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, an estimation procedure for the mean life 0 having a small relative error is presented, with some pre-assigned confidence 1 a, within a certain percentage (100 6 percent) of the true but unknown mean life.
Abstract: In this note we find an estimation procedure for the mean life 0 having a small relative error. Put more precisely, we show how to get an estimate which is, with some preassigned confidence 1 a, within a certain percentage (100 6 percent) of the true but unknown mean life 0. In the language of life testing, this will require observing a suitably large number of failures, r. The exact solution of the problem involves considerations like those in [1]. Suppose that a fixed number, r, of failures is observed, and suppose that the associated total life is T , then it can be shown that the "best" estimator of 0 in the minimax sense corresponding to the loss function
5 citations
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TL;DR: It is shown that the Legendre-Fenchel transform of the maximized cumulant function yields a lower bound for the minimized large deviations rate function, and that in many cases this bound is tight.
Abstract: Large deviations theory is used to analyze the exponential rate of decrease of error probabilities for a sequence of decisions based on a test statistics sequence (T/sub n/). It is assumed that (for a given statistical hypothesis) the distributions of T/sub n/ are determined by some unknown member of a class of probability distributions. The worst case, or least favorably exponential rate of error probability decrease over this class, is sought. It is shown that the Legendre-Fenchel transform of the maximized cumulant function yields a lower bound for the minimized large deviations rate function, and that in many cases this bound is tight. Application of the result is illustrated by a detailed consideration of i.i.d memoryless detection with an epsilon -contamination distribution family. >
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TL;DR: In this article, a general framework for characterizations of a distribution by zero (or constant) regression properties of arbitrary degree polynomial statistics on the sample mean is presented, and various practical steps collected from the relevant literature are put together in this framework into a comprehensive guideline for constructing such characterizations.
Abstract: Characterizations of a distribution by zero (or constant) regression properties of arbitrary degree polynomial statistics on the sample mean are discussed. Various practical steps collected from the relevant literature are put together in this framework into a comprehensive guideline for constructing such characterizations. Applications are provided for natural exponential families (NEF’s). In particular, two reciprocal NEF’s associated with the continuous time symmetric Bernoulli random walk are characterized using this guideline. Moreover, a class of infinitely divisible NEF’s having some polynomial variance function structure is discussed in this framework.
5 citations
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5 citations