Multivariate lack of memory property, Characterizations, Modelling, Hazard gradient, Singular component, Shock models,

The two parameter Gamma distribution is widely used for modeling lifetime distributions in reliability theory. There is much literature on the inference on the individual parameters of the Gamma distribution, namely the shape parameter k and the scale parameter theta when the other parameter is known. However, usually the reliability professionals have a major interest in making statistical inference about the mean lifetime mu, which equals the product thetak for the Gamma distribution. The problem of inference on the mean mu when both parameters theta and k are unknown has been less attended in the literature for the Gamma distribution. In this paper we review the existing methods for interval estimation of mu. A comparative study in this paper indicates that the existing methods are either too approximate and yield less reliable confidence intervals or are computationally quite complicated and need advanced computing facilities. We propose a new simple method for interval estimation of the Gamma mean and compare its performance with the existing methods. The comparative study showed that the newly proposed computationally simple optimum power normal approximation method works best even for small sample sizes.

In many statistical studies involving failure data, mean residual life function is of prime importance. The bivariate mean residual life function has received relatively less attention in the literature. In this article we use a simple nonparametric estimator for a bivariate mean residual life function. The estimator is shown to be uniformly strongly consistent and, on proper normalization, converges weakly to a zero-mean bivariate Gaussian process. Numerical studies demonstrate that the estimator performs well even for moderate sample sizes. Results are applied to a real dataset related to cancer recurrence. A few supporting results in connection with weak convergence proved in Appendix C may be of independent interest.

We propose a simple close-to-normal approximation to a Weibull random variable (r.v.) and consider the problem of estimation of upper prediction limit (UPL) that includes at leastlout ofmfuture observations from a Weibull distribution at each ofrlocations, based on the proposed approximation and the well-known Box-Cox normal approximation. A comparative study based on Monte Carlo simulations revealed that the normal approximation-based UPLs for Weibull distribution outperform those based on the existing generalized variable (GV) approach. The normal approximation-based UPLs have markedly larger coverage probabilities than GV approach, particularly for small unknown shape parameter where the distribution is highly skewed, and for small sample sizes which are commonly encountered in industrial applications. Results are illustrated with a real dataset for practitioners.

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