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On generalized order statistics from Kumaraswamy distribution.

1. Introduction

The distributions of the product and ratio of random variables find an important place in the literature and much work is done when the random variables are independent and come from a particular probability distribution.

If the random variables [X.sub.1], [X.sub.2], ..., [X.sub.n] are arranged in ascending order of magnitudes and then written as [X.sub.(1)] [less than or equal to] [X.sub.(2)],...[less than or equal to] [X.sub.(n)], then [X.sub.(i)] is called the [i.sup.th] order statistics (i=1,2, ...,n) and the ordered random variables are necessarily dependent. The distribution of product and quotient of the extreme order statistics and that of consecutive order statistics are useful in ranking and selection problems. Subrahmaniam (16) has made the study of product and quotient of order statistics from uniform distribution and exponential distribution, whereas Malik and Trudel (14) studied the cases when the order statistics are from Pareto, power and Weibull distributions.Recently the author (21) has studied order statistics from Kumaraswami distribution.

The subject of order statistics has been further generalized and the concept of generalized order statistics is introduced and studied by Kamps in a series of papers and books (26), (27), (28), (29). The order statistics, record values and sequential order statistics are special cases of generalized order statistics. This concept is widely studied by many research workers namely Ahsanullah (17), (18), (19), (20), El-Baset, Ahmed and Al-Matrofi (2), Cramer and Kamps (7), (8), Cramer, Kamps and Rychlik (9), (10), (11), (12), Kamps and Cramer (29) and Reiess (25).

In the present paper we shall obtain the joint distribution, distribution of product and distribution of ratio of two generalized order statistics from the family of distributions known as Kumaraswamy distribution (24).

2. Definitions

(i) Generalized Order Statistics

Let F(x) denote an absolutely continuous distribution function with density function f(x) and [X.sub.1;n,m,k], [X.sub.2;n,m,k],..., [X.sub.n;n,m,k] (k[less than or equal to]1, m is a real number) be 'n' generalized order statistics. Then the joint probability density function (p.d.f.) [f.sub.1, ..., n] ([x.sub.1], ..., [x.sub.n]) can be written as (26)

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] (1)

where,

[[gamma].sub.j] = k + (n-j)(m + 1) and f(x) = [[dF(x)]/[dx]]

If m=0 and k=1 it gives the joint p.d.f. of 'n' ordinary order statistics [X.sub.1,n] [less than or equal to] ... [less than or equal to] [X.sub.n,n]. If m = -1 and k = 1, it gives the joint p.d.f. of the first 'n' upper records of the independent and identically distributed random variables. Various distributional properties of generalized order statistics are studied by Kamps (27) and that of record values by Ahsanullah (17), (19), Arnold, Balakrishnan and Nagaraja (4) and Raqab (24).

Further integrating out [x.sub.1], ..., [x.sub.[r-1]], [x.sub.[r+1]], ..., [x.sub.n] from (1), we get p.d.f. [f.sub.r,n,m,k] of [X.sub.r;n,m,k] (26) as

[f.sub.r,n,m,k](x) = [[C.sub.r]/[(r-1)!]][[1-F([x.sub.r])].sup.[[gamma].sub.r-1]][g.sub.m.sup.[r-1]][F([x.sub.r])] f([x.sub.r]) (2)

where

[C.sub.r] = [r.[product].[j=1]][[gamma].sub.j], [[gamma].sub.r] = k + (n-r)(m + 1) (3)

and

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] (4)

Since [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]

We shall write

[g.sub.m] (x) = [1/[m + 1]][1-[(1-x).sup.[m + 1]]] for all x [member of] (0,1) and for all m

with

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] (5)

Also the joint distribution of of ith and jth generalized order statistics is given by

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] (6)

symbols [C.sub.j], [[gamma].sub.j], [g.sub.m](x) are as defined above.

The result (6), on taking m=0 and k=1 reduces to the joint p.d.f. of ith and jth order statistics as given in David (13).

(ii) Kumaraswamy Distribution

In this distribution, the probability density function of a random variable X is given by

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] (7)

with the cumulative density function (or distribution function), given as

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] (8)

In probability theory Kumaraswamy's double bounded distribution is as versatile as the beta distribution, but much simpler to use especially in simulation studies as it has a simple closed form for both the p.d.f. and c.d.f.

(iii) The Mellin Transform

Let ([X.sub.1], [X.sub.2]) be a two dimensional random variable having the joint probability density function f([x.sub.1], [x.sub.2]) that is positive in the first quadrant and zero elsewhere. The Mellin transform of f([x.sub.1], [x.sub.2]) is defined by Fox (5) as

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] (9)

with the inverse

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] (10)

under the appropriate conditions discussed by Fox.

In this paper, we are interested in the following two particular cases [16].

If Y = [X.sub.1] [X.sub.2], then h(y), the p.d.f. of Y, has the Mellin transform

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] (11)

and if Z = [X.sub.1]/[X.sub.2], then g(z), the p. d. f. of Z, has the Mellin transform

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] (12)

(iv) Fox H- function

We shall require the following definition of Fox H-function [6]

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] (13)

where [omega]=[square root of (-1)], x( [not equal to] 0) is a complex variable and [x.sup.s] = exp[s{log |x| + [omega]arg x}],

[theta](s) = [[m.[product].[j=1]][GAMMA]([b.sub.j]-[[beta].sub.j]s)[n.[product].[j=1]][GAMMA](1-[a.sub.j] + [a.sub.j]s)]/[[q.[product].[j=[m+1]]][GAMMA](1-[b.sub.j] + [[beta].sub.j]s)[p.[product].[j=[n+1]]][GAMMA]([a.sub.j]-[a.sub.j]s)]] (14)

m, n, p and q are non-negative integers satisfying, 0 [less than or equal to] n [less than or equal to] p, 1 [less than or equal to] m [less than or equal to] q; [[alpha].sub.j] (j = 1, ..., p) and [[beta].sub.j] (j = 1, ..., q) are assumed to be positive quantities for standardization purpose.

The definition of the H-function given by (13) will however have meaning even if some of these quantities are zero, giving us in tern simple transformation formulas. The nature of contour L, a set of sufficient conditions for the convergence of this integral, the asymptotic expansion, some of its properties and special cases can be referred to in the book by Srivastava, Gupta and Goyal (15).

3. Joint Distribution and Distributions of Product and Ratio of Two Generalized Order Statistics

Theorem 1. Let [X.sub.i;n,m,k] and [X.sub.j;n,m,k] be ith and jth generalized order statistics with (i<j), based on a random sample of size n from the Kumaraswamy distribution. The joint p.d.f of these generalized order statistics is given by:

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] (15)

provided that a, b > 0, 1 [less than or equal to] i < j [less than or equal to] n, m is a real number, k [greater than or equal to] 1 and n > 1, 0 [less than or equal to] [x.sub.i] < [x.sub.j] [less than or equal to] 1 and [C.sub.j] and [[gamma].sub.j] are defined by (3).

Proof. The result can easily be established on substituting the values of f(x), F(x), and [g.sub.m](x) from equations (7), (8) and (4) respectively in the equation (6) and expressing the values of [[[g.sub.m](F([X.sub.i]))].sup.[i-1]] and [[[g.sub.m](F([X.sub.j]))-[g.sub.m](F([X.sub.i]))].sup.[j-i-1]] in their series forms.

Theorem 2. Let [X.sub.i;n,m,k] and [X.sub.j;n,m,k] denote the ith and jth generalized order statistics from a random sample of size 'n' drawn from Kumaraswamy distribution defined by (7), then the probability density function of the product

Y = [X.sub.i;n,m,k][X.sub.j;n,m,k] (16)

and the ratio

Z = [[X.sub.i;n,m,k]/[X.sub.j;n,m,k]] (17)

are given by

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] (18)

and

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] (19)

where H[z] is the Fox H-function defined by (14) and j-i + [l.sub.1]-[l.sub.2] >0, a > 0, b>0, 1[less than or equal to]i < j[less than or equal to]n, k[greater than or equal to]1, m and k are real numbers and the symbols [[gamma].sub.j] and [C.sub.j] are defined by (3).

Proof. To find the p.d.f. of the product Y, we take double Mellin transform of eq.(15) and evaluate the integrals with the help of known result [1, p.311, eq.(31)]. Now using (11) we obtain Mellin transform of g(y) as

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] (20)

where [beta][a,b] is usual beta function. Taking inverse Mellin transform of above equation w.r.t. 's' and using a known result [22, p.115] we arrive at (23).

To obtain the p.d.f. of the ratio i.e. [h.sub.i,j;n,m,k](z), we use (12) with (9) and (15) to get Mellin transform of h(z) as

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] (21)

Taking Mellin inversion of the above result and interpretating with the help of definition of H-function given by (13), we get the desired result (19).

4. Special Cases

Corollary 1. If we take a = b = 1 in Theorems 1 and 2, we get the joint p.d.f. and p.d.f. of product and ratio of ith and jth generalized order statistics from uniform distribution. The results are given by

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] (22)

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] (23)

and

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] (24)

where [G.sub.[2,2].sup.[1,1]][z] is Meijer G function [3].

Corollary 2. If we take j = i + 1 in Theorems 1 and 2, we get the joint distribution and distribution of product and ratio of consecutive generalized order statistics based on a random sample of size n from the Kumaraswamy distribution and are given by

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] (25)

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] (26)

and

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] (27)

Corollary 3. If we take i = 1, j = n in Theorems 1 and 2, we get the joint distribution and distribution of product and ratio of the extreme generalized order statistics based on a random sample of size n, from Kumaraswamy distribution, and are given by

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] (28)

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] (29)

and

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] (30)

Corollary 4. If we take n to be odd say 2p+1 then putting i = p+1 and j = 2p+1 in Theorem 2. we get the p.d.f. of the product and ratio of peak to median of a random sample of size 2p+1 of generalized order statistics as

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] (31)

and

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] (32)

Remark. If we take m = 0 and k = 1 in Theorems 1 and 2, then generalized order statistics reduces into order statistics and we get the joint distribution and distribution of product and ratio of order statistics [X.sub.i,n] and [X.sub.n,n] from a sample of size n from Kumaraswamy distribution as obtained recently by the author (21).

Acknowledgements

The author is thankful to anonymous referee for useful suggestions which led to the present form of the paper.

References

(1) A. Erdelyi, W. Magnus, F. Oberhetting and, F. G. Tricomi, Table of Integral Transform, McGraw-Hall, New York, Toronto and London, 1954.

(2) Abd. EI. Baset, A. Ahmad and, Bakheet N. Al. Matrofi, Reccurence relation for moments and conditional moments of generalized order statistics., Univ. of Sharjah J. of Pure and Appl. Sci., 3(3)(2006), 61-81.

(3) A. M. Mathai and R. K. Saxena, Generalized Hypergeometric Functions with Applications in Statistics and Physical Sciences. Springer-Verlag, Lecture Notes No. 348, Heildeliteerg, 1973.

(4) B. C. Arnold, N. Balakrishnan and H. N. Nagaraja, Records, John Wiley, New York, 1998.

(5) C. Fox, Some applications of mellin transforms to the theory of bivariate statistical distributions, Proc. Cambridge Philos. Soc., 53(1957), 620-628.

(6) C. Fox., The G-and H-function as symmetrical Fourier kernels, Trans. Amer. Math. Soc. 98(1961), 395-429.

(7) E. Cramer and U. Kamps, Relations for expectations of functions of generalized order statistics, J. Statist. Plann. and Inference, 89(1-2)(2000), 79-89.

(8) E. Cramer and U. Kamps, Marginal distributions of sequential and generalized order statistics, Metrika, 53(3)(2003), 289-310.

(9) E. Cramer, U. Kamps and, T. Rychlik, Moments of generalized order statistics, In: H. Langseth and B. Lindqvist (Hrsg.), Third International Conference on Mathematical Methods in Reliability, Methodology and Practice, S(2002), 165-168, Trondheim: NTNU.

(10) E. Cramer, U. Kamps and, T. Rychlik, On the existence of moments of generalized order statistics, Statist. and aprobability Letters, 59(2002), 397-404.

(11) E. Cramer, U. Kamps and, T. Rychlik, Evaluations of expected generalised order statistics in various scale units, Applicationes Mathematicae, 29(3)(2002), 285-295.

(12) E. Cramer, U. Kamps and, T. Rychlik, Unimodality of uniform generalised order statistics with application to mean bounds, Ann. Inst. Statist. Math., 56(1)(2004), 183-192.

(13) H. A. David, Order Statistics, John Wiley, New York, 1981.

(14) H. Malik and R. Trudel, Distributions of product and quotient of order statistics, University of Guelph statistical Series, 1975-30(1976), 1-25.

(15) H. M. Srivastava, K. C. Gupta and S. P. Goyal, The H-Function of One and Two Variables with Applications, South Asian Publishers, New Delhi and Madras, 1982.

(16) K. Subrahmaniam, On some applications of Mellin transforms to statistics: Dependent random variables, SIAM J. Appl. Math., 19(4)(1970), 658-662.

(17) M. Ahsanullah, Record Statistics, Nova Science Publishers, Inc. Commack, New York, 1995.

(18) M. Ahsanullah, Generalised order statistics from exponential distribution, Journal of Statistical Planning and Inference, 85(2000), 85-91.

(19) M. Ahsanullah, Record Values Theory and Application, University Press of America, London and Maryland, 2004.

(20) M. Ahsanullah, The generalised order statistics from exponential distribution, Pak. J. Statist., 22(2)(2006), 121-128.

(21) M. Garg, On distribution of order statistics from Kumaraswami distribution, Kyungpook Math. J. (to appear).

(22) M. D. Springer, The Algebra of Random Variables, John Wiley & Sons. New York, 1979.

(23) P. Kumaraswamy, A generalised probability density function for double bounded random variables, J. of Hydrology, 46(1980), 79-88.

(24) Raqab, M. Z., Inferences for generalised exponential distribution based on record statistics, J. Statist. Plann. and Inference, 104(2)(2002) 339-350.

(25) R. D. Reiess, Approximate Distributions of Order Statistics, Springer, New York, 1989.

(26) U. Kamps, A Concept of Generalized Order Statistics, B. G. Teubner, Stuttgart, Germany, 1995.

(27) U. Kamps, A concept of generalised order statistics, J. Statist. Plann. Inference, 48(1995), 1-23.

(28) U. Kamps, Subranges of generalised order statistics from exponential distributions, Fasciculi Mathematici, 28(1998), 63-70.

(29) U. Kamps, and E. Cramer, On the distributions of generalised order statistics, Statist. 35(2001), 269-280.

Mridula Garg [dagger]

Department of Mathematics,University of Rajasthan, Jaipur-302055, India

Received June 25, 2007, Accepted March 31, 2009.

* 2000 Mathematics Subject Classification. 60E99, 33C60.

[dagger] E-mail: gargmridula@gmail.com
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Author:Garg, Mridula
Publication:Tamsui Oxford Journal of Mathematical Sciences
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