Classical estimation of hazard rate and mean residual life functions of Pareto distribution

Abstract

In this paper we find the maximum likelihood estimates (MLEs) of hazard rate and mean residual life functions (MRLF) of Pareto distribution, their asymptotic non degenerate distribution, exact distribution and moments. We also discuss the uniformly minimum variance unbiased estimate (UMVUE) of hazard rate function and MRLF. Finally, two numerical examples with simulated data and real data set, are presented to illustrate the proposed estimates.     

The umvue and bayes estimate of reliability of mixed failure time distribution

We study the reliability estimates of the non-standard mixture of degenerate (degenerated at zero) and exponential distributions. The Uniformly Minimum Variance Unbiased Estimator (UMVUE) and Bayes estimator of the reliability for some selective prior when the mixing proportion is known and unknown are derived. The Bayes risk is computed for each Bayes estimator of the reliability. A simulated study is carried out to assess the performance of the estimators alongwith the true and Maximum Likelihood Estimate (MLE) of the reliability. An example from Vannman (1991) is also discussed at the end of the paper.     

Estimation of functions of the parameters of a normal distribution

Several authors have considered the problem of estimating parameters of a distribution after some fixed Gaussian inducing transformation has been applied to the observations. This paper extends this work to the situation where the observations represent a noisy version of a true process, the parameters of the latter requiring estimation     

Estimation of Parameters of Mixed Failure Time Distribution Based on an Extended Modified Sampling Scheme

Abstract

The problem of estimation of parameters of a mixture of degenerate (at zero) and exponential distribution is considered by Dixit and Prasad [Dixit, V. U. (Nee: Jayade, V. D.), Prasad, M. S. (1990 Dixit, V. U. and Prasad, M. S. 1990. Estimation of parameters of mixed failure time distribution. Commun.in Statist.-Theory Meth, 19(12): 4667–4678. (Nee: Jayade, V. D.) [Google Scholar]). Estimation of parameters of mixed failure time distribution. Commun.in Statist.-Theory Meth., 19(12):4667–4678]. The sampling scheme proposed in it is extended to k positive observations in Dixit [Dixit, V. U. (1993 Dixit, V. U. 1993. “Statistical Inference for AR (1) Process with Mixed Errors”. Kolhapur, , India: Shivaji University. Unpublished Ph.D. thesis [Google Scholar]). Statistical Inference for AR (1) Process with Mixed Errors. Unpublished Ph.D. thesis, Shivaji University Kolhapur, India] and moment estimator, MLE and UMVUE based on it are obtained and their finite sample and asymptotic properties are studied. These results are presented in this paper. It is interesting to mention that the sampling scheme proposed by Shinde and Shanubhogue [Shinde, R. L., Shanubhogue, A. (2000 Shinde, R. L. and Shanubhogue, A. 2000. Estimation of parameters and the mean life of a mixed failure time distribution. Commun. Statist.-Theory Meth, 29(11): 2621–2642. [Taylor & Francis Online], [Web of Science ®] , [Google Scholar]). Estimation of parameters and the mean life of a mixed failure time distribution. Commun. Statist.-Theory Meth. 29(11):2621–2642] is a particular case of the sampling scheme proposed in Dixit [Dixit, V. U. (1993 Dixit, V. U. 1993. “Statistical Inference for AR (1) Process with Mixed Errors”. Kolhapur, , India: Shivaji University. Unpublished Ph.D. thesis [Google Scholar]). Statistical Inference for AR (1) Process with Mixed Errors. Unpublished Ph.D. thesis, Shivaji University Kolhapur, India] for n = k.     

Reliability estimation for the exponential distribution

This paper is concerned with classical statistical estimation of the reliability function for the exponential density with unknown mean failure time θ, and with a known and fixed mission time τ. The minimum variance unbiased (MVU) estimator and the maximum likelihood (ML) estimator are reviewed and their mean square errors compared for different sample sizes. These comparisons serve also to extend previous work, and reinforce further the nonexistence of a uniformly best estimator. A class of shrunken estimators is then defined, and it produces a shrunken quasi-estimator and a shrunken estimator. The mean square errors for both these estimators are compared to the mean square errors of the MVU and ML estimators, and the new estimators are found to perform very well. Unfortunately, these estimators are difficult to compute for practical applications. A second class of estimators, which is easy to compute is also developed. Its mean square error properties are compared to the other estimators, and it outperforms all the contending estimators over the high and low reliability parameter space. Since, for all the estimators, analytical mean square error comparisons are not tractable, extensive numerical analyses are done in obtaining both the exact small sample and large sample results.     

Limiting distributions of MLE and UMVUE in the biparametric uniform distribution

In this paper, we study the asymptotic distributions of MLE and UMVUE of a parametric functionh(θ₁, θ₂) when sampling from a biparametric uniform distributionU(θ₁, θ₂). We obtain both limiting distributions as a convolution of exponential distributions, and we observe that the limiting distribution of UMVUE is a shift of the limiting distribution of MLE.     

Estimating parameters of a selected Pareto population

Let Π₁,…,Π_k be k populations with Π_i being Pareto with unknown scale parameter α_i and known shape parameter β_i;i=1,…,k. Suppose independent random samples (X_i1,…,X_in), i=1,…,k of equal size are drawn from each of k populations and let X_i denote the smallest observation of the ith sample. The population corresponding to the largest X_i is selected. We consider the problem of estimating the scale parameter of the selected population and obtain the uniformly minimum variance unbiased estimator (UMVUE) when the shape parameters are assumed to be equal. An admissible class of linear estimators is derived. Further, a general inadmissibility result for the scale equivariant estimators is proved.     

Estimation of Parameters of a Mixed Failure Time Distribution Which is a Mixture of Degenerate (Degenerate at Zero) and I. G. Distribution

The problem of estimation of parameters of a mixture of degenerate (at zero) and exponential distribution is considered by Jayade and Prasad (1990). The sampling scheme proposed in it is extended in this paper to a mixture of degenerate and Inverse Gaussian distribution. The Inverse Gaussian distribution is very relevant for studying reliability and life-testing problems. The inverse Gaussian being the first passage time distribution for Wiener process makes it particularly appropriate for failure or reaction time data analysis.     

A note on the common location parameter of several exponential populations

The problem of estimation of an unknown common location parameter of several exponential populations with unknown and possibly unequal scale parameters is considered. A wide class of estimators, including both a modified maximum likelihood estimator (MLE), and the uniformly minimum variance unbiased estimator (Umvue) proposed by ghosh and razmpour(1984), is obtained under a class of convex loss functions.