Characterization of Distribution by Conditional Expectation of Lower Record Values
1Department of Statistics and Operations Research, Aligarh Muslim University, Aligarh, India
It is widely known that the problem of characterizing a distribution an important problem which has recently attracted the attention of many researchers. Thus various characterizations have been established in many directions. In this paper, a general form of continuous probability distribution is characterized through conditional expectation of contrast of lower record statistics, conditioned on a non-adjacent record statistics and some of its deductions are also discussed.
Keywords: characterization, conditional expectation, continuous distributions, lower record values
American Journal of Applied Mathematics and Statistics, 2014 2 (1),
Received November 31, 2013; Revised December 02, 2013; Accepted January 03, 2013Copyright: © 2013 Science and Education Publishing. All Rights Reserved.
Cite this article:
- Khan, M. I., and M. Faizan. "Characterization of Distribution by Conditional Expectation of Lower Record Values." American Journal of Applied Mathematics and Statistics 2.1 (2014): 7-9.
- Khan, M. I. , & Faizan, M. (2014). Characterization of Distribution by Conditional Expectation of Lower Record Values. American Journal of Applied Mathematics and Statistics, 2(1), 7-9.
- Khan, M. I., and M. Faizan. "Characterization of Distribution by Conditional Expectation of Lower Record Values." American Journal of Applied Mathematics and Statistics 2, no. 1 (2014): 7-9.
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The record values were introduced by . 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 .
Characterizing the distributions via their record statistics has a long history. For excellent review one may refer to [7-15] 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 non-adjacent lower record statistics.
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
and the joint of two lower records and, , is
Then the conditional of given is
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
if and only if
where are real numbers satisfying , for some and is a non-increasing and differentiable function of such that is a .
Proof: First we will prove (4.2) implies (4.1). We have from  for
hence the ‘if’ part.
To prove the sufficiency part, we have
Integrating left hand side of (4.5) by parts, we get
Now from (3.3), we have
Comparing (4.7) and (4.8), we get
and hence the Theorem.
Remark: Putting and in Theorem , we get the characterizing result as obtained by .
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 non-adjacent 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.
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