Estimation of parameters of inverse Weibull distribution and application to multi-component stress-strength model

The problem of estimation of the parameters of two-parameter inverse Weibull distributions has been considered. We establish existence and uniqueness of the maximum likelihood estimators of the scale and shape parameters. We derive Bayes estimators of the parameters under the entropy loss function. Hierarchical Bayes estimator, equivariant estimator and a class of minimax estimators are derived when shape parameter is known. Ordered Bayes estimators using information about second population are also derived. We investigate the reliability of multi-component stress-strength model using classical and Bayesian approaches. Risk comparison of the classical and Bayes estimators is done using Monte Carlo simulations. Applications of the proposed estimators are shown using real data sets.     

Maximal Invariant and Weakly Equivariant Estimators

Equivariant functions can be useful for constructing of maximal invariant statistic. In this article, we discuss construction of maximal invariants based on a given weakly equivariant function under some additional conditions. The theory easily extends to the case of two or more weakly equivariant functions. Also, we derive a maximal invariant statistic when the group contains a sharply transitive and a characteristic subgroup. Finally, we consider the independence of invariant and weakly equivariant functions under some special conditions.     

Sufficiency of a sufficient statistic for equivariant estimation

Suppose an estimation problem is invariant under a group of transformations and one is interested in finding an optimal equivariant estimator. The usual proactice is to confine attention to non-randomized equivariant estimators based on a minimal sufficient statistic. A justification of this restriction to a smaller clas of estimators is given in this paper under certain conditions.     

Estimation of the smaller and larger of two exponential location parameters

Assume independent random samples are drawn from two populations which are exponentially distributed with unknown location parameters and a common known scale parameter. We want to estimate the maximum and the minimum of the unknowo location paremeters. In this paper several estimators are proposed which are better than the natural estimations in terms of absolute bias and /or meaqn squared error.     

Estimating powers of the generalized variance under the Pitman closeness criterion

For estimating powers of the generalized variance under a multivariate normal distribution with an unknown mean, the inadmissibility of the closest affine equivariant estimator is shown for the Pitman closeness criterion.     

Combined estimate of a location parameter

In this paper, we revisit the problem of combining estimates of location considered by Cohen (1976). Our results unify and strengthen the results of Cohen (1976), Bhattacharya (1981) and Akai (1982).     

Equivariant prediction in finite populations

A theory of equivariant prediction is developed for predicting the population total in finite populations. Minimum risk equivariant predictors (MREP) are derived under the location, scale and locationscale superpopulation models. Under the general linear model, it is shown that the best(linear) unbiased predictor (B(L)UP) is an MREP.     

Estimating the mean of the selected uniform population

Let π_i(i=1,2,…K) be independent U(0,?_i) populations. Let Y_i denote the largest observation based on a random sample of size n from the i-th population. for selecting the best populaton, that is the one associated with the largest ?_i, we consider the natural selection rule, according to which the population corresponding to the largest Y_i is selected. In this paper, the estimation of M. the mean of the selected population is considered. The natural estimator is positively biased. The UMVUE (uniformly minimum variance unbiased estimator) of M is derived using the (U,V)-method of Robbins (1987) and its asymptotic distribution is found. We obtain a minimax estimator of M for K≤4 and a class of admissible estimators among those of the form cY_max. For the case K = 2, the UMVUE is improved using the Brewster-Zidek (1974) Technique with respect to the squared error loss function L₁ and the scale-invariant loss function L₂. For the case K = 2, the MSE'S of all the estimators are compared for selected values of n and ρ=?₁/(?₁+?₂).     

Estimating Average Worth of the Selected Subset from Two-Parameter Exponential Populations

ABSTRACT

Suppose independent random samples are available from k(k ≥ 2) exponential populations ∏₁,…,∏ _k with a common location θ and scale parameters σ₁,…,σ _k , respectively. Let X _i and Y _i denote the minimum and the mean, respectively, of the ith sample, and further let X = min{X ₁,…, X _k } and T _i  = Y _i  ? X; i = 1,…, k. For selecting a nonempty subset of {∏₁,…,∏ _k } containing the best population (the one associated with max{σ₁,…,σ _k }), we use the decision rule which selects ∏ _i if T _i  ≥ c max{T ₁,…,T _k }, i = 1,…, k. Here 0 < c ≤ 1 is chosen so that the probability of including the best population in the selected subset is at least P* (1/k ≤ P* < 1), a pre-assigned level. The problem is to estimate the average worth W of the selected subset, the arithmetic average of means of selected populations. In this article, we derive the uniformly minimum variance unbiased estimator (UMVUE) of W. The bias and risk function of the UMVUE are compared numerically with those of analogs of the best affine equivariant estimator (BAEE) and the maximum likelihood estimator (MLE).     

A note on Pitman's measure of closeness with balanced loss function

In this note, we consider the problem of estimating an unknown parameter θ in the sense of the Pitman's measure of closeness (PMC) using the balanced loss function (BLF). We show that the PMC comparison of estimators under the BLF can be reduced to the PMC comparison under the usual absolute error loss. The Pitman-closest estimators of the location and scale parameters under BLF are also characterized. Illustrative examples are given to show the broad range applications of the obtained results.