On Unbiased Estimation of Positive Integral Powers of the Natural Parameter in Exponential Families

We explore the structure of one‐parameter exponential families admitting an unbiased estimator for a positive integral power of the natural parameter. It is seen that only exponential families dominated by Lebesgue measure can have this property. It is outlined that similar results can be obtained for other functions of the natural parameter.     

On the optimality of the sample variance

The present paper explores the structure of linear exponential families for which the sample variance is a uniformly minimum variance unbiased estimator.     

A unified approach to minimum variance unbiased estimation of a probability function belonging to an exponential family

The problem of minimum variance unbiased estimation of the probability density function of a random variable belonging to an exponential family is considered. The method of estimation proposed in this paper requires the solution of a certain integral equation. For many probability distributions the solution of this equation is given by a known result in integral transform theory.     

The Lindsay transform of natural exponential families

Let μ be an infinitely divisible positive measure on R. If the measure ρ_μ is such that x^-2[ρ_μ(dx)—ρ_μ({0})δ₀(dx)] is the Lévy measure associated with μ and is infinitely divisible, we consider for all positive reals α and β the measure T_α,β(μ) which is the convolution of μ*^α and ρ_μ*β. For example, if μ is the inverse Gaussian law, then ρ_μ is a gamma law with paramter 3/2. Then T_α,β(μ) is an extension of the Lindsay transform of the first order, restricted to the distributions which are infinitely divisible. The main aim of this paper is to point out that it is possible to apply this transformation to all natural exponential families (NEF) with strictly cubic variance functions P. We then obtain NEF with variance functions of the form □ΔP(□Δ), where A is an affine function of the mean of the NEF. Some of these latter types appear scattered in the literature.     

Conditionally Reducible Natural Exponential Families and Enriched Conjugate Priors

Consider a standard conjugate family of prior distributions for a vector-parameter indexing an exponential family. Two distinct model parameterizations may well lead to standard conjugate families which are not consistent, i.e. one family cannot be derived from the other by the usual change-of-variable technique. This raises the problem of finding suitable parameterizations that may lead to enriched conjugate families which are more flexible than the traditional ones. The previous remark motivates the definition of a new property for an exponential family, named conditional reducibility. Features of conditionally-reducible natural exponential families are investigated thoroughly. In particular, we relate this new property to the notion of cut, and show that conditionally-reducible families admit a reparameterization in terms of a vector having likelihood-independent components. A general methodology to obtain enriched conjugate distributions for conditionally-reducible families is described in detail, generalizing previous works and more recent contributions in the area. The theory is illustrated with reference to natural exponential families having simple quadratic variance function.     

Multivariate stable exponential families and Tweedie scale

In this paper, we characterize the multivariate stable natural exponential families by a property of homogeneity of the cumulant function of some basis, and by a property of homogeneity of the variance function. We also extend the definition of a Tweedie scale to a finite dimensional space and we give a class of natural exponential families belonging to this scale on the space of symmetric matrices.     

Approximate mean squared errors of estimations of reliability in the two-parameter exponential case

Combining the method used i n Chao (1981) and conditional procedure, we extend our previous results for one-parameter exponential to two-parameter case, i .e. we provide simple approximation formulas for the mean squared errors of the maximum likelihood and minimum variance unbiased estimators of reliability of general k-out-of-m systems when the component lifetimes are independent and follow a two-parameter exponential distribution.     

Multiparameter estimation of discrete exponential distributions

Let X₁, …, X_p be independent random variables, all having the same distribution up to a possibly varying unspecified parameter, where each of the p distributions belongs to the family of one parameter discrete exponential distributions. The problem is to estimate the unknown parameters simultaneously. Hudson (1978) shows that the minimum variance unbiased estimator (MVUE) of the parameters is inadmissible under squared error loss, and estimators better than the MVUE are proposed. Essentially, these estimators shrink the MVUE towards the origin. In this paper, we indicate that estimators shifting the MVUE towards a point different from the origin or a point determined by the observations can be obtained.     

Progressive type II censored order statistics from exponential distributions

In the model of progressive type II censoring, point and interval estimation as well as relations for single and product moments are considered. Based on two-parameter exponential distributions, maximum likelihood estimators (MLEs), uniformly minimum variance unbiased estimators (UMVUEs) and best linear unbiased estimators (BLUEs) are derived for both location and scale parameters. Some properties of these estimators are shown. Moreover, results for single and product moments of progressive type II censored order statistics are presented to obtain recurrence relations from exponential and truncated exponential distributions. These relations may then be used to compute all the means, variances and covariances of progressive type II censored order statistics based on exponential distributions for arbitrary censoring schemes. The presented recurrence relations simplify those given by Aggarwala and Balakrishnan (1996)     

Estimation of reliability for a multicomponent survival stress-strength model based on exponential distributions

Uniformly minimum variance unbiased estimator (UMVUE) of reliability in stress-strength model (known stress) is obtained for a multicomponent survival model based on exponential distributions for parallel system. The variance of this estimator is compared with Cramer-Rao lower bound (CRB) for the variance of unbiased estimator of reliability, and the mean square error (MSE) of maximum likelihood estimator of reliability in case of two component system.     

A note on umvu estimation in the transformed chi-square family

The transformed chi-square family includes many common one-parameter continuous distributions. In that family, we give conditions under which a given function of the mean admits a minimum variance unbiased estimator and an orthogonal expansion for this estimator in terms of the generalized Laguerre polynomials. We show that such expansion is useful for obtaining bounds for the variance and for the study of the asymptotic properties of the unbiased estimators.     

Estimation of probabilities in the class of modified power series distributions

In this paper we consider the class of modified power series distribution introduced by GUPTA (1974) and derive a minimum variance unbiased estimator (MVUE) of the probability function for this class. these results are then applied to obtain MVUE of the probability function for the generalized negative binomial distributions, the generalized poisson distribution, the generalized logarithmic series distribution and the lost game distribution. A large number of results in the literature follow trivially from out results as special cases.     

A note on the estimation of pr{y<x} for truncation parameter distributions

In this note a relationship in the treatment of the lower and upper truncations considered in Beg (1980) is pointed out and the minimum variance unbiased estimator of P = Pr{Y<X) for the (upper) truncated exponential distribution is obtained.     

Estimation of the variance when kurtosis is known

The unbiased estimator of a population variance σ², S ² has traditionally been overemphasized, regardless of sample size. In this paper, alternative estimators of population variance are developed. These estimators are biased and have the minimum possible mean-squared error [and we define them as the “minimum mean-squared error biased estimators” (MBBE)]. The comparative merit of these estimators over the unbiased estimator is explored using relative efficiency (RE) (a ratio of mean-squared error values). It is found that, across all population distributions investigated, the RE of the MBBE is much higher for small samples and progressively diminishes to 1 with increasing sample size. The paper gives two applications involving the normal and exponential distributions.     

On Minimum Variance Unbiased Estimators

The problem of finding minimum variance unbiased estimators of various parameters for parametric distributions is an important one in statistics. This article gives analytical formulas for the minimum variance unbiased estimators of parametric functions, which are usually used in a classroom, for two types of densities. The first type is the one-parameter regular exponential family, and the second is a two-parameter family of a continuous random variable whose range depends on the unknown parameters.     

The use of the weighted likelihood in the natural exponential families with quadratic variance

The authors propose a weighted likelihood concept for the estimation of parameters in natural exponential families with quadratic variance. They apply the results to both simulated and real data.     

On generalized variance of product of powered components and multiple stable Tweedie models

A flexible family of multivariate models, named multiple stable Tweedie (MST) models, is introduced and produces generalized variance functions which are products of powered components of the mean. These MST models are built from a fixed univariate stable Tweedie variable having a positive value domain, and the remaining random variables given the fixed one are also real independent Tweedie variables, with the same dispersion parameter equal to the fixed component. In this huge family of MST models, generalized variance estimators are explicitly pointed out by maximum likelihood method and, moreover, computably presented for the uniform minimum variance and unbiased approach. The second estimator is brought from modified Lévy measures of MST which lead to some solutions of particular Monge–Ampère equations.     

Efficient sequential estimation for compound poisson processes

This paper discusses uniformly minimum variance, unbiased sequential estimation of a real-valued parametric function for a compound Poisson process when the compounding random variables belong to the exponential family. The characterization of Cramer-Rao efficient plans by Stefanov (1982 b) is shown to be incomplete by obtaining a new efficient plan for the compound Poisson-Bernoulli process. This new plan completes the characterization of Cramer-Rao efficient plans. The class of Bhattacharyya efficient estimators of order two is determined for all the efficient sampling schemes.     

Umvu estimators for the population mean and variance based on random effects models for lognormal data

Cross-classified data are often obtained in controlled experimental situations and in epidemiologic studies. As an example of the latter, occupational health studies sometimes require personal exposure measurements on a random sample of workers from one or more job groups, in one or more plant locations, on several different sampling dates. Because the marginal distributions of exposure data from such studies are generally right-skewed and well-approximated as lognormal, researchers in this area often consider the use of ANOVA models after a logarithmic transformation. While it is then of interest to estimate original-scale population parameters (e.g., the overall mean and variance), standard candidates such as maximum likelihood estimators (MLEs) can be unstable and highly biased. Uniformly minimum variance unbiased (UMVU) cstiniators offer a viable alternative, and are adaptable to sampling schemes that are typiral of experimental or epidemiologic studies. In this paper, we provide UMVU estimators for the mean and variance under two random effects ANOVA models for logtransformed data. We illustrate substantial mean squared error gains relative to the MLE when estimating the mean under a one-way classification. We illustrate that the results can readily be extended to encompass a useful class of purely random effects models, provided that the study data are balanced.