On General Exponential Weight Functions and Variation Phenomenon
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On General Exponential Weight Functions and Variation Phenomenon C´elestin C. Kokonendji and Aboubacar Y. Tour´e Universit´e Bourgogne Franche-Comt´e, Besan¸con, France
Rahma Abid University of Sfax, Sfax, Tunisia and University Paris-Dauphine Tunis, Tunis, Tunisia Abstract General weighted exponential distributions including modified exponential ones are widely used with great ability in statistical applications, particularly in reliability. In this paper, we investigate full exponential weight functions and their extensions from any nonnegative continuous reference weighted distribution. Several properties and their connections with the recent variation phenomenon are then established. In particular, characterizations, weightening operations and dual distributions are set forward. Illustrative examples and concluding remarks are extensively discussed. AMS (2000) subject classification. 60E05, 62E10, 62E15.. Keywords and phrases. Dual distribution, equi-variation, exponential dispersion models, exponential distribution, life distribution, over-variation, reliability, under-variation, weighted distribution.
1 Introduction The variation phenomenon was recently introduced by Abid et al. (2020) as a measure of departure from the standard exponential law of nonnegative continuous distributions. This measure is dedicated by the exponential variation index, so-called Jørgensen variation index, for nonnegative continuous random variable Y on [0, ∞) and defined as the ratio of variance to the squared expectation. More precisely, this positive quantity is written by VI(Y ) :=
VarY 1, (EY )2
(1)
i.e., Y is over-, equi- and under-varied compared to the exponential random variable with the same expectation EY if VarY > (EY )2 , VarY = (EY )2 and
2
C. C. Kokonendji et al.
VarY < (EY )2 , respectively. It can be viewed as the squared of the standard coefficient of variation. Within the framework of reliability, an increasing/decreasing failure rate on the average (IFRA/DFRA) distribution, which stands also for the corresponding IFR/DFR one, implies VI ≤ (≥) 1; that is under- (over-) variation in the large sense (i.e., including equi-variation). See, e.g., Barlow and Proschan (1981) in the sense of the coefficient of variation. Note that IFR distributions are more frequent than DFR, but over-varied data are not the most frequent situation in reliability. We briefly mention here that the relative variation index, namely RVI, has been also introduced as the ratio of two VIs for changing the standard reference. We refer to Kokonendji et al. (2020b) for more details in multivariate setup. The probability density function (pdf) of the standard reference which is the exponential random variable X ∼ E(μ) with parameter μ > 0 is (2) denotes the indicator function of any given event A. Its mean and where variance are equal to 1/μ and 1/μ2 , respectively. However, it is always equivaried. It may happen for the sample variance to be greater or smaller than the squared sample mean, which are referred to as over-variation and undervariation
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