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In this work improved point and interval estimation of the smallest scale parameter of independent gamma distributions with known shape parameters are studied in an integrated fashion. The approach followed is based on formulating the model in such a way that enables us to treat the estimation of the smallest scale parameter as a problem of estimating an unrestricted scale parameter in the presence of a nuisance parameter. The class of improved point and interval estimators is enriched. Within this class, a subclass of generalized Bayes estimators of a simple form is identified.

We describe a simple approach for estimating the ratio ρ  =  σ 2 / σ 1 of the scale parameters of two populations from a decision theoretic point of view. We show that if the loss function satisfies a certain condition, then the estimation of ρ reduces to separately estimating σ 2 and 1/ σ 1 . This implies that the standard estimator of ρ can be improved by just employing an improved estimator of σ 2 or 1/ σ 1 . Moreover, in the case where the loss function is convex in some function of its argument, we prove that such improved estimators of ρ are further dominated by corresponding ones that use all the available data. Using this result, we construct new classes of double-adjustment improved estimators for several well-known convex as well as non-convex loss functions. In particular, Strawderman-type estimators of ρ in general models are given whereas Shinozaki-type estimators of the ratio of two normal variances are briefly treated.

This article proposes new approaches for optimal planning of step-stress accelerated life testing models. The experiment considered is time constrained with the tested items not monitored continuously but inspected at particular time points instead. The inspection points are primarily the points of stress level change and the experiment’s termination point, but the inclusion of additional intermediate inspection points is possible. The underlying lifetimes in each stress level follow a general-scale family of distributions having, among others, the exponential and the Weibull as special cases. For this model, the optimal allocation of the inspection points is studied in terms of the classical A-, C-, D- and E-optimality criteria, as well as in the context of minimizing the probability of nonexistence of the maximum likelihood estimators of the model’s parameters. For the determination of the inspection intervals’ length, a deterministic and a hazard rate-based approach are introduced. Simulation study results indicate that these new approaches outperform the standard ones of equal spacing and equal probability. All the considered designs are comparatively evaluated and discussed on the basis of simulation studies.

Suppose that there is a typical (scale equivariant) improved estimator of the variance, σ², of a univariate normal distribution with unknown mean at our disposal. Using this estimator, in this work we construct in a very simple way an improved estimator of the covariance matrix, Σ, of a multivariate normal distribution with unknown mean and another improved estimator of the generalized variance, |Σ|. The data is a sample of i.i.d. observations from this distribution. The loss function is the entropy loss or the quadratic loss in the case of Σ and a general loss satisfying a certain condition in the case of |Σ|. These novel results reduce, in a specific way, the problems of estimating Σ or |Σ| to the univariate problem of estimating σ². As a consequence, Stein-type, Brewster and Zidek-type, Strawderman-type and Maruyama-type improved estimators for Σ and |Σ| are directly constructed from their univariate counterparts. This work unifies and extends previously obtained results on the estimation of Σ and |Σ|.