A special case of the well known T – X family of distributions proposed by Ayman Alzaatreh et al. (2012) called Pareto – Rayleigh (P – R) distribution is considered. Its scale parameter is estimated using a single order statistic in small samples. Optimal criterion for the choice of single order statistic in a given small sample is worked out. Comparison is made with the corresponding optimal choice of single order statistic with respect to the criterion of asymptotic variance. The results are extended to estimate parametric functions like reliability and hazard rate.
Subba Rao et al. (2015) studied maximum likelihood method estimation, modified maximum likelihood method estimation of the parameter θ for a given shape parameter. He also suggested method of estimation of θ based on single order statistic corresponding to minimum asymptotic variance. In this paper we attempt to propose estimation of θ based on single order statistic with respect to optimum small sample variance along with the asymptotic relative efficiency. The procedure is extended to estimate of reliability function as well as hazard function adopting the invariance property for estimation using order statistics. The paper is organized with Section – 2 dealing with estimation of parameters and Section – 3 dealing with estimation of parametric functions.
II. PARAMETRIC ESTIMATION
We know that the cdf of P – R model is given by
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