Showing papers on "Natural exponential family published in 1968"

Life Distributions Derived from Stochastic Hazard Functions

Abstract: Most of the familiar time-to-failure distributions used today are derived from hazard functions whose parameters are assumed constant. An unconditional time-to-failure distribution is derived here by assuming that a parameter of a classical failure distribution (viz., exponential and Weibull) is a random variable with a known distribution. With the use of the derived compound distributions and Bayesian techniques, it is possible to join the test data with prior information to arrive at a combined, and possibly superior, estimate of reliability. The prior distributions considered here are the two-point, the uniform, and the gamma. Conceptually, such a scheme may be a more realistic model for describing failure patterns under specific conditions.

Some results on tests of separate families of hypotheses

Abstract: SUMMARY Tests of the log normal distribution versus the exponential distribution were proposed by Cox (1961, 1962), who gave their large-sample distributions. We investigate the adequacy of the asymptotic results and give power functions of the tests. We then use Cox's general results to derive tests for the log normal distribution versus the gamma distribution. Finally, we compare the performance of the separate family tests of the exponential versus the log normal distribution with that of other tests for departure from the exponential distribution.

Reliability Applications of a Bivariate Exponential Distribution

Abstract: This paper examines some two-unit systems in which the lifetimes of the two units in service are not independent but depend upon one another in a particular way. This dependence is characterized by the bivariate exponential distribution of Marshall and Olkin, which has exponential marginal distributions and other physically motivating properties. Two measures of reliability are determined: the first is the distribution and mean of the time to system failure (i.e., when all units are failed) and the second gives steady-state probabilities of the number of working units. Some graphical results are given to illustrate the deviation of these quantities from the values obtained under the classical assumption of independent lifetimes.

Large deviations theory in exponential families

Abstract: 0. Summary. We consider repeated independent sampling from one member of an exponential family of probability distributions. The probability that the sample mean of n such observations falls into some set Sn is, by definition, a "large deviations", "small deviations", or "medium deviations" problem depending on the location of the set S. relative to the expectation of the distribution. We present a theorem which allows the accurate approximation of all such probabilities under a wide variety of circumstances. These approximations are shown to yield simple and numerically accurate expressions for the small sample power functions of hypothesis tests in the exponential family. Various large sample properties of exponential families are presented, many of which are seen to be extensions and refinements of familiar large deviations results. The method employed is to replace the given exponential family by a suitably modified normal translation family, which is shown to approximate the original family uniformly well over any bounded subset of the parameter space. The simple and tractable nature of normal translation families then provides our results. 1. Introduction and an outline of the paper. Exponential families of probability distributions play a dominant role in parametric statistical analysis, embracing almost all of the common univariate and multivariate distributions. In this paper we represent a d parameter, or d dimensional, exponential family by

On the linear exponential family

Abstract: In this paper we give a characterization theorem for a subclass of the exponential family whose probability density function is given bywhere a(x) ≥ 0, f(ω) = ∫a(x) exp (ωx) dx and ωx is to be interpreted as a scalar product. The random variable X may be an s-vector. In that case ω will also be an s-vector. For obvious reasons we will call (1) as the linear exponential family. It is easy to verify that the moment generating function (m.g.f.) of (1) is given by

Some Distribution Problems of Order Statistics From Exponential and Power Function Distributions1

Abstract: This paper gives alternative straightforward and simpler proofs of some of the results of Laurent's [10], and Likes' [11], [13]. The derivation of the results is simplified by using the theory of Dirichlet's multiple integral and the transformation used to derive this multiple integral. Some applications of Dirichlet's transformation to order statistic theory from gamma, and normal populations, have been already given by Kabe [7].