Topic
Natural exponential family
About: Natural exponential family is a research topic. Over the lifetime, 1973 publications have been published within this topic receiving 60189 citations. The topic is also known as: NEF.
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TL;DR: In this paper, the authors derived asymptotic approximations of the mean squared error (MSE) of the constrained empirical Bayes (CEB) estimators and provided second-order unbiased estimators of MSE based on the parametric bootstrap method.
Abstract: Empirical Bayes (EB) estimates in general linear mixed models are useful for the small area estimation in the sense of increasing precision of estimation of small area means. However, one potential difficulty of EB is that the overall estimate for a larger geographical area based on a (weighted) sum of EB estimates is not necessarily identical to the corresponding direct estimate such as the overall sample mean. Another difficulty is that EB estimates yield over-shrinking, which results in the sampling variance smaller than the posterior variance. One way to fix these problems is the benchmarking approach based on the constrained empirical Bayes (CEB) estimators, which satisfy the constraints that the aggregated mean and variance are identical to the requested values of mean and variance. In this paper, we treat the general mixed models, derive asymptotic approximations of the mean squared error (MSE) of CEB and provide second-order unbiased estimators of MSE based on the parametric bootstrap method. These results are applied to natural exponential families with quadratic variance functions. As a specific example, the Poisson-gamma model is dealt with, and it is illustrated that the CEB estimates and their MSE estimates work well through real mortality data.
8 citations
TL;DR: In this article, the positive dependence of a subclass of multivariate exponential distributions is examined, characterized by an index vector k and a parameter vector $\lambda$, which are used as an ordering to yield degrees of positive dependence.
Abstract: The positive dependence of a subclass of multivariate exponential distributions is examined. This class is characterized by an index vector k and a parameter vector $\lambda$, which are used as an ordering to yield degrees of positive dependence. The results presented have a direct implication on the reliability function of a system and the survival probability function of a shock model, and consequently on the optimal assembly of systems.
8 citations
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TL;DR: In this article, the assumption of unimodality for variables with distributions belonging to the two-parameter exponential family with known or constant dispersion parameter was discussed and a non-parametric method based on monotonicity properties was proposed.
Abstract: In this paper we discuss statistical methods for curve-estimation under the assumption of unimodality for variables with distributions belonging to the two-parameter exponential family with known or constant dispersion parameter. We suggest a non-parametric method based on monotonicity properties. The method is applied to Swedish data on laboratory verified diagnoses of influenza and data on inflation from an episode of hyperinflation in Bulgaria.
8 citations
TL;DR: Two probability distributions are analyzed which are formed by compounding inverse Weibull with zero-truncated Poisson and geometric distributions and are found to exhibit both monotone and non-monotone failure rates.
Abstract: In this paper two probability distributions are analyzed which are formed by compounding inverse Weibull with zero-truncated Poisson and geometric distributions. The distributions can be us...
8 citations
TL;DR: In this article, the authors proposed a distribution called Odds Exponential Log Logistic Distribution (OELLD), which is an odds family of distribution and hazard rate is an increasing and decreasing function based on the value of the parameter.
Abstract: We propose a distribution called Odds Exponential Log Logistic Distribution (OELLD), which is an odds family of distribution. Its hazard rate is an increasing and decreasing function based on the value of the parameter. Explicit expressions for the ordinary moments, L-moments, quantile, generating functions, Bonferroni Curve, Lorenz Curve, Gini's index and order statistics are derived. The parameters of the proposed distribution are estimated by using maximum likelihood method and also illustrated by a lifetime data set.
8 citations