Switching record and order statistics via random contractions

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Abstract

In this paper, we investigate a random contraction scheme proposed by Wesolowski and Ahsanullah (2004. Austral. NZ J. Statist. 46, 297–303). Random contraction is used in the record and order statistics settings. Some distributional recurrence relations were found for the probability density functions and distribution functions of record and order statistics. These recurrence relations lead to new characterizations of some distributions. Our results are extensions of works by Pakes (1992. Austral. J. Statist. 34, 323–339), Wesolowski and Ahsanullah (2004. Austral. NZ J. Statist. 46, 297–303) and Alzaid and Ahsanullah (2003. Comm. Statist. Theory Methods 32, 2101–2108). Some new characterizations for the Weibull and Pareto distributions and one special distribution are provided in this study.

Introduction

Let X1,X2,, be a sequence of independent and identically distributed random variables with an absolutely continuous distribution function F and a corresponding probability density function f. For n1, we denote the order statistics of X1,X2,,Xn by X1:nX2:nXn:n. Define U(1)=1,U(n+1)=min{j:j>U(n),Xj>XU(n)}.The sequence {XU(n)} ({U(n)}) is known as upper record statistics (record times). Record and order statistics are of importance in many real-life situations involving data relating to sports, economics, weather and life-tests.

In this study, we consider random contractions of switching record and order statistics, which were discussed by Nevzorov (2001) and Wesolowski and Ahsanullah (2004) for order statistics. The general random contraction setting can be described in the following way: Let U be a random variable with a distribution concentrated on (0,1) or (1,), and let X be a positive random variable which is independent of U. Then the distribution of XU is a random contraction of the distribution of X. Assume that Y is a random variable such that the distributions of X and Y are somehow related and consider the equation Y=dXU,where =d denotes equality in distribution. The main question here is the characterization of distributions when X and/or Y are some order statistics or record values and U is some given contractor. Interesting results in this context have been obtained by Nevzorov (2001) and Wesolowski and Ahsanullah (2004). Similar characterization schemes were considered in the literature for the cases of length-biasing equality and equality of convolutions (see Gather et al. (1998), Pakes, 1992, Pakes, 1994, Pakes et al. (1996), Kotz and Steutel (1988), Riedel and Rossberg (1994), Rossberg et al. (1997), Yeo and Milne (1991)).

For one-sided contraction, Wesolowski and Ahsanullah (2004) have shown that if U has the Power(1,α) distribution for some α>0 and is independent of X1,,Xn which are independent, identically distributed positive random variables, and if Xk:n=dXk:n-1U(respectively,Xk:n=dXk+1:nU)for an arbitrary but fixed k{1,2,,n-1}, then there exists a>0 such that X1 has the Power(a,α/n) (respectively, Power(a,α/k)) distribution. Here and hereafter, the Power(a,α) distribution with positive parameters a and α is defined by the probability density functionf(x)=αa-αxα-1I(0,a)(x),where I is the indicator function.

Wesolowski and Ahsanullah (2004) also considered the two-sided power random contraction Xk:n-1U1=dXk+1:nU2,where Ui has a Power(1,αi) distribution for some αi>0. In this case, the authors showed that if α1/n=α2/k=α, then there exists a>0 such that X1 has a Power(a,α) distribution, and conversely.

Characterizations of the Gumbel distribution using contractions of records and order statistics are presented in recent work by Alzaid and Ahsanullah (2003).

In this paper, we use the Pareto distribution Pareto(α,μ) with the probability density functionf(x)=αμαx-(α+1)I(μ,)(x)(α>0,μ>0)and the Weibull distribution Weibull(λ,β) with the probability density functionf(x)=λβxβ-1exp{-λxβ}I(0,)(x)(λ>0,β>0).

The following section consists of the main results without proofs, and in the third section we give some remarks about distributions of order statistics and record values. Proofs of theorems are presented in the last section.

Section snippets

Main results

First, we give a new characterization of the Pareto distribution. The contracting random variable also has the Pareto distribution with parameters (α,1).

Theorem 1

Let U have the Pareto(α,1) distribution for some α>0, with U being independent of X1,X2, which are positive, independent identically distributed random variables. For any fixed k{1,2,}, the necessary and sufficient condition for a random variable X1 to have the Pareto(α,μ) distribution with some μ>0 is thatXU(k+1)=dXU(k)U.

If U has the Pareto

Distributional recurrences for record and order statistics

Let X1,,Xn be a random sample from an absolutely continuous distribution function F. Let XU(1),XU(2),,XU(n), be record statistics and X1:n,X2:n,,Xn:n be order statistics. Denote by Fk:n and fk:n the distribution function and probability distribution function of the order statistic Xk:n (k=1,,n), respectively. The distribution function Fk(x) and probability density function fk(x) of kth upper record value XU(k) is given byFk(x)=P(XU(k)x)=-xRk-1(u)(k-1)!dF(u)andfk(x)=1(k-1)!Rk-1(x)f(x),

Proofs of Theorems

Proof of Theorem 1

Necessity: Let α,μ>0, let X1 has the Pareto(α,μ) distribution and let U has the Pareto(α,1) distribution. By (2) the probability density function of X1 is f(x)=αμαx-α-1 and R(x)=-αln(μ/x) (x>μ).

From (10) the distribution function of XU(k) is Fk(x)=1-(μ/x)αj=0k-1(-αln(μ/x))j/j! and the distribution function of XU(k)U isP(XU(k)U<x)=1x/μFk(x/u)αu-α-1du=1x/μ[1-(μu/x)αj=0k-1(-αln(μu/x))j/j!]αu-α-1du=1-(μ/x)α-α(μ/x)αj=0k-1(-α)jj!1x/μlnj(μu/x)u-1du.Noting that ddulnj+1(μu/x)j+1=lnj(μu/x)u-1in

Acknowledgements

The authors would like to thank an anonymous referee for detailed comments and suggestions.

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