This paper deals with the estimation of stress-strength reliability parameter, R = P (Y < X ), based on progressively type II censored samples when stress, strength are two independent generalized Pareto random variables. The maximum likelihood estimators, their asymptotic distributions, asymptotic confidence intervals, bootstrap based confidence intervals and Bayes estimators are derived for R. Using Monte Carlo simulations, the MSE, Bayes risk estimators, credible sets and coverage probabilities are computed and compared.
In this paper, we investigate a multivariate regression model with multivariate heavy-tailed stable errors. Since in heavy-tailed data, especially in multivariate stable distributions, some moments do not exist; classical multivariate regression methods do not perform well. We suggest using an effective property of the existence of some moments of order statistics stable distribution. We propose a method for trimming the data set using this property. Then, we estimate the regression coefficients based on the rest of the ordered data. We calculate the trimmed data set based on the error’s tail index and skewness parameters. Also, we analytically compute the bias and variance of the introduced estimators of the regression parameters. Finally, we study the performance of the proposed methods with ordinary least squares via a simulation study and a real data set.
Classical regression approaches are not robust when errors are heavy-tailed or asymmetric. That may be due to the non-existence of the mean or variance of the error distribution. Estimation based on trimmed data, which ignored outlier or leverage points, has an old history and frequently used. This procedure chooses fixed cut-off points. In this work, we use this idea recently applied for initial estimates of regression coefficients with heavy-tailed stable errors. We propose an effective procedure to calculate the cut-off points based on the tail index and skewness parameters of errors. We use the property of the existence of some moments of stable distribution order statistics. Data are trimmed based on ordered residuals of a least square regression. However, the trimmed data’s optimal number is determined based on the number of error order statistics whose variance exists. Then, we use the rest of the ordered data to estimate the regression coefficients. Based on these order statistics’ joint distribution, we analytically compute the bias and variance of the introduced estimator of regression parameters that was impossible for regression with stable errors.
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