One-parameter Exponential Family Of Distributions Research Articles

Theory and inference for a class of nonlinear models with application to time series of counts

This paper studies theory and inference related to a class of time series models that incorporates nonlinear dynamics. It is assumed that the observations follow a one-parameter exponential family of distributions given an accompanying process that evolves as a function of lagged observations. We employ an iterated random function approach and a special coupling technique to show that, under suitable conditions on the parameter space, the conditional mean process is a geometric moment contracting Markov chain and that the observation process is absolutely regular with geometrically decaying coefficients. Moreover the asymptotic theory of the maximum likelihood estimates of the parameters is established under some mild assumptions. These models are applied to two examples; the first is the number of transactions per minute of Ericsson stock and the second is related to return times of extreme events of Goldman Sachs Group stock.

A model of prior ignorance for inferences in the one-parameter exponential family

This paper proposes a model of prior ignorance about a scalar variable based on a set of distributions M. In particular, a set of minimal properties that a set M of distributions should satisfy to be a model of prior ignorance without producing vacuous inferences is defined. In the case the likelihood model corresponds to a one-parameter exponential family of distributions, it is shown that the above minimal properties are equivalent to a special choice of the domains for the parameters of the conjugate exponential prior. This makes it possible to define the largest (that is, the least-committal) set of conjugate priors M that satisfies the above properties. The obtained set M is a model of prior ignorance with respect to the functions (queries) that are commonly used for statistical inferences; it is easy to elicit and, because of conjugacy, tractable; it encompasses frequentist and the so-called objective Bayesian inferences with improper priors. An application of the model to a problem of inference with count data is presented.

The construction of the uniformly minimum variance unbiased estimator

For a one-parameter exponential family of distributions, a method to find the uniformly minimum variance unbiased (UMVU) estimator based on the complete sufficient statistic is given in Jani and Dave [1] by change of the expression of the unbiasedness condition. But, it heavily depends on the concrete form of the distribution of the statistic in obtaining indeed the UMVU estimator. In this paper, from the different point of view, the construction of the UMVU estimator for a one-parameter exponential families of distributions and certain two-parameter family of distributions is discussed. Some examples are also given.

Deficiency of Two Estimators in One-Parameter Exponential Family of Distributions

The concept of deficiency was introduced by Hodges and Lehmann [Hodges, J. L., Lehmann, E. L. (1970). Deficiency. Ann. Math. Statist. 41(3):783–801] which can be used to discriminate two asymptotically efficient (BAN) estimators. One-parameter exponential family of distributions depending on the parameter θ is considered. The mean square errors of the maximum likelihood (ML) estimator and the uniformly minimum variance unbiased (UMVU) estimator of an estimable function g(θ) are obtained up to the order n −2, where n is the sample size. The deficiency between the two estimators is considered. The restriction imposed by Hwang and Hu [Hwang, T.-Y., Hu, C.-Y. (1990). More comparisons of MLE with UMVUE for exponential families. Ann. Inst. Statist. Math. 42(1):65–75] is relaxed. An illustrative example is discussed.

Umvu estimation for certain family of exponential distributions

In this paper we study certain properties of estimable and UMUV-estimable functions in a subfamily of the one-parameter exponential family of distributions for which there exists a sufficient and complete statistic following a Gamma distribution. These results are applied to the problem of estimation in the transformed chi-square family.

Empirical Bayes sequential estimation for exponential families: The untruncated component

We consider the empirical Bayes decision problem where the component problem is the sequential estimation of the mean θ of one-parameter exponential family of distributions with squared error loss for the estimation error and a cost c>0 for each observation. The present paper studies the untruncated sequential component case. In particular, an untruncated asymptotically pointwise optimal sequential procedure is employed as the component. With sequential components, an empirical Bayes decision procedure selects both a stopping time and a terminal decision rule for use in the component with parameter θ. The goodness of the empirical Bayes sequential procedure is measured by comparing the asymptotic behavior of its Bayes risk with that of the component procedure as the number of past data increases to infinity. Asymptotic risk equivalence of the proposed empirical Bayes sequential procedure to the component procedure is demonstrated.

A Characterization of One-Parameter Exponential Family of Distributions

A characterization through moments given by Khatri (1959) for power series distributions (p. s. d.) and by Ahsanullah (1992) for modified power series distributions (m.p.s.d.) bas been extended for the wider class viz one-parameter exponential family of distributions.

Imprecise conjugate prior densities for the one-parameter exponential family of distributions

Reconsidering generalizations of the original Bayesian framework that have been suggested during the last three decades, imprecise conjugate prior densities are proposed for members of the one-parameter exponential family of distributions.

Testing and Adjusting for Departures from Nominal Dispersion in Generalized Linear Models

SUMMARY In this paper we describe a score test of the hypothesis of no departure from nominal dispersion in a generalized linear model. We also give a method for adjusting the nominal variance-covariance matrix of the estimated regression coefficients when overdispersion is suspected. This procedure is an alternative to the traditional method of adjusting each element of the variance-covariance matrix by the same factor. We illustrate our method for the one-parameter exponential family of distributions in a logistic analysis of a factorial experiment and a Poisson regression analysis of bioassay data.

Non-existence of unbiased estimators of ordered parameters

For a certain general class of distributions and for the one-parameter exponential family of distributions, it is shown that ordered parameters do not admit unbiased estimators. It is also seen that these classes of distributions, for which non-existence of un¬biased estimators of ordered parameters is exhibited, can be enlarged

Trimmed estimates in simultaneous estimation of parameters in exponential families

Let X 1,…, X p be p (≥ 3) independent random variables, where each X i has a distribution belonging to the one-parameter exponential family of distributions. The problem is to estimate the unknown parameters simultaneously in the presence of extreme observations. C. Stein ( Ann. Statist. 9 (1981), 1135–1151) proposed a method of estimating the mean vector of a multinormal distribution, based on order statistics corresponding to the | X i |'s, which permitted improvement over the usual maximum likelihood estimator, for long-tailed empirical distribution functions. In this paper, the ideas of Stein are extended to the general discrete and absolutely continuous exponential families of distributions. Adaptive versions of the estimators are also discussed.

A.P.O. rules are asymptotically non deficient for estimation with squared error loss

The problem considered is sequential estimation of the mean θ of a one-parameter exponential family of distributions with squared error loss for estimation error and a cost c>0 for each of an i.i.d. sequence of potential observations X 1, X 2,...A Bayesian approach is adopted, and natural conjugate prior distributions are assumed. For this problem, the asymptotically pointwise optimal (A.P.O.) procedure continues sampling until the posterior variance of θ is less than c(r0+n), where n is the sample size and r 0 is the fictitous sample size implicit in the conjugate prior distribution. It is known that the A.P.O. procedure is Bayes risk efficient, under mild integrability conditions. In fact, the Bayes risk of both the optimal and A.P.O. procedures are asymptotic to 2V 0 √c, as c→0, where V 0 is the prior expectation of the standard deviation of X 1 given θ. Here the A.P.O. rule is shown to be asymptotically non-deficient, under stronger regularity conditions: that is, the difference between the Bayes risk of the A.P.O. rule and the Bayes risk of the optimal procedure is of smaller order of magnitude than c, the cost of a single observation, as c→0. The result is illustrated in the exponential and Bernoulli cases, and extended to the case of a normal distribution with both the mean and variance unknown.