Showing papers on "Natural exponential family published in 1976"
TL;DR: In this paper, the use of approximate posterior distributions resulting from operational prior distributions chosen with regard to the realized likelihood function is proposed, including mixed-type prior distributions with positive probabilities on singular subsets, and a new approximation is also given relating such distributions to absolutely continuous distributions with high local concentrations of density.
Abstract: This paper proposes the use of approximate posterior distributions resulting from operational prior distributions chosen with regard to the realized likelihood function. L.J. Savage's “precise measurement” is generalized for approximation in terms of an arbitrary operational prior density, including mixed-type prior distributions with positive probabilities on singular subsets. A new approximation is also given relating such distributions to absolutely continuous distributions with high local concentrations of density. Mixed-type distributions constructed from the natural conjugate prior distributions are proposed and illustrated in the normal-sampling case for unified Bayesian inference in testing and estimation contexts.
61 citations
45 citations
TL;DR: In this paper, the distribution functions of 3, fj+ and 1- for the exponential case are derived from general results for order statistics, and computationally efficient approximations to these distribution functions are obtained.
Abstract: SUMMARY Let 1, b+ and 13- denote Kolmogorov-Smirnov type one-sample statistics to test goodness of fit in the presence of unknown nuisance parameters; then the distributions of b, fj+ and i- depend on the population sampled and the estimator used. Simulation has been the primary tool for studying these statistics. Recently, Durbin obtained the distributions of D1, + and b- in terms of a Fourier transform for a wide class of underlying populations, and produced explicit results for the exponential case. In this paper, the distribution functions of 3, fj+ and 1- for the exponential case are derived from general results for order statistics, and computationally efficient approximations to these distribution functions are obtained. In the course of this derivation, Bonferroni inequalities of Kounias, and Sobel & Uppuluri are generalized. Certain problems of goodness-of-fit testing in the presence of nuisance parameters, whose solutions make use of existing tables, are also discussed. These problems include the Pareto, Rayleigh, power function, and uniform distributions.
40 citations
TL;DR: In this paper, it was shown that there exist at least two algebraically independent quantities among several values of the exponential function, and by using certain additional considerations, the author obtains a result concerning the algebraic independence of the values of both the exponential and the elliptic functions.
Abstract: With the aid of Gel'fond's method [1, 2], which makes it possible to show that there exist at least two algebraically independent quantities among several values of the exponential function, and by using certain additional considerations, the author obtains a result concerning the algebraic independence of the values of the exponential and the elliptic functions.
37 citations
TL;DR: For scale mixtures of distributions it is possible to prescribe simple moment measures of distance, such as the kurtosis and half the squared coefficient of variation minus one as discussed by the authors, which are exhibited as bounds on the uniform metric for the distance between distribution functions.
18 citations
14 citations
12 citations
12 citations
TL;DR: In this article, the posterior distributions and the posterior bounds of the reliability functions have been derived for the one and two-parameter exponential distributions. And the posteriors are tabulated and plotted and their robustness studied.
Abstract: The posterior distributions and the posterior bounds of the reliability functions have been derived for the one and twoparameter exponential distributions. Using Grubbs' (1971) data the posteriors are tabulated and plotted and their robustness studied
11 citations
8 citations
TL;DR: The robustness of the power function of the standard one-sample parametric test for the mean of the negative exponential distribution is examined in this paper, where the main departure from the exponential assumption is a mixture of negative exponential components although an alternative Gamma distribution is also examined.
Abstract: The robustness of the power function of the standard one-sample parametric test for the mean of the negative exponential distribution is examined The main form of departure from the exponential assumption is a mixture of negative exponential components although an alternative Gamma distribution is also examined It is found that the test is sensitive to these departures although the effect of mixtures with short tails is less dramatic than those with long tails
TL;DR: In this article, a class of exponential type distributions with special exponential parameters is defined, and it is assumed that the exponential parameters vary according to some known (known) probability law.
Abstract: A class of exponential type distributions with special exponential parameters is defined. It is assumed that the exponential parameters vary according to some (known) probability law. It has been shown in this paper that the compound distribution can be easily represented in form involving moment generating function of the mixing distribution. The results obtained in this paper provide an efficient and simple method of obtaining compound failure time distribution with known mixing distributions (uniform, exponential, gamma).
01 Jan 1976
TL;DR: In this paper, the authors review the basic theory of graph plotting and the interpretation of certain equations from a graphical point of view, and examine the graphs of this function and combination functions containing trigonometric and exponential components.
Abstract: In this chapter we shall review the basic theory of graph plotting and the interpretation of certain equations from a graphical point of view. In the process we shall look again at graphs of trigonometric functions, study a new topic — the exponential function — and examine the graphs of this function and combination functions containing trigonometric and exponential components.
TL;DR: In this article, Roy's union intersection principle is used to develop a test procedure to test the equality of scale parameters of several exponential distributions and the test statistic for two and three exponential distributions are tabulated and an illustrative simulated example is qiven.
Abstract: Roy's union-intersection principle is used to develop a test procedure to test the equality of scale parameters of several exponential distributions. Upper five and one percent values of the test statistic for two and three exponential distributions are tabulated and an illustrative simulated example is qiven.
01 Mar 1976
TL;DR: This paper will investigate the estimators of the parameter beta of the EMA1 process, and some basic properties of the epsilon(i), and then extend these results to the EmaK process.
Abstract: : Properties of a stationary sequence of random variables chi(i) which have exponential marginal distributions and random linear combinations of order one of an i.i.d. exponential sequence epsilon(i) were discussed by Lawrence and Lewis (1976); they called this model the EMA1 (exponential moving average of order one) point process. This paper will investigate the estimators of the parameter beta of the EMA1 process, and some basic properties of the EMA2 process, and then extend these results to the EMAK process.
TL;DR: In this paper, it was shown that the corresponoence between exponential and geometric distributions expressed by the lack of memory property is essentially still retained if an auxiliary random variable confined to the range 40,1) is added to the original exponential one.
Abstract: : It is shown that the corresponoence between exponential and geometric distributions expressed by the lack of memory property is essentially still retained if an auxiliary random variable confined to the range 40,1) is added to the original exponential one. (Author)
TL;DR: In this paper, the authors derived an estimator for failure-law parameters for the components of a series system under an attribute life test situation where the parameters of the failure laws are unequal.
Abstract: Estimators for failure-law parameters are derived for the components of a series system under an attribute life test situation where the parameters of the failure laws are unequal. Exponential, Rayleigh, mixed exponential and Rayleigh, and a special case of the Weibull failure models are treated. An example uses simulated data for a 10-component system.