Let X1, ..., Xn be a random sample from a normal distribution with unknown mean q and known variance s2. The usual estimator of the mean, i.e., sample mean, is the maximum likelihood estimator which under squared error loss function is minimax and admissible estimator. In many practical situations, q is known in advance to lie in an interval, say [-m, m] for some m>0. In this case, the maximum likelihood estimator changes and dominates X- but it is no longer admissible. Minimax and some other estimators for this problem have been studied by some researchers. In this paper, a new estimator is proposed and the risk function of it is compared with some other competitors. According to our findings, the use of X- and the maximum likelihood estimator is not recommended when some information are accessible about the finite bounds on [-m, m] in advance. Based on the values taken by q in [-m, m], the appropriate estimator is suggested.

In a subclass of the scale-parameter exponential family, we consider the sequential point estimation of a function of the scale parameter under the loss function given as the sum of the weighted squared error loss and a linear cost. For a fully sequential sampling scheme, second order expansions are obtained for the expected sample size as well as for the regret of the procedure. The former researches on Gamma and Exponential distributions can be deduced from our general results.

In the present paper, we study shrinkage testimation for the unknown scale parameter q>0 of the exponential distribution based on record data under the asymmetric squared log error loss function. A minimum risk unbiased estimator within the class of the estimators of the form cTm is derived, where Tm is the maximum likelihood estimate of q. Some shrinkage testimators are proposed and their risks are computed. The relative efficiencies of the shrinkage testimators with respect to a minimum risk unbiased estimator of the form cTm under the squared log error loss function are calculated for the comparison purposes. An illustrative example is also presented.

The problem of estimating the parameter q, when it is restricted to an interval of the form [m,1], in a class of discrete distributions, including Binomial(k, q), Negative Binomial(r, q), discrete Weibull(q) and etc., is considered. We give necessary and sufficient conditions for which the Bayes estimator of q, with respect to a two points boundary supported prior is minimax under squared log error loss function. For some of the distributions in this class, we give numerical values of the smallest values of m for which the corresponding Bayes estimator of q is minimax.

In risk analysis based on Bayesian framework, premium calculation requires specification of a prior distribution for the risk parameter in the heterogeneous portfo-lio. When the prior knowledge is vague, the E-Bayesian and robust Bayesian analysis can be used to handle the uncertainty in specifying the prior distribution by consid-ering a class of priors instead of a single prior. In this paper, we study the E-Bayes and robust Bayes premium estimation and prediction in exponential model under the squared log error loss function. A prequential analysis in a simulation study is carried out to compare the proposed predictors. Finally, a real data example is included for illustrating the results.