Estimation of the smallest scale parameter of two-parameter exponential distributions

Improved point and interval estimation of the smallest scale parameter of n independent populations following two-parameter exponential distributions are studied. The model is formulated in such a way that allows for treating the estimation of the smallest scale parameter as a problem of estimating an unrestricted scale parameter in the presence of a nuisance parameter. The classes of improved point and interval estimators are enriched with Stein-type, Brewster and Zidek-type, Maruyama-type and Strawderman-type improved estimators under both quadratic and entropy losses, whereas using as a criterion the coverage probability, with Stein-type, Brewster and Zidek-type, and Maruyama-type improved intervals. The sampling framework considered incorporates important life-testing schemes such as i.i.d. sampling, type-II censoring, progressive type-II censoring, adaptive progressive type-II censoring, and record values.     

Improved estimation of the scale parameter, the hazard rate parameter and the ratio of the scale parameters in exponential distributions: An integrated approach

We give new classes of Strawderman-type improved estimators for the scale parameter σ₂ and the hazard rate parameter 1/σ₁ of the exponential distributions E(μ₂,σ₂) and E(μ₁,σ₁) under the entropy loss. We then use these estimators to construct improved estimators for the ratio ρ=σ₂/σ₁. The sampling framework we consider integrates important life-testing schemes separately studied in the literature so far, namely, (i) i.i.d. sampling, (ii) Type-II censoring, (iii) progressive Type-II censoring and adaptive progressive Type-II censoring and (iv) record values data. Furthermore, we establish simple identities connecting the risk functions of the estimators of σ₂ and 1/σ₁ and those of ρ that have a direct impact on studying the risk behavior of the latter estimators. Finally, we indicate that no matter which of the above life-testing schemes is employed for the estimation of σ₂, 1/σ₁ or ρ, the corresponding improved estimator, which may be of Stein-type or Brewster and Zidek-type or Strawderman-type, will offer the same improvement over the usual estimator as long as the number of observed complete failure times is the same for each scheme. Our results unify and extend existing results on the estimation of exponential scale parameters in one or two populations.     

Minimax estimation of a lower-bounded scale parameter of a gamma distribution for scale-invariant squared-error loss

Let X have a gamma distribution with known shape parameter θr;aL and unknown scale parameter θ. Suppose it is known that θ ≥ a for some known a > 0. An admissible minimax estimator for scale-invariant squared-error loss is presented. This estimator is the pointwise limit of a sequence of Bayes estimators. Further, the class of truncated linear estimators C = {θ_ρ|θ_ρ(x) = max(a, ρ), ρ > 0} is studied. It is shown that each θ_ρ is inadmissible and that exactly one of them is minimax. Finally, it is shown that Katz's [Ann. Math. Statist., 32, 136–142 (1961)] estimator of θ is not minimax for our loss function. Some further properties of and comparisons among these estimators are also presented.     

Estimation of the scale parameter of the half-logistic distribution with multiply type II censored sample

In this paper, we consider the maximum likelihood and Bayes estimation of the scale parameter of the half-logistic distribution based on a multiply type II censored sample. However, the maximum likelihood estimator(MLE) and Bayes estimator do not exist in an explicit form for the scale parameter. We consider a simple method of deriving an explicit estimator by approximating the likelihood function and discuss the asymptotic variances of MLE and approximate MLE. Also, an approximation based on the Laplace approximation (Tierney & Kadane, 1986) is used to obtain the Bayes estimator. In order to compare the MLE, approximate MLE and Bayes estimates of the scale parameter, Monte Carlo simulation is used.     

On the improved estimation of a function of the scale parameter of an exponential distribution based on doubly censored sample

In the present article, we have studied the estimation of entropy, that is, a function of scale parameter ln⁡σ of an exponential distribution based on doubly censored sample when the location parameter is restricted to positive real line. The estimation problem is studied under a general class of bowl-shaped non monotone location invariant loss functions. It is established that the best affine equivariant estimator (BAEE) is inadmissible by deriving an improved estimator. This estimator is non-smooth. Further, we have obtained a smooth improved estimator. A class of estimators is considered and sufficient conditions are derived under which these estimators improve upon the BAEE. In particular, using these results we have obtained the improved estimators for the squared error and the linex loss functions. Finally, we have compared the risk performance of the proposed estimators numerically. One data analysis has been performed for illustrative purposes.