Partially parametric interval estimation of Pr{Y>X}

https://doi.org/10.1016/j.csda.2005.12.007Get rights and content

Abstract

Let X and Y be two independent continuous random variables. Three techniques to obtain confidence intervals for ρ=Pr{Y>X} are discussed in a partially parametric framework. One method relies on the asymptotic normality of an estimator for ρ; the remaining methods involve empirical likelihood and combine it with maximum likelihood estimation and with full parametric likelihood, respectively. Finite-sample accuracy of the confidence intervals is assessed through a simulation study. An illustration is given using a data set on the detection of carriers of Duchenne Muscular Dystrophy.

Section snippets

Introduction and motivation

Let X and Y be two independent random variables (RVs) with continuous cumulative distribution functions (CDFs) G and F, respectively. A wide range of problems, especially in engineering and medical research, involves making inference about the quantity ρ=Pr{Y>X}.

In reliability contexts, evaluation of, and inference on, ρ is known as the stress-strength problem (see Kotz et al., 2003, as a general reference). Take X to be the stress potentially affecting a component, and take Y to be the

Methodologies

Consider a general parametric model {F(y;θ);θ∈Θ} for the variable Y. Here, F(y;θ)=Pr{Y⩽y;θ} denotes the CDF, which depends on an unknown parameter θ belonging to some set Θ⊆Rq,q⩾1. Let S(y;θ)=1-F(y;θ) denote the survival function corresponding to F(y;θ). We do not make any parametric assumption about the distribution of the variable X. We only assume that X is independent of Y. In this setting, ρ=Pr{Y>X;θ}.

Let X1,X2,…,Xn be a random sample of size n from X and Y1,Y2,…,Ym a random sample of size

Some simulation results

In this section, we report the results of a simulation study carried out to assess the finite-sample accuracy of the confidence intervals obtained by using the techniques discussed in Section 2.

For three levels of nominal coverage 1-γ, Table 1, Table 2, Table 3 give the estimated coverage probabilities of the confidence intervals based on the profile combined log likelihood ratio lP(ρ) (CL), the adjusted empirical log likelihood ratio l˜(ρ) (AEL), and the asymptotic normality of tρ^, with t(·)

An example

Duchenne muscular dystrophy is one of the most prevalent types of muscular dystrophy and is characterized by rapid progression of muscle degeneration that occurs early in life. It is a genetically transmitted disease, which is passed from a mother to her children. Unfortunately, no cure has yet been discovered, so that the screening of females who could be potential carriers is of great importance.

Andrews and Herzberg (1985) report some data collected during a program run at the Hospital for

Conclusions

In this paper, we have considered the problem of making inference on ρ in a partially parametric framework. In particular, we have discussed three techniques to obtain confidence intervals. Simulation results have shown a substantial comparability of the three procedures, at least so far as accuracy of the corresponding confidence intervals is concerned. Overall, we have observed a good agreement between nominal and actual coverages. Confidence intervals seem to have comparable length, too.

Acknowledgements

This work was supported by MIUR under Grant No. 133820, 2003.

References (17)

There are more references available in the full text version of this article.
View full text
Copyright © 2006 Elsevier B.V. All rights reserved.