Preservation of ageing classes in deterioration models with independent increments

Carmen Sangüesa,Ji Hwan Cha,Francisco German Badía

doi:10.1186/1029-242x-2014-200

Carmen Sangüesa, Ji Hwan Cha + Show 1 more

Open Access

https://doi.org/10.1186/1029-242x-2014-200

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Abstract

Abstract In the present paper we consider ageing properties in a deterioration model in which the stochastic process measuring deterioration is a process with independent increments. Preservation of increasing and decreasing failure rates, as well as decreasing reversed hazard rate, is considered. We also take into account the preservation of log-concave and log-convex densities. Our main results are based on technical results concerning preservation of log-concave and log-convex functions by positive linear operators, and they include the study of stochastic ordering properties among the random variables in the process. MSC:60G51, 60E15, 60K10, 26A51.

Highlights

Deterioration models belong to the topics of interest in reliability theory
The so-called shock models are appropriate if deterioration is caused due to external shocks occurring at certain instants in time
Our aim in this paper is to address this question for processes with independent increments, including Lévy processes

Summary

Introduction

Deterioration models belong to the topics of interest in reliability theory. They aim to describe how a mechanism deteriorates with age. If FY is log-concave, the random variable is said to be increasing failure rate (IFR), whereas if it is log-convex, we have the decreasing failure rate property (DFR). We use the representation given in ( ) and apply techniques based on the preservation of log-convexity and log-concavity by positive linear operators (see [ , ]) These techniques involve the study of stochastic order properties of the random variables in the process. ) that for a given process (X(t), t ≥ ) in the IPSI class, the time-transformed process (X(a (t)), t ≥ ), with a being an increasing and convex function, belongs to the IPII class, whereas if a is increasing and concave, the process belongs to the IPDI class We will use this fact in order to obtain results concerning non-homogeneous Poisson processes J∗ := {t ≥ | P(X(t) ∈ J) > } is an interval

Proof To show the assertion we will prove that

We have

Xi is a compound

Note that