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.     

Identifiability of Finite Mixtures of Elliptical Distributions

Abstract.  We present general results on the identifiability of finite mixtures of elliptical distributions under conditions on the characteristic generators or density generators. Examples include the multivariate t -distribution, symmetric stable laws, exponential power and Kotz distributions. In each case, the shape parameter is allowed to vary in the mixture, in addition to the location vector and the scatter matrix. Furthermore, we discuss the identifiability of finite mixtures of elliptical densities with generators that correspond to scale mixtures of normal distributions.     

Existence of Maximum Likelihood Estimates for Multi-dimensional Exponential Families

This paper deals with the existence of maximum likelihood estimators for multi-dimensional exponential families, including curved exponential families. It first gives an algorithm for determining the MLE from the data. Then it establishes that when the parameter set is either open or relatively closed in the natural parameter set, the MLE of the parameter exists in the sense of Hoffmann-Jorgensen.     

Semiparametric Mixtures of Symmetric Distributions

We consider in this paper the semiparametric mixture of two unknown distributions equal up to a location parameter. The model is said to be semiparametric in the sense that the mixed distribution is not supposed to belong to a parametric family. To insure the identifiability of the model, it is assumed that the mixed distribution is zero symmetric, the model being then defined by the mixing proportion, two location parameters and the probability density function of the mixed distribution. We propose a new class of M‐estimators of these parameters based on a Fourier approach and prove that they are ‐consistent under mild regularity conditions. Their finite sample properties are illustrated by a Monte Carlo study, and a benchmark real dataset is also studied with our method.     

Optimal Multiple Constant-Stress Accelerated Life Tests for Generalized Exponential Distribution

Recently generalized exponential distribution has been discussed by many authors. In this article, we study the optimal constant-stress accelerated life tests with complete sample for the generalized exponential distribution. The problem of choosing the optimal proportions of test units allocated to each stress level is addressed by using V-optimality as well as D-optimality criteria. Some interesting conclusions are obtained. Finally, real data example and numerical examples have been analyzed to illustrate the proposed procedures.     

Trace of the Variance-Covariance Matrix in Natural Exponential Families

In this article, we introduce the notion of trace variance function which is the trace of the variance-covariance matrix. Under some conditions, we prove that this trace variance function characterizes the Natural Exponential Family (NEF). We apply this characterization in order to estimate the distribution which belongs to some NEFs. Therefore, we introduce the estimator of this trace variance function. We give the asymptotic properties of this estimator. Finally, we illustrate our results using a simulation study.     

Independence Structure of Natural Conjugate Densities to Exponential Families and the Gibbs' Sampler

In this paper the independence between a block of natural parameters and the complementary block of mean value parameters holding for densities which are natural conjugate to some regular exponential families is used to design in a convenient way a Gibbs' sampler with block updates. Even when the densities of interest are obtained by conditioning to zero a block of natural parameters in a density conjugate to a larger "saturated" model, the updates require only the computation of marginal distributions under the "unconditional" density. For exponential families which are closed under marginalization, including both the zero mean Gaussian family and the cross-classified Bernoulli family such an implementation of the Gibbs' sampler can be seen as an Iterative Proportional Fitting algorithm with random inputs.     

Exponential Family Techniques for the Lognormal Left Tail

Let X be lognormal(μ,σ²) with density f(x); let θ > 0 and define . We study properties of the exponentially tilted density (Esscher transform) f_θ(x) = e^?θxf(x)/L(θ), in particular its moments, its asymptotic form as θ→∞ and asymptotics for the saddlepoint θ(x) determined by . The asymptotic formulas involve the Lambert W function. The established relations are used to provide two different numerical methods for evaluating the left tail probability of the sum of lognormals S_n=X₁+?+X_n: a saddlepoint approximation and an exponential tilting importance sampling estimator. For the latter, we demonstrate logarithmic efficiency. Numerical examples for the cdf F_n(x) and the pdf f_n(x) of S_n are given in a range of values of σ²,n and x motivated by portfolio value‐at‐risk calculations.     

Generic Identifiability of Linear Structural Equation Models by Ancestor Decomposition

Linear structural equation models, which relate random variables via linear interdependencies and Gaussian noise, are a popular tool for modelling multivariate joint distributions. The models correspond to mixed graphs that include both directed and bidirected edges representing the linear relationships and correlations between noise terms, respectively. A question of interest for these models is that of parameter identifiability, whether or not it is possible to recover edge coefficients from the joint covariance matrix of the random variables. For the problem of determining generic parameter identifiability, we present an algorithm building upon the half‐trek criterion. Underlying our new algorithm is the idea that ancestral subsets of vertices in the graph can be used to extend the applicability of a decomposition technique.     

Generalized Mixtures of Gamma and Exponentials and Reliability Properties of the Maximum from Friday and Patil Bivariate Exponential Model

The minimum and maximum order statistics from many of the common bivariate exponential distributions are predominantly generalized mixtures of exponentials; however, the maximum from the Friday and Patil bivariate exponential (FPBVE) model is either a generalized mixture of three or fewer exponentials or a generalized mixture of gamma and exponentials. In this article, we obtain conditions based on the weights and parameters of the generalized mixtures of gamma and one or two exponential distributions that yield legitimate probability models. Furthermore, we analyze properties of the failure rate of the maximum from the FPBVE model. This answers a question raised in Baggs and Nagaraja (1996 Baggs , G. E. , Nagaraja , H. N. ( 1996 ). Reliability properties of order statistics from bivariate exponential distributions . Commun. Statist. Stochastic Mod. 12 : 611 – 631 .[Taylor & Francis Online] , [Google Scholar]).     

Test of fit for Marshall–Olkin distributions with applications

Marshall and Olkin [1967. A multivariate exponential distribution. J. Amer. Statist. Assoc. 62, 30–44], introduced a bivariate distribution with exponential marginals, which generalizes the simple case of a bivariate random variable with independent exponential components. The distribution is popular under the name ‘Marshall–Olkin distribution’, and has been extended to the multivariate case. L2-type statistics are constructed for testing the composite null hypothesis of the Marshall–Olkin distribution with unspecified parameters. The test statistics utilize the empirical Laplace transform with consistently estimated parameters. Asymptotic properties pertaining to the null distribution of the test statistic and the consistency of the test are investigated. Theoretical results are accompanied by a simulation study, and real-data applications.     

A Class of Improved Estimators for the Scale Parameter of a Mixture Model of Exponential Distribution with Unknown Location

Under Stein's loss, a class of improved estimators for the scale parameter of a mixture of exponential distribution with unknown location is constructed. The method is analogous to Maruyama's (1998 Maruyama , Y. ( 1998 ). Minimax estimators of a normal variance . Metrika 48 : 209 – 214 .[Crossref], [Web of Science ®] , [Google Scholar]) construction for the variance of a normal distribution and also an extension of the result produced in Petropoulos and Kourouklis (2002 Petropoulos , C. , Kourouklis , S. ( 2002 ). A class of improved estimators for the scale parameter of an exponential distribution with unknown location . Commun. Statist. Theor. Meth. 31 : 325 – 335 .[Taylor & Francis Online], [Web of Science ®] , [Google Scholar]). Also, robustness properties are considered.     

Estimation op parameters in finite mixtures of exponential families from censored data

A general successive substitutions' scheme is developed to estimate parameters in a finite mixture of distributions from the exponential family, based on censored data. It is assumed that the data can be grouped in the first class and the number of observations in each of the remaining classes are known Examples from Poisson Exponential and Normal distributions are given A small simulation exercise has also been carried out for the mixture of two one parameter exponential population.     

Normalization of the origin-shifted exponential distribution for control chart construction

This study demonstrates that a location parameter of an exponential distribution significantly influences normalization of the exponential. The Kullback–Leibler information number is shown to be an appropriate index for measuring data normality using a location parameter. Control charts based on probability limits and transformation are compared for known and estimated location parameters. The probabilities of type II error (β-risks) and average run length (ARL) without a location parameter indicate an ability to detect an out-of-control signal of an individual chart using a power transformation similar to using probability limits. The β-risks and ARL of control charts with an estimated location parameter deviate significantly from their theoretical values when a small sample size of n≤50 is used. Therefore, without taking into account of the existence of a location parameter, the control charts result in inaccurate detection of an out-of-control signal regardless of whether a power or natural logarithmic transformation is used. The effects of a location parameter should be eliminated before transformation. Two examples are presented to illustrate these findings.     

A Study of Interval Censoring in Parametric Regression Models

Parametric models for interval censored data can now easily be fitted with minimal programming in certain standard statistical software packages. Regression equations can be introduced, both for the location and for the dispersion parameters. Finite mixture models can also be fitted, with a point mass on right (or left) censored observations, to allow for individuals who cannot have the event (or already have it). This mixing probability can also be allowed to follow a regression equation.Here, models based on nine different distributions are compared for three examples of heavily censored data as well as a set of simulated data. We find that, for parametric models, interval censoring can often be ignored and that the density, at centres of intervals, can be used instead in the likelihood function, although the approximation is not always reliable. In the context of heavily interval censored data, the conclusions from parametric models are remarkably robust with changing distributional assumptions and generally more informative than the corresponding non-parametric models.