Showing papers on "Natural exponential family published in 1984"
TL;DR: It is shown that additional information about the distributions is more important for the interarrival-time than for the service time, and one kind of shape constraint — assuming that the distribution is a mixture of exponential distributions — can be examined analytically.
Abstract: To evaluate queueing approximations based on a few parameters (e.g., the first two moments) of the interarrival-time and service-time distributions, we examine the set of all possible values of the mean queue length given this partial information. In general, the range of possible values given such partial information can be large, but if in addition shape constraints are imposed on the distributions, then the range can be significantly reduced. The effect of shape constraints on the interarrival-time distribution in a GI/M/1 queue was investigated in Part II (see "On Approximations for Queues, II: Shape Constraints," this issue) by restricting attention to discrete probability distributions with probability on a fixed finite set of points and then solving nonlinear programs. In this paper we show how one kind of shape constraint — assuming that the distribution is a mixture of exponential distributions — can be examined analytically. By considering GI/G/1 queues in which both the interarrival-time and service-time distributions are mixtures of exponential distributions with specified first two moments, we show that additional information about the distributions is more important for the interarrivai time than for the service time.
81 citations
TL;DR: In this paper, the Double Exponential Distribution (DED) was used to find a maximum likelihood estimator for a given set of probability distributions, using the calculus of the double exponential distribution.
Abstract: (1984). The Double Exponential Distribution: Using Calculus to Find a Maximum Likelihood Estimator. The American Statistician: Vol. 38, No. 2, pp. 135-136.
68 citations
TL;DR: In this paper, a particular class of p-dimensional exponential distributions have Laplace transforms |I + VT|−1, V positive definite or positive semi-definite and T = diagonal (t1,…, tp).
67 citations
TL;DR: The Libby-Novick class of three-parameter generalized beta distributions as discussed by the authors provides a rich class of prior distributions for the binomial model that removes some of the restrictions of the...
Abstract: The Libby-Novick class of three-parameter generalized beta distributions is shown to provide a rich class of prior distributions for the binomial model that removes some of the restrictions of the ...
66 citations
TL;DR: In this paper, the Tarski-Seidenberg theorem for the field of reals with exponentiation was extended to the case of exponential rings, a special class of ring types, where the underlying ring has no nilpotents φ 0 and has characteristic a prime.
Abstract: Introduction. An exponential ring, or E-ring for short, is a pair (i?, E) with R a ring—in this paper always commutative with 1—and E a morphism of the additive group of R into the multiplicative group of units of R, that is, E(x + y) = E(x)E(y) for all x9 y in i?, and E(0) = 1. Examples are (R, a), a any positive real, and (C, e). Of course, any ring R can be expanded to an £-ring (R, E) by putting E(x) — 1 for all x\\ such brings will be called trivial. Ken Manders observed that an Zί-ring whose underlying ring has no nilpotents φ 0 and has characteristic a prime/? > 0 is trivial: in such a ring each x satisfies 1 = E(0) — E(px) = E{x), so (E(x) \\y = 0, which implies E(x) = 1. Related notions of exponential ring have been considered by M. Beeson, by B. Dahn and Wolter, and by A. Wilkie, all in connection with the longstanding open problem of A. Tarski on the decidability of the field of reals with exponentiation. An effective positive solution to this problem seems unlikely without major advances in transcendental number theory: such a solution would give us a decision method to answer any question: is e = p/q>> where/?, q are positive integers. Of course there is such a decision method, but, as we don't know yet whether e is rational, we don't know how it works. Now in mathematical practice it is less the effectiveness of Tarski's decision method for the real field which matters—though this aspect is interesting—but rather the information the method provides on the algebraic-topological nature of the definable sets in R, and on the asymptotic behavior of definable functions. For example in semi-algebraic and real algebraic geometry this use is formalized in the Tarski-Seidenberg theorem (in an inconstructive version) and in a result like the finiteness of the number of connected components of a semi-algebraic set. Parts of this use of Tarski's work on the elementary theory of the reals offer more hope of being generalized to the E-ήng (R, e). The following
54 citations
Book•
01 Dec 1984
40 citations
TL;DR: In this paper, it was shown that the HNBUE family of life distributions is closed under weak convergence and that weak convergence within this family is equivalent to convergence of each moment sequence of positive order to the corresponding moment of the limiting distribution.
Abstract: We show that the HNBUE family of life distributions is closed under weak convergence and that weak convergence within this family is equivalent to convergence of each moment sequence of positive order to the corresponding moment of the limiting distribution. A necessary and sufficient condition for weak convergence to the exponential distribution is given, based on a new characterization of exponentials within the HNBUE family of life distributions. EXPONENTIAL DISTRIBUTIONS
36 citations
Journal Article•
35 citations
TL;DR: The results show that it is possible to combine these two methods to compute exponential divided differences accurately, and a hybrid algorithm is presented for which the error bound grows quite slowly with the order of the divided difference.
Abstract: : The traditional recurrence for the computation of exponential divided differences, along with a new method based on the properties of the exponential function, are studied in detail in this paper. Our results show that it is possible to combine these two methods to compute exponential divided differences accurately. A hybrid algorithm is presented for which our error bound grows quite slowly with the order of the divided difference. (Author)
33 citations
TL;DR: In this paper, the product moments of order statistics from a doubly truncated exponential distribution were obtained for all sample sizes and the relation between the moments and the order of the order was obtained.
Abstract: Recurrence relations for the product moments of order statistics from a doubly truncated exponential distribution are obtained. These relations allow us to evaluate the product moments for all sample sizes.
23 citations
TL;DR: In this article, the unique minimum variance unbiased estimator (UMVUE) was proposed along with an expression for its variance, and the asymptotic distribution was given for a special case and a generalized Bayes property was exhibited.
Abstract: Consider the problem of estimating the common location parameter of two exponential distributions when censored samples are taken. The unique minimum variance unbiased estimator (UMVUE) is found along with an expression for its variance. The asymptotic distribution is given for a special case and a generalized Bayes property is exhibited. Extensions include the case of k > 2 populations. Also the UMVUE is found for P(Y > X) and certain reliability functions.
TL;DR: In this paper, the authors studied the characterizations of the exponential distribution and hence of the Poisson process by certain properties of the distributions of Ut and Vt on the nonnegative real line.
Abstract: Consider a renewal proess on the nonnegative real line with distribution function F(x). Then the backward or forward recurrence times Ut and Vt are , in general, non–independent r.v.s. and are indenpendence iff F(.) is exponential. Several authors have studied the characterizations of the exponential distribution and hence of the Poisson process by certain properties of the distributions of Ut and Vt.
TL;DR: In this article, an easily programmed recursive formula for the evaluation of the distribution function of ratios of linear combinations of independent exponential random variables is developed, which yields the probability that one team beats another in a contest we call the special gladiator game.
Abstract: Summary
An easily programmed recursive formula for the evaluation of the distribution function of ratios of linear combinations of independent exponential random variables is developed. This formula is shown to yield the probability that one team beats another in a contest we call the special gladiator game. This game generates tournaments which exhibit nontransitive dominance and have some surprising consequences. Similar results are obtained for a recursive formula based on the geometric distribution.
TL;DR: In this article, the finite sample distribution of the likelihood ratio sta-tistic is obtained for testing independence, given marginal homo-geneity, in the absolutely continuous bivariate exponential distri-bution of Block and Basu (1974).
Abstract: The finite sample distribution of the likelihood ratio sta-tistic is obtained for testing independence, given marginal homo-geneity, in the absolutely continuous bivariate exponential distri-bution of Block and Basu (1974). This test is discussed in light of the analysis of Gross and Lam (1981) on times to relief of head-aches for standard and new treatments on ten subjects.
TL;DR: The authors constructed prediction intervals for the l-th smallest observation and the mean lifetime of a future sample taken from a 2-parameter exponential distribution based on incomplete data, that is, where data may be missing from both extremes and/or elsewhere.
Abstract: This paper constructs prediction intervals for the l-th smallest observation and the mean lifetime of a future sample taken from a 2-parameter exponential distribution. The intervals are based on incomplete data, that is, where data may be missing from both extremes and/or elsewhere. These results are extended to some other distributions and provide conservative prediction intervals for the class of distributions with increasing failure rate.
01 Jan 1984
TL;DR: In this article, it was shown that there are exactly min(n+1, m) different positive values of γ with this property, where m is the number of positive values γ.
Abstract: Let Rn/m(z, γ)=Pn(z; γ)/(1-γz)m be a restricted rational approximation to exp(z), zeℂ, of order n for all real γ. In this paper we discuss how γ can be used to obtain fitting at a real non-positive point z1. It is shown that there are exactly min(n+1, m) different positive values of γ with this property.
TL;DR: In this paper, an asymptotic expansion of the distribution of the statistic for testing the equality of p two-parameter exponential distributions is obtained upto the order n −4 with the second term of the order N −3 where n is the size of the sample drawn from the i th exponential population.
01 Jun 1984
TL;DR: In this article, the Von-Neumann rejection approach via the exponential can be modified to become an efficient algorithm for generating a normal and then present a method for generating normal order statistics.
Abstract: : This paper indicates how exponential random variables can be effeciently used in a variety of simulation problems. One of the problems is the simulation of order statistics from a normal population. The authors discuss the general problem of simulating order statistics and then consider the normal case. They start by showing how the Von-Neumann rejection approach via the exponential can be modified to become an efficient algorithm for generating a normal and then present a method for generating normal order statistics. They show how to use the exponential to efficiently simulate random permutations with weights. They consider the problem of simulating a 2-dimensional Poissin process both for a homogeneous and nonhomogeneous Poisson process. (Author)
TL;DR: In this paper, the idea of Stein's estimator is extended to the general discrete and absolutely continuous exponential families of distributions, where adaptive versions of the estimators are also discussed, and a method of estimating the mean vector of a multinormal distribution, based on order statistics corresponding to the |Xi|'s, which permitted improvement over the usual maximum likelihood estimator, for long-tailed empirical distribution functions.
01 Jan 1984
TL;DR: In this article, the estimation of parameters of asymptotic distributions of extreme order statistics from exponential type parents is studied, and moment and maximum likelihood methods are compared relative to the latter for the top ten order statistics.
Abstract: This paper is concerned with the estimation of parameters of asymptotic distributions of extreme order statistics from exponential type parents. Moment and maximum likelihood methods are discussed; the asymptotic efficiency of the former method relative to the latter is evaluated for the top ten order statistics.
TL;DR: In this article, the existence of a statistic of the form with distribution independent of X(1) and depending on θ is characterized, and the case k=1 is of particular interest and an application in this case to uniformly most powerful unbiased tests is presented.
Abstract: Let be the order statistics of a sample of size n from the left truncated k-parameter exponential family of distributions with unknown truncation parameter, The densities of this family are of the form . Let and gn be a vector valued function. The existence of a statistic of the form with distribution independent of X(1) and depending on θ is characterized. The case k=1 is shown to be of particular interest and an application in this case to uniformly most powerful unbiased tests is presented. Right and two-sided truncation models are also discussed.
TL;DR: For a doubly truncated exponential distribution, the probability density function of a quasi-range is derived in this paper, and the density of sample range is obtained as a special case.
Abstract: For a doubly truncated exponential distribution, the probability density function of a quasi-range is derived. From this the density of sample range is obtained as a special case. Expressions for the mean and variance of the range are also obtained.