A Study on Mixture of Exponentiated Rayleigh and Exponentiated Exponential Distributions Based on Order Statistics

Mixtures of lifetime distributions occur when two different causes of failure arc present, each with the same parametric form of lifetime distributions. This paper is considered with the mixture model of exponentiated Rayleigh and exponentiated exponential distributions. The author＇s objectives are finding the statistical properties of the model and estimating the parameters of the model by using point estimation and interval estimation methods. First, some properties of the model with some graphs of the density function are discussed. Next, the maximum likelihood method of estimation is used for estimating scale and shape parameters of the model. Estimating the parameters is studied under complete and type II censored samples for different sample sizes. Asymptotic Fisher information matrix of the estimators for complete samples is founded with different sample sizes. The asymptotic variances of the maximum likelihood estimates are derived. Based on the asymptotic variances of the maximum likelihood estimates, interval estimates of the parameters are obtained. Some of the equations in this paper are solved by using numerical iteration such as Newton Raphson method by using Mathematica 7.0. The performance of findings in the paper is showed by demonstrating some numerical illustrations through Monte Carlo simulation study based on absolute relative bias and mean square error.

Sharp Error Estimate for Maximum Likelihood Estimator of Nonstationary Diffusion Processes

We consider the maximum likelihood estimator of the unknown parameter in aclass of nonstationary diffusion processes. We give further a precise estimate for the error of theestimator.