Partially parametric interval estimation of
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 .
In reliability contexts, evaluation of, and inference on, 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 for the variable Y. Here, denotes the CDF, which depends on an unknown parameter belonging to some set . Let denote the survival function corresponding to . 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, .
Let be a random sample of size n from X and 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 , Table 1, Table 2, Table 3 give the estimated coverage probabilities of the confidence intervals based on the profile combined log likelihood ratio (CL), the adjusted empirical log likelihood ratio (AEL), and the asymptotic normality of , with
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 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.
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