Estimation of the stress-strength parameter, R = Pr(X < Y), is perhaps one of the challenging concepts in the reliability analysis. The estimation of R often criticized for its lack of stability and robustness against the presence of outliers and extreme values. The issue of estimating R under the presence of outliers is considered in this contribution for independently distributed random variables X and Y by the Pareto-based models. It is assumed that X has the Pareto distribution in the presence of outliers, whereas the random variable Y follows uncontaminated Pareto distribution. Under various assumptions on the parameters of the model, the maximum likelihood, method of moments, least squares, and modified maximum likelihood estimators are obtained. The shrinkage estimate of the stress-strength reliability parameter is also derived for each case using a prior guess, R0. We conduct a Monte Carlo simulation study to compare the proposed methods of estimation. Finally, the performance of the postulated methodology is illustrated by analyzing two real-world datasets in the physical and insurance studies.

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