Keywords: characterization, conditional expectation, continuous distributions, lower record values
American Journal of Applied Mathematics and Statistics, 2014 2 (1),
pp 79.
DOI: 10.12691/ajams212
Received November 31, 2013; Revised December 02, 2013; Accepted January 03, 2013
Copyright © 2013 Science and Education Publishing. All Rights Reserved.
1. Introduction
The record values were introduced by ^{[1]}. Suppose that is a sequence of independent and identically distributed random variables with common distribution function and common probability density function . Set for . We say is a lower (upper) record values of this sequence if for . By definition is a lower as well as upper record values and with denote the times of lower record values.
Record values are found in many situations of daily life as well as in many statistical applications. Often we are interested in observing new records, e.g. Olympic records. It is also useful in reliability theory, meteorology, hydrology, seismology, mining. For a more specific example, consider the situation of testing the breaking strength of wooden beams as described by ^{[2]}.
For comprehensive accounts of the theory and applications of record values, we refer the readers to ^{[3, 4, 5, 6]}.
2. Objective
Characterizing the distributions via their record statistics has a long history. For excellent review one may refer to [715]^{[7]} amongst others.
The aim of this paper is to characterize a general class of distributions via the contrast of the conditional expectation of function of lower record statistics, conditioned on nonadjacent lower record statistics.
3. Method
Let be the first lower record statistics from a population whose probability density function is and the distribution function is. Let Then the of ,is
 (3.1) 
and the joint of two lower records and, , is
 (3.2) 
where .
Then the conditional of given is
 (3.3) 
4. Characterization Result
Theorem: Let be an absolutely continuous random variable with the and the on the support, where and may be finite or infinite. Then for
 (4.1) 
if and only if
 (4.2) 
where are real numbers satisfying , for some and is a nonincreasing and differentiable function of such that is a .
Proof: First we will prove (4.2) implies (4.1). We have from ^{[14]} for
Therefore,
 (4.3) 
hence the ‘if’ part.
To prove the sufficiency part, we have
 (4.4) 
or,
 (4.5) 
Integrating left hand side of (4.5) by parts, we get
 (4.6) 
That is,
 (4.7) 
Now from (3.3), we have
Therefore,
 (4.8) 
Comparing (4.7) and (4.8), we get
implying
and hence the Theorem.
Remark: Putting and in Theorem , we get the characterizing result as obtained by ^{[14]}.
Table 4.1. Examples based on the distribution function F(x)=e^{ah(x)}, a>0
5. Discussion
The purpose of this paper was to characterize a general classs of probability distribution through the conditional expectation based on lower record statistics conditioned on nonadjacent lower record statistics using the contrast technique. We hope that findings of this paper will useful for the researcher in various fields. Further advancement of research in distribution theory, lower record theory and their application.
References
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 In article  

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