Notes on Entropy for Concomitants of Record Values in Farlie-Gumbel-Morgenstern (FGM) Family

Saeid Tahmasebi

doi:10.6339/jds.2013.11(1).1104

Abstract

Let {(Xi , Yi), i ≥ 1} be a sequence of bivariate random variables from a continuous distribution. If {Rn, n ≥ 1} is the sequence of record values in the sequence of X’s, then the Y which corresponds with the nth record will be called the concomitant of the nth-record, denoted by R[n] . In FGM family, we determine the amount of information contained in R[n] and compare it with amount of information given in Rn. Also, we show that the Kullback-Leibler distance among the concomitants of record values is distribution-free. Finally, we provide some numerical results of mutual information and Pearson correlation coefficient for measuring the amount of dependency between Rn and R[n] in the copula model of FGM family.

Highlights

Let (X1, Y1), (X2, Y2), · · · be a sequence of bivariate random variables from a continuous distribution
If {Rn, n ≥ 1} is the sequence of record values in the sequence of X’s, the Y which corresponds with the nth-record will be called the concomitant of the nth-record, denoted by R[n]
The concomitants of record values arise in a wide variety of practical experiments such as industrial stress testing, life time experiments, meteorological analysis, sporting matches and some other experimental fields

Summary

Introduction

Let (X1, Y1), (X2, Y2), · · · be a sequence of bivariate random variables from a continuous distribution. The cumulative distribution function (cdf) for the FGM family is given by Johnson and Kotz (1975) as FX,Y (x, y) = FX (x)FY (y)[1 + α(1 − FX (x))(1 − FY (y))], −1 ≤ α ≤ 1, (1). Houchens (1984) has obtained the probability density function (pdf) of concomitant of nth-record value for n ≥ 1 arising in (1) as. In. Section 2, we determine the amount of information contained in R[n] and compare it with amount of information given in Rn. In Section 3, we show that the Kullback-Leibler distance between concomitants of nth- and mth- record values in FGM family is free from marginal distributions. We present analytical expressions and some numerical results of mutual information and Pearson correlation coefficient between Rn and R[n] in the copula model of FGM family

Entropy for Concomitants of Record Values in FGM Family

Kullback-Leibler Distance