Switching record and order statistics via random contractions
Introduction
Let , 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 , we denote the order statistics of by . Define The sequence () 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 or , 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 where 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 distribution for some and is independent of which are independent, identically distributed positive random variables, and if for an arbitrary but fixed , then there exists such that has the (respectively, distribution. Here and hereafter, the distribution with positive parameters a and is defined by the probability density functionwhere I is the indicator function.
Wesolowski and Ahsanullah (2004) also considered the two-sided power random contraction where has a distribution for some . In this case, the authors showed that if , then there exists such that has a 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 with the probability density functionand the Weibull distribution with the probability density function
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 . Theorem 1 Let U have the distribution for some , with U being independent of which are positive, independent identically distributed random variables. For any fixed , the necessary and sufficient condition for a random variable to have the distribution with some is that
If U has the
Distributional recurrences for record and order statistics
Let be a random sample from an absolutely continuous distribution function F. Let be record statistics and be order statistics. Denote by and the distribution function and probability distribution function of the order statistic , respectively. The distribution function and probability density function of th upper record value is given byand
Proofs of Theorems
Proof of Theorem 1 Necessity: Let , let has the distribution and let U has the distribution. By (2) the probability density function of is and . From (10) the distribution function of is and the distribution function of isNoting that in
Acknowledgements
The authors would like to thank an anonymous referee for detailed comments and suggestions.
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