On some properties of the hazard rate function for compound distributions
- PDF / 287,383 Bytes
- 9 Pages / 439.37 x 666.142 pts Page_size
- 94 Downloads / 227 Views
On some properties of the hazard rate function for compound distributions Hassan S. Bakouch1 · Marcelo Bourguignon2 · Christophe Chesneau3 Received: 17 September 2019 / Accepted: 25 April 2020 © African Mathematical Union and Springer-Verlag GmbH Deutschland, ein Teil von Springer Nature 2020
Abstract The hazard rate function plays a crucial role to extract some informations of several random life phenomena. In this paper, we investigate important properties satisfied by the hazard rate function for a general power series class of distributions, as monotonicity properties, sharp bounds and convexity properties. In particular, we highlight how these properties are related to those of the baseline distribution and the corresponding probability-generating function of discrete power series distributions. Keywords Compound distributions · Hazard rate function · Power series distributions Mathematics Subject Classification 60E05 · 62E15
1 Introduction Several aspects of an absolutely continuous distribution can be seen more clearly from the hazard rate function (hrf) than from either the cumulative distribution or probability density functions. That is, it reveals to be an important function characterizing life phenomena. Some mathematical basics are recalled below. Let X be a random variable with probability density function (pdf) f (x) and cumulative distribution function (cdf) F(x). Then, the hrf of X is defined by the following ratio: f (x)/F(x), where F(x) = 1 − F(x) denotes the survival function. The hrf may be increasing, decreasing, constant, upside-down bathtub (unimodal), bathtub-shaped or indicate a more complicated process. In many applications, there is a qualitative information about the hrf shape, which can help in selecting a specified model.
B
Christophe Chesneau [email protected] Hassan S. Bakouch [email protected] Marcelo Bourguignon [email protected]
1
Department of Mathematics, Faculty of Science, Tanta University, Tanta, Egypt
2
Departamento de Estatística, Universidade Federal do Rio Grande do Norte, Natal, Brazil
3
LMNO, University of Caen-Normandy, Caen, France
123
H. S. Bakouch et al.
In the last few years, several classes of distributions were proposed by compounding some useful lifetime and discrete distributions. The compounding procedure follows the pioneering work of [8]. Then, [3] defined the exponential power series (EPS) class of distributions, which contains as special cases involving the exponential geometric [1], exponential Poisson [6] and exponential logarithmic [16] distributions. Also, [9] defined the Weibull power series (WPS) class which includes the EPS distribution as a sub-model. The WPS distributions can have increasing, decreasing and upside down bathtub hrf. The generalized exponential power series distributions were proposed by [7]. As notable works, [12] studied the extended Weibull power series family of distributions, which includes as special models the EPS and WPS classes of distributions, [2,13] proposed the Birnb
Data Loading...
