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SYNOPTIC ABSTRACTIn the Bayesian approach to statistical analyses we incorporate prior information about the parameter of the model with observed data. This prior information is in the form of a prior distribution of the parameter. If the prior information is available as a constant value of the parameter rather than its prior distribution, the Bayesian approach cannot be pursued. However, there are estimation methods that incorporate such prior information with the observed data. The expectation is that the incorporation of such additional information in the estimation process would result in a better estimator than that based on the observed data alone. In some cases this may be true, but in many other cases the risk of worse consequences cannot be ruled out. This paper studies the performance of the observed data based unrestricted estimator (UE), and both observed data and prior information based preliminary test estimator (PTE) of the univariate normal mean under the linex loss function. The risk functions of both UE and PTE are derived. The moment generating function (MGF) of PTE is derived which turns out to be a component of the risk function. From the MGF the first two moments of PTE are obtained and found to be identical to those obtained using different approaches in Khan and Saleh (2001) and Zellner (1986). Under the linex loss criterion the performance of the PTE is compared with that of UE. It is revealed that if the uncertain non-sample prior information about the value of the mean is not too far from its true value, PTE outperforms UE.

The mean and the variance of the time S(t) spent by a system below a random threshold until tare obtained when the system level is modelled by the current value of a sequence of independent and identically distributed random variables appearing at the epochs of a nonhomogeneous Poisson process. In the case of the homogeneous Poisson process, the asymptotic distribution of S(t)/tas t? 8 is derived.

The classical martingale characterizations of the Poisson process were obtained for point process or purely discontinuous martingale i.e. under additional assumptions on properties of trajectories. Here our aim is to search for related characterizations without relying on properties of trajectories. Except for a new martingale characterization, results based on conditional moments jointly involving the past and the nearest future are presented.

The problem of identification of uniform mixtures via posterior means is studied. For linear posterior means a complete solution is given. It determines a family of prior distributions involving beta of both kinds and gamma. Identifiability via any consistent posterior mean is also investigated.

Summary. Relations between the univariate gamma, multivariate Dirichlet and Liouville distributions are investigated.

Uniqueness of specification of a bivariate distribution by a Pareto conditional and a consistent regression function is investigated. New characterizations of the Mardia bivariate Pareto distribution and the bivariate Pareto conditionals distribution are obtained.

A random vector has a multivariate Pareto distribution if one of its univariate conditional distribution is Pareto and some of its marginals are identically distributed.A general method developed in the course of the proof of this result is applied also to characterize the multivariate Student (Cauchy) measure by one univariate Student conditional distribution.

A constant regression of the quotient on the sum of two i.i.d. non-degenerate positive random variables is a characteristic property of the gamma distribution.