Characterizations of Distributions via Conditional Expectation of Generalized Order Statistics

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1 Marquette University Mathematics, Statistics and omputer Science Faculty Research and Publications Mathematics, Statistics and omputer Science, Department of haracterizations of Distributions via onditional Expectation of Generalized Order Statistics M. Ahsanullah Rider University-Lawrenceville Gholamhossein Hamedani Marquette University, Mehdi Maadooliat Marquette University, Published version. International Journal of Statistics and Probability, Vol. 4, No : DOI. 205 The Authors. Used with permission.

2 International Journal of Statistics and Probability; Vol. 4, No. 3; 205 ISSN E-ISSN Published by anadian enter of Science and Education haracterizations of Distributions via onditional Expectation of Generalized Order Statistics M. Ahsanullah, G.G. Hamedani 2 and M. Maadooliat 2 Department of Management Sciences, Rider University, Lawrenceville, NJ, USA 2 Department of Mathematics, Statistics and omputer Science, Marquette University, Milwaukee, WI, USA orrespondence: M. Maadooliat, Department of Mathematics, Statistics and omputer Science, Marquette University, Milwaukee, WI , USA. Tel: mehdi@mscs.mu.edu Received: May 22, 205 Accepted: June 5, 205 Online Published: July 0, 205 doi:0.5539/ijsp.v4n3p2 Abstract URL: haracterizations of probability distributions by different regression conditions on generalized order statistics has attracted the attention of many researchers. We present here, characterizations of certain continuous distributions based on the conditional expectation of generalized order statistics. Mathematics Subject lassification: 62E0; 60E05; 60E5 Keywords: generalized order statistics, probability distribution. Introduction Kamps 995 introduced generalized order statistics gos in terms of their joint density function. The record values, k-record values, order statistics are special cases of the gos. The random variables X, n, m, k, X 2, n, m, k,..., X n, n, m, k, k > 0, m R, are n gos from an absolutely continuous distribution F with density f if their joint density, f,2,...,n x, x 2,..., x n, is given by f,2,...,n x, x 2,..., x n = k Π j n = γ m ] j Π n j = F x j f x j F xn k f xn, F 0+ < x < x 2 <... < x n < F,. where x+ x stands for the rightleft-sided limit at x, = and γ j = k + n j m + for all j, j n, k is a positive integer and m. For k = and m = 0, X r, n, m, k will be the rth order statistic and for k = and m =, it will be the upper record value of the i.i.d. random variables with common distribution F and density f. From. the density of X r, n, m, k, f r,n,m,k, is where c r = Π r j = γ j and f r,n,m,k x = c r Γ r γr f x h r m,.2 h m x = ] x m+, f or all x 0, and all m with m + h x = lim h m x = ln x. m The joint density of X r, n, m, k and X r + 2, n, m, k, r n 2, is Kamps, 995 f r, r+2,n,m,k x, y = Πi r+2 = γ r+2 i i=r+ ar r i= a i γi f x f y Fy γi Fy Fx i r + 2, x y, 2

3 International Journal of Statistics and Probability Vol. 4, No. 3; 205 where a i = a i r = Π r j= γ j γ i j i and a r i s = Π s j=r+ j i γ j γ i, r + i r + 2 as borrowed from Bieniek and Szynal onsequently, the conditional density of X r + 2, n, m, k given X r, n, m, k = x, for m, is f r +2 r,n,m,k y x = γ r+ γ r+2 m + ] γ r+ f y F y ] γ r+2 ] m+ F y ] m+, y > x..3 The conditional density of X r, n, m, k given X r + 2, n, m, k = y, for m, is f r r+2,n,m,k x y = r r + m + ] m+ r F y ] m+ r ] m+ ] m+ F y f x ] m, y > x..4 Different regression conditions on generalized order statistics have been employed to characterize distributions e.g., see Bieniek & Szynal, 2003; Bieniek, 2009; ramer, Kamps & Keseling, 2004, to name a few. In Bieniek and Szynal 2003, the authors consider all distributions F for which the following linearity of regression holds: E X r + l, n, m, k X r, n, m, k] = a X r, n, m, k + b. They conclude that only exponential, Pareto and power function distributions satisfy this equation. Using this result they obtain characterizations of these distributions based on sequential order statistics, records and progressive type II censored order statistics. In the following section, we consider a certain class of gos and present two characterization results. 2. haracterization Results ramer et al. 2004, point out that characterizations of distributions based on linear regressions E s X r + l, n, m, k X r, n, m, k = ] = g 2. have been studied extensively for order statistics and record values r N, l = and set up a comprehensive solution related to characterization problems. They solved the case of gos l = and pointed out that for larger l the calculations become more difficult. They were able to obtain an explicit result when g is a linear function. They concluded that the linearity of the conditional expectation provides a characterization of generalized Pareto distribution. Ahsanullah and Hamedani 203 presented characterizations of continuous distributions based on 2. for l = but without the assumptions of monotonicity of s and linearity of g. In what follows we present our characterizations for l = 2 with the assumption of monotonicity of s but without assumption of linearity of g. Theorem 2.. Let X : Ω a, b be a random variable with a twice differentiable distribution function and with density function f x. Let s x be a non-negative, non-decreasing and twice differentiable function on a, b such that s a = f a > 0, s a = 0 and lim x b s x =. Then E s X r + 2, n, m, k X r, n, m, k = x] = γr+ γ r+2 s x ], 2.2 γ r+ + γ r γ r+ γ r+2 22

4 International Journal of Statistics and Probability Vol. 4, No. 3; 205 implies Proof. Suppose 2.2 holds and let A = = γ r+ + γ r+2 + s x + ] /γ r+ + γ r+2 +, a x b. 2.3 γ r+ γ r+2 γ r+ + γ r+2 ++γ r+ γ r+2 and B = γ r+ + γ r+2 ++γ r+ γ r+2, then γ r+ γ r+2 m + b x s y f y F y ] γ r+2 ] m+ F y ] m+ dy = As x + B ] γ r Differentiating both sides of 2.4 with respect to x, we have γ r+ γ r+2 f x ] b m s y f y F y ] γ r+2 dy = As x ] γ r+ As x + B γ r+ f x ] γ r+. Upon simplification, we get x b s y f y F y ] γ r+2 A s x F ] γr+2 + dy = x + x γ r+ γ r+2 f x As x + B ] γ r γ r+2 Differentiating both sides of 2.5 with respect to x, we obtain Simplifying 2.6 results in s x f x ] γ r+2 ] γr+2 = As x + B f x + A γ r+ + γ r+2 + ] γr+2 s x γ r+ γ r+2 A d s x F ] γr+2 + x. 2.6 γ r+ γ r+2 dx f x A s x f x ] 2 + A γ r+ + γ r+2 + s ] x γ r+ γ r+2 A d s x B f x γ r+ γ r+2 dx f x ] 2 = Substituting for A and B in 2.7 and regrouping terms we arrive at d γ r+ + γ r+2 + dx + f x ] 2 = 0. s x d dx s x f x 23

5 International Journal of Statistics and Probability Vol. 4, No. 3; 205 Integrating both sides of the above equation from a to x and using the assumptions s a = f a > 0 and s a = 0, we obtain or s x γ r+ + γ r+2 + s x + f x ] = 0, f x = s x γ r+ + γ r+2 + s x Integrating both sides of 2.8 from a to x and in view of the assumption lim x b s x =, we obtain the distribution function = γ r+ + γ r+2 + s x + ] /γ r+ + γ r+2 +. Remark 2.. For different functions s satisfying the conditions of Theorem 2., one can obtain characterizations of various well-known distributions. We will mention some of these distributions below. For s x = eλγ r+ +γ r+2 +x γ r+ +γ r+2 +, λ > 0, x 0, 2.3 presents = e λx, i.e., X has an exponential distribution. x αγr+ +γ r+2 + β 2 For s x = γ r+ +γ r+2 +, α, β > 0 and x β, 2.3 presents = x/β α, x β, i.e., X has a Pareto distribution. 3 For s x = exp x λ p γ r+ +γ r+2 + γ r+ +γ r+2 +, p, λ > 0, x 0, 2.3 presents = e x λ p, x 0, i.e., X has Weibull also Rayleigh distribution. xµ +ξ σ ] γr++γ r+2+ ξ 4 For s x = γ r+ +γ r+2 +, ξ, σ > 0, x µ 2.3 presents = + ξ ] xµ /ξ σ, x µ, i.e., X has a generalized Pareto distribution. αγ r+ +γ r+2 + x β 5 For s x = γ r+ +γ r+2 +, α > 0, 0 x β, 2.3 presents = α, β x 0 x β,, i.e., X has a power function distribution. 6 For s x = +xp αγ r+ +γ r+2 + expαγ r+ +γ r+2 + x p γ r+ +γ r+2 +, α, p > 0, x 0, 2.3 presents = + x p α e αxp, x 0. For α = p =, X has a generalized gamma distribution and for α =, p = 2, X has a generalized normal distribution. Theorem 2.2. Let X : Ω a, b be a random variable with a twice differentiable distribution function Fx and with density function f x. Let s x be a non-positive, non-decreasing and twice differentiable function on a, b such that for some x 0 a < x 0 < b, f x 0 > 0, lim x b s x = 0 and lim x a s x =. Then implies = E s X r, n, m, k X r + 2, n, m, k = x] = s x, 2.9 ] /2r+ /m+ 2r + m + s x, a x b, 2.0 where, which depends on x 0, is a positive normalizing constant. Proof. Suppose 2.9 holds, then we have 24

6 International Journal of Statistics and Probability Vol. 4, No. 3; 205 x r r + m + s y f y F y ] m F y ] m+ r a ] m+ ] m+ ] F y dy = s x ] m+ r+ 2. Differentiating both sides of 2. with respect to x, we get r r + m + f x ] x m m + s y f y F y ] m F y ] m+ r dy = s x a ] m+ r+ + r + m + s x f x ] m ] m+ r. Upon simplification, we have x a s y f y F y ] m F y ] m+ r dy = s x f x F x ] m ] m+ r+ r r + m r m + s x ] m+ r. 2.2 Differentiating both sides of 2.2 with respect to x and then simplifying, we obtain or d dx s x + 2 r + m + s x ] m f x ] + ms x = 0, m+ d s x dx f x s x f x f x ] m + 2 r + m + ] + m f x = 0. m+ Integrating the above equation from x 0 to x, we arrive at s x = f x ] m ] m+ 2r Integrating both sides of 2.3 from x to b and using the assumption lim x b s x = 0, we obtain and s x = = ] ] m+ 2r, 2r + m + ] /2r+ /m+ 2r + m + s x. 25

7 International Journal of Statistics and Probability Vol. 4, No. 3; 205 Remark 2.2. Similar to Remark 2., for different functions s satisfying the conditions of Theorem 2.2, one can obtain characterizations of various well-known distributions of which, we mention some of them below. e λm+x] 2r+, x > 0, 2.0 presents = e λx, i.e., X has an 7 For s x = 2r+m+ exponential distribution. 8 For sx = 2r+m+ i.e., X has a Pareto distribution. 9 For sx = 2r+m+ X has Weibull also Rayleigh distribution. 0 For sx = 2r+m+ x/β αm+] 2r+, α, β > 0 and x > β, 2.0 presents = x/β α, e x λ p m+ ] 2r+, p, λ > 0, and x > 0, 2.0 presents = e x λ p, i.e., + ξ xµ + ξ xµ ] ξ σ, i.e., X has generalized Pareto distribution. For sx = 2r+m+ x α, β i.e., X has power function distribution. Acknowledgments σ ] m+ 2r+ ξ, ξ, σ > 0, and x > µ, 2.0 presents = x αm+ ] 2r+ β, α > 0, and 0 < x β, 2.0 presents = The authors would like to thank the Editor and anonymous referees for their comments and suggestions. References Ahsanullah, M., & Hamedani, G. G haracterizations of continuous distributions based on conditional expectation of generalized order statistics. ommunications in Statistics - Theory and Methods, 429, Bieniek, M., & Szynal, D haracterizations of distributions via linearity of regression of generalized order statistics. Metrika, 583, Bieniek, M A note on characterizations of distributions by regressions of non-adjacent generalized order statistics. Australian & New Zealand Journal of Statistics, 5, ramer, E., Kamps, U., & Keseling, haracterizations via linear regression of ordered random variables: A unifying approach. ommunications in Statistics - Theory and Methods, 332, Kamps, U A concept of generalized order statistics. Teubner Skripten zur Mathematischen Stochastik Teubner Texts on Mathematical Stochastics]. B. G. Teubner, Stuttgart. opyrights opyright for this article is retained by the authors, with first publication rights granted to the journal. This is an open-access article distributed under the terms and conditions of the reative ommons Attribution license 26