Estimation of Stress-Strength Reliability for Some General Bivariate Distributions

Abstract

In the reliability literature, the term “stress-strength” mainly refers to a component having a random strength X₁ , and is exposed to a random stress X₂. The component works (does not fail), whenever X_1>X_2. Thus R=P(X_1>X_2) is a measure of component reliability, which is commonly referred to as the "stress-strength reliability parameter". However, X₁ and X₂ do not necessarily always represent the strength of a component and the stress imposed on it. Actually, X₁ and X₂ may represent other variables of interest. For example, X₁ and X₂ may represent two types of a certain product, or two types of medical treatment of a certain disease. R in these cases is a measure of comparison. The stress-strength models have been widely used in physics and engineering, and are spread across different disciplines, such as quality control, genetics, psychology, and economics. Many researchers have derived and estimated R when X₁ and X₂ are independent variables belonging to the same univariate family of distributions. However, there are practical situations in which X₁ and X₂ are associated in some way or another. So, in this thesis we study R and its estimation when X₁ and X₂ are not independent but are associated. The model we use for describing the association between X₁ and X₂, is the Farlie-Gumbel-Morgenstern copula model, which is one of the most common copulas. This model is applicable in many practical situations, for example in a hydrology X₁ and X₂ may be represent rainfall intensity and depth (see Barmalzan et. al. (2017)). We assume that X₁ and X₂, have a bivariate Farlie-Gumbel-Morgenstern distribution where the marginal distribution functions are members of some general families, namely: the general exponential form family (GEF) and the general inverse exponential form family (IEF). Many distributions in the literature belong to these two families of distributions. So, consequently, the results obtained in this thesis are applicable to a great number of distributions.