Showing papers on "Natural exponential family published in 1975"
TL;DR: The discrete Weibull distribution is defined to correspond with the continuous time Weibell distribution in continuous time as mentioned in this paper, and a few properties of the discrete Webull distribution are discussed.
Abstract: The discrete Weibull distribution is defined to correspond with the Weibull distribution in continuous time. A few properties of the discrete Weibull distribution are discussed.
389 citations
TL;DR: In this article, exact expressions for the first and second order moments of order statistics from the truncated exponential distribution, when the proportion 1-P of truncation is known in advance, are presented.
Abstract: Exact expressions for the first and second order moments of order statistics from the truncated exponential distribution, when the proportion 1–P of truncation is known in advance, are presented in this paper. Tables of expected values and variances-covariances are given for P = 0.5 (0.1) 0.9 and n = 1 (1) 10.
50 citations
TL;DR: The exponential distribution is often characterized as the only distribution with lack of memory as mentioned in this paper, and the exponential is the only one distribution that is sometimes forgetful, but this is not always true.
Abstract: The exponential distribution is often characterized as the only distribution with lack of memory. This note points out a stronger result: the exponential is the only distribution that is occasionally forgetful.
47 citations
TL;DR: In this article, it is shown that the claim distribution is well described by distribution functions where this condition is not satisfied (cf. the frequent use of Pareto distributions in this context, see below).
Abstract: Almost all analytical studies of the probability of ruin in risk business have been based on the assumption that the moment generating function of the claim distribution is finite for some positive real argument. But it is well known that in many practical cases, the claim distribution is well described by distribution functions where this condition is not satisfied (cf. the frequent use of Pareto distributions in this context, see below).
35 citations
TL;DR: In this paper, the decay of the maximum of the successive partial sums of independent and identically distributed random variables is asymptotically equal to c. Under conditions typical for complete exponential convergence, G(x)-G(x) is the limiting distribution function.
Abstract: Let Gn(x) be the distribution function of the maximum of the successive partial sums of independent and identically distributed random variables and G(x) its limiting distribution function. Under conditions, typical for complete exponential convergence, the decay of Gn(x) - G(x) is asymptotically equal to c.H(x)n- Qrn as n-* oo where c and v are known constants and H(x) is a function solely depending on x. COMPLETE EXPONENTIAL CONVERGENCE; MAXIMUM OF SUMS OF INDEPENDENT AND IDENTICALLY DISTRIBUTED RANDOM VARIABLES; SUPREMUM OF PROCESSES WITH STATIONARY INDEPENDENT INCREMENTS; WAITING TIME DISTRIBUTION; EXPONENTIAL RATE OF DECAY
29 citations
Journal Article•
TL;DR: In this paper, a class of models generalizing exponential families via the algebraic structure of the sufficient statistics is defined via a sequence of sufficient statistics from successive repetitions of experiments corresponding to a general exponential model.
Abstract: : A class of models generalizing exponential families is defined via the algebraic structure of the sufficient statistics. The maximum likelihood estimate for the unknown parameter is shown to exist and be unique. The sequence of sufficient statistics from successive repetitions of experiments corresponding to a general exponential model is shown to form an extreme family of Markov chains as defined by Lauritzen (1974).
23 citations
TL;DR: In this article, a family of bivariate Dirichlet distributions for which both conditional distributions are beta is evaluated, and some multivariate extensions are briefly discussed, where the assumptions that both conditional distribution and one marginal distribution are beta are assumed.
Abstract: Distributions having beta conditional distributions arise in connection with the generation of distributions for random proportions which do not necessarily possess neutrality properties (see [3]). The family of bivariate distributions for which both conditional distributions are beta is evaluated, and some multivariate extensions are briefly discussed. The bivariate Dirichlet distribution is characterized by the assumptions that both conditional distributions and one marginal distribution are beta.
18 citations
TL;DR: In this paper, it was shown that if a random variable has an unknown exponential family distribution with a natural parameter, then the moment generating function of the random variable is uniquely specified.
Abstract: It is shown that if $\mathbf{T}$ has an unknown exponential family distribution with natural parameter $\mathbf{\theta}$, then $\mathbf{G(\theta)} = \mathbf{ET}$ uniquely specifies the moment generating function The converse is proved, namely, if $\{\mathbf{T_\theta}\}$ is a family of random variables with moment generating functions of a certain form, then it must be an exponential family Moreover, several necessary and sufficient conditions are given so that a function can be the mean value function of an exponential family distribution
17 citations
TL;DR: The moment distributions of a nonnegative random variable are defined and their applications in life length studies are indicated in this paper, where some properties of the moment distributions are employed to characterize the discrete distributions in the class of modified power series distributions introduced by the author.
Abstract: The moment distributions of a nonnegative random variable are defined and their applications in life length studies are indicated. Some properties of the moment distributions are employed to characterize the discrete distributions in the class of modified power series distributions introduced by the author (1974). In particular, characterisations of Poisson, binomial and negative binomial distributions are obtained.
12 citations
TL;DR: Comparison with the method of Moore and Yalcin shows that this method gives a more accurate data analysis when used with their technique to determine the exponential time constant.
10 citations
01 Jan 1975
TL;DR: Several characterizations of the exponential distribution in terms of the variances and the covariances of order statistics in a random sample of size n (n ≥ 2) are made as mentioned in this paper.
Abstract: Several characterizations of the exponential distribution in terms of the variances and the covariances of order statistics in a random sample of size n (n ≥ 2) are made. Analogous characterizations hold for the positive exponential distribution.
TL;DR: In this article, the problem of testing for the exponential distribution (with scale, or both location and scale parameters unknown) against Weibull alternatives is considered, and upper bounds for the power of any invariant test are presented.
Abstract: The problem of testing for the exponential distribution (with scale, or both location and scale parameters unknown) against Weibull alternatives is considered. Upper bounds for the power of any invariant test are presented. Tests based upon the maximum likelihood estimator of the shape parameter, or a modification of it, are given which virtually achieve these bounds.
TL;DR: In this paper, the authors considered the exponential function with respect to exponential functions and showed that it can be computed in polynomial time, and they proposed an exponential function for exponential functions.
Abstract: (1975). On the Exponential Function. The American Mathematical Monthly: Vol. 82, No. 8, pp. 842-844.
TL;DR: In this paper, it was shown that the posterior pdf tends to the multivariate normal in exponential families and certain associated convex functions, using basic properties of exponential family and convex function.
Abstract: Let $f(x)$ be a $\operatorname{pdf}$ of exponential form with respect to the measure $\mu$. Suppose a prior $\operatorname{pdf}$ $\pi$ has been placed on the natural parameter space $\Omega$, where $\pi$ is a density (with respect to $m$-dimensional Lebesgue measure) which is both positive and continuous at $\tau^\ast$, the true but unknown parameter value. Using basic properties of exponential families and certain associated convex functions, it is shown that the posterior pdf tends to the multivariate normal.
TL;DR: In this paper, the matrix exponential technique is adapted as a direct integration procedure for predicting the dynamic response due to discrete inputs, and a sample problem whose exact solution is known is solved analytically, and the response is compared with the Wilson method, as well as matrix exponential.
TL;DR: Recently, many optimal or almost optimal Jackson-Müntz theorems on the approximation properties of the A-polynomials (3) for the interval [0, 1] have been published as mentioned in this paper.
Abstract: (3) P,(*)= £ bkx \\ bkER, s = l , 2 , . fc=i Recently, many optimal or almost optimal Jackson-Müntz theorems on the approximation properties of the A-polynomials (3) for the interval [0, 1] have been published (cf. J. Bak and D. J. Newman [1] and M. v. Golitschek [2]). Considering intervals [a, 1], a > 0, one would expect that the A-polynomials have even better approximation properties than on [0, 1], as the \"singular\" point* = 0 might have less influence. Theorems 1 and 2 prove this conjecture.
TL;DR: In this article, the authors considered the class of delay distibutions in G1/G/1 systems with concave service time distributions and derived moment inequalities for M/D/1 and M/M/1 delay distributions which agree with the former in terms of the first two moments.
TL;DR: In this paper, a fundamental loss function is introduced, for which the optimality property of the maximum probability estimators yields the classical result of R.A. Fisher on the asymptotic efficiency of the M.p.
Abstract: In 1966–1969L. Weiss andJ. Wolfowitz developed the theory of „maximum probability” estimators (m.p.e.'s). M.p.e.'s have the property of minimizing the limiting value of the risk (see (2.10).) In the present paper, therfore, after a short description of the new method, a fundamental loss function is introduced, for which—in the so-called regular case—the optimality property of the maximum probability estimators yields the classical result ofR.A. Fisher on the asymptotic efficiency of the maximum likelihood estimator. Thereby it turns out that the m.p.e.'s possess still another important optimality property for this loss function. For the latter the parameters of the exponential distribution—in the one-and the two-dimensional case—are estimated by the new method; for the estimation of the location parameter aWeibull distribution — in a more general sense — is taken as a basis. This shows that the maximum likelihood estimators involve a greater risk than the m.p.e.'s.