; j ẳ 0; 1; r < s; Xr ỵ j; n; m; kị ẳ Xs; n; m; kị Yrỵj:s1 ; j ẳ 0; 1; ð2:5Þ th where Yr+j:sÀ1 is the (r + j) order statistic from a sample of size (s À 1) drawn from pow(a) distribution and is independent of X(s, n, m, k) if and only if X1 $ genpow(a) Proof Here the product X(s, n, m, k) Ỉ Yr+j:sÀ1 in (2.5) is called random contraction of X(s À j, n, m, k) (Beutner and Kamps [3]) Since th where YsÀrÀj:sÀ1 is the (s À r À j) order statistic from a sample of size (s À 1) drawn from exp(a) distribution and is independent of XL(s) if and only if X1 $ Gum(a), as obtained by Khan and Shah [7] d log X r; n; m; kị ẳ À log Xà ðs; n; m; kÞ À log YsÀr:sÀ1 implies d Xr; n; m; kị ẳ Xs; n; m; kÞ Á Yr:sÀ1 Remark 2.5 Alzaid and Ahsanullah [2] have shown that in view of Remarks 1.1 and 1.2 and the result follows h d XLrị ẳ XLrỵ1ị ỵ V Remark 2.8 Beutner and Kamps [3] have shown that for adjacent generalized order statistics where V $ exp(r) if and only if X1 $ Gum(1) Corollary 2.1 Let X*(s, n, m, k) be the sth m-dgos from a sample of size n drawn from a continuous population with the pdf f(x) and the df F(x), then for r < s n, d X r ỵ j; n; m; kị ẳ X s; n; m; kị Ysrj:s1 ; j ẳ 0; 1; 2:4ị where Ysrj:s1 is the (s À r À j)th order statistic from a sample of size (s À 1) drawn from Par(a) distribution and is independent of X*(s,n,m,k) if and only if X1 $ genPar(a) Proof Here the product X*(s, n, m, k) Ỉ YsÀrÀj:sÀ1 in (2.4) is called random dilation of X*(s, n, m, k) (Beutner and Kamps [3]) Note that if d d Xr; n; m; kị ẳ Xr ỵ 1; n; m; kÞ Á V where V $ pow(ra) if and only if X1 $ genpow(a) Remark 2.9 We can get the corresponding characterizing results for the order statistics at m = as: Let Xr:n be the rth order statistic from a sample of size n drawn from continuous population with the pdf f(x) and the df F(x), then for r < s n, d Xrỵj:n ẳ Xs:n Xrỵj:s1 ; j ẳ 0; 1; where Xr+j:sÀ1 is independent of Xs:n if and only if X1 $ pow(a), as given by Khan and Shah [7] For adjacent order statistics one may also refer to Ahsanullah [1] and Wesolowski and Ahsanullah [8] à log X ðr; n; m; kị ẳ log X s; n; m; kị ỵ log Ysr:s1 then d X r; n; m; kị ẳ Xà ðs; n; m; kÞ Á YsÀr:sÀ1 in view of Remarks 1.1 and 1.2 and the result follows h Remark 2.10 The corresponding result for the lower records as m fi À is: 214 A.H Khan et al Let XU(s) be the sth upper record from a continuous population with the pdf f(x) and the df F(x), then d XUrỵjị ẳ XUsị Yrỵj:s1 ; j ẳ 0; 1; r < s; where Yr+j:sÀ1 is the (r + j)th order statistic from a sample of size (s À 1) drawn from pow(a) distribution and is independent of XU(s) if and only if X1 $ Wei(a), as obtained by Khan and Shah [7] Acknowledgment The authors are thankful to the Referees and the Editor for their fruitful suggestions and comments References [1] M Ahsanullah, A characterization of the uniform distribution by dual generalized order statistics, Commun Statist Theor Meth 33 (2004) 2921–2928 [2] A.A Alzaid, M Ahsanullah, A characterization of the Gumbel distribution based on record values, Commun Statist Theor Meth 32 (2003) 2101–2108 [3] E Beutner, U Kamps, Random contraction and random dilation of generalized order statistics, Commun Statist Theor Meth 37 (2008) 2185–2201 [4] M Burkschat, E Cramer, U Kamps, Dual generalized order statistics, Metron LXI (2003) 13–26 [5] A Castan˜o-Martı´nez, F Lo´pez-Bla´zquez, B Salamanca-Min˜o, Random translations, contractions and dilations of order statistics and records, Statistics (2010) 1–11 [6] U Kamps, A Concept of Generalized Order Statistics, Teubner, Stuttgart, 1995 [7] A.H Khan, Imtiyaz A Shah, Distributional properties of order statistics and record statistics, Pak J Statist.Oper.Res (3) (2012) 293–301 [8] J Wesolowski, M Ahsanullah, Switching order statistics through random power contractions, Aust N.Z.J Statist 46 (2) (2004) 297–303