Computation and estimation of reliability for some bivariate copulas with Pareto marginals

ABSTRACT

In this article, we consider an expression for the probability $R=P(Y<X)$, where X and Y are random variables denoting the strength and stress respectively. We assume that X and Y follow two-parameter Pareto distributions and model their dependency by a copula with the dependency parameter θ. We obtain expression for R for four copula functions. We estimate R by plugging in the estimates of the marginal parameters and θ in its expression. The estimates of the marginal parameters are based on the marginal likelihoods. The estimates of θ are obtained from two different methods: one is based on the conditional likelihood and the other on the method of moments using Blomqvist's beta. Results of a simulation study show that the estimates based on Blomqvist's beta are better. We plot the graph of R versus θ to study the effect of dependency on R.

KEYWORDS:

• Blomqvist's beta
• Gaussian error function
• hypergeometric function
• Monte Carlo method
• ProductLog function
• stress–strength models
• two-stage estimation procedure

Disclosure statement

No potential conflict of interest was reported by the authors.