Showing papers on "Natural exponential family published in 1974"
TL;DR: Two distinct versions of the generalized beta distribution of the second kind are considered in this article, and they compare favorably with the commonly used gamma and log normal distributions in their ability to fit selected sets of accumulated streamflow and precipitation amount data.
Abstract: Two distinct versions of the generalized beta distribution of the second kind are considered. These beta-type distributions compare favorably with the commonly used gamma and log normal distributions in their ability to fit selected sets of accumulated streamflow and precipitation amount data. The comparisons are based on empirical results associated with three different goodness of fit criteria. Since the cumulative distribution functions of these beta-type distributions are in closed form, they possess unique computational advantages over the gamma and log normal distributions.
92 citations
TL;DR: In this paper, it was shown that the multivariate distribution of a set of random variables has exponential minimums if the minimum over each subset of the variables has an exponential distribution, and that the life length of the system has an increasing hazard rate average distribution.
Abstract: The multivariate distribution of a set of random variables has exponential minimums if the minimum over each subset of the variables has an exponential distribution. Such distributions are shown equivalent to the more strongly structured multivariate exponential distributions described by Marshall and Olkin in 1967 in the sense that a multivariate exponential distribution can be found that gives the same marginal distribution for each minimum. The basic application of the result is that in computing the reliability of a coherent system a joint distribution for the component life lengths with exponential minimums can be replaced by a multivariate exponential distribution. It follows that the life length of the system has an increasing hazard rate average distribution. Other applications include characterizations of multivariate exponential distributions and the derivation of a positive dependence condition for multivariate distributions with exponential minimums.
54 citations
TL;DR: In this paper, a fixed-width confidence interval procedure is developed for estimating the minimum life of a sequence of independent and identically distributed (r.i.d.) random variadles with density in (l.l).
Abstract: where oc. < fL < oc. , 0 < a < oc are two unknown parameters. Consider a sequence XI> X2, ••• of independent and identically distributed (i.i.d.) random variadles (r.v.'s) with density in (l.l). Our object is to estimate JL, the minimum life. In section 2 we derive a sequential procedure to estimate fL pointwise with minimum risk, which is shown to be 'asymptotically risk efficient' (Starr (1966) ). Basu (1971) considered the same problem, but with a less general loss structure. We point out two gross mistakes in his paper: (i) the proof of his theorem 3 is incorrect; (ii) applicability of his sequential procedure in practice is not substantiated by his computations, since he used an algorithm of J. E. Moyal (given in Robbins (1959)) which is not applicable in this case. Since there is no algorithm available to compute the exact distribution of the random sample size, we study the moderate sample size behaviour of our procedure by Monte-Carlo methods using pseudo-random exponential deviates (i.e. density is .f(x ; 0, 1) ). In section 3, a sequential fixed-width confidence interval procedure is developed for estimating fL· The same is shown to be 'asymptotically consistent' and 'asymptotically efficient' in the Chow-Robbins' (1965) sense. For this problem also, moderate sample size behaviour of our procedure is studied.
52 citations
TL;DR: In this article, the Bhattacharyya bounds are considered for the unbiased estimation of a parametric function when the sampling distribution is a member of an exponential family of distributions.
Abstract: SUMMARY Bhattacharyya bounds are considered for the unbiased estimation of a parametric function when the sampling distribution is a member of an exponential family of distributions. It is shown that the Bhattacharyya bounds converge to the variance of the best unbiased estimator. The application of this result to variance determination is demonstrated with examples from the negative binomial distribution and from the exponential distribution in a reliability theory context.
30 citations
TL;DR: In this paper, the authors examined the family whose probability generating functions have the form of the generalized hypergeometric function, pFq [(a); (b); λ(s-1)].
Abstract: This paper examines the family whose probability generating functions have the form of the generalized hypergeometric function, pFq [(a); (b); λ(s-1)] . It includes a number of matching distributions as well as many classic discrete distributions. Properties may be derived from the differential equations satisfied by the various generating functions e.g. useful recurrence formulae for probabilities, cumulants, and moments about an arbitrary point can be obtained.
28 citations
TL;DR: In this paper, the use of Patnaik's chi-square approximation to the noncentral Chi-square distribution and the Wilson-Hilferty transformation of chi-squares to approximate normality is explored as a simple, efficient means of finding k-orderstatistic confidence bounds on parameters and reliability of the one-and two-parameter negative exponential distributions.
Abstract: The use of Patnaik's [28] chi-square approximation to the noncentral chi-square distribution and the Wilson-Hilferty [32] transformation of chi-square to approximate normality are explored as a simple, efficient means of finding k-order-statistic confidence bounds on parameters and reliability of the one- and two-parameter negative exponential distributions. An important implication of the results leads to obtaining simple closed-form approximations to percentiles of the beta distribution either with integer or noninteger parameters. The general methodology is also used to approximate confidence or prediction intervals for the time of the kth exponential failure based on r < k sample failure times.
21 citations
TL;DR: In this article, it was shown that given a family of probability measures on the real line which are equivalent to Lebesgue measure, a locally Lipschitz function of size n > 1$ variables yields sufficient data reduction only if the given family is exponential.
Abstract: Using a locally Lipschitz function $T$ of $n > 1$ variables one can reduce data consisting of a sample of size $n$ to one real number. If we are given a family of probability measures on the real line which are equivalent to Lebesgue measure then $T$ yields a sufficient data reduction only if the given family is exponential. This result is compared with the results of Brown (1964) and Denny (1970).
21 citations
TL;DR: In this article, series expansions and recurrence relations suitable for numerical computation are developed for the generalized exponential integral functions, and tables of these functions are presented in the microfiche section of this issue.
Abstract: Series expansions and recurrence relations suitable for numerical computation are developed for the generalized exponential integral functions. Tables of these functions are presented in the microfiche section of this issue.
19 citations
17 citations
14 citations
TL;DR: In this paper, a characterization of exponential distributions based on the property that the waiting time variable Z = Tn-Y has the same distribution as each one of the independent random variables, provided Y has a uniform prior is presented.
Abstract: A characterization of exponential distributions based on the property that the waiting time variable Z = Tn-Y has the same distribution as each one of the independent random variables , provided Y has a uniform prior is presented.
TL;DR: In this article, the authors considered the nonparametric selection problems under the more general two-way layout models and proposed a more general selection procedure based on the ranks of the observations.
Abstract: The nonparametric selection problems in analysis of variance have been mainly developed for one-way layout models. For example, Lehmann [4], Puri and Puri [6] and Alam and Thompson [1] have respectively discussed the selection procedures based on the ranks of the observations. Randles [7] has also emphasized the use of the Hodges-Lehmann estimates for the same models to eliminate the difficulties concerning the least favorable configuration (cf. Rizvi and Woodworth DO. Only a work for the two-way layouts is seen in Hollander [3]. We consider some selection problems under the more general two-way layout models. Let Xia be the random observation on the i-th treatment IIi in the a-th block and suppose that
TL;DR: In this article, a two-parameter Weibull distribution is used to estimate the reliability of an engineering device, and a Bayesian estimate of reliability is developed by assuming that a value β 0 of the shape parameter is known.
TL;DR: In this paper, a generalization of the work dealing with the normal and Poisson cases is presented, where a class of sequential procedures is proposed and bounds on the error probabilities are obtained.
Abstract: The estimation of restricted parameters by fixed sample size rules has been considered by Hammersley [1]. A sequential solution to the problem of estimating the mean of a normal distribution when it is some unknown integer and a general method for solving problems of this sort were studied by Robbins in [4]. Based on the work of Robbins, a sequential procedure for estimating the parameter of a Poisson distribution when it is known to be an integer was given in [3]. The results obtained herein represent a generalization of the work dealing with the normal and Poisson cases. A class of sequential procedures is proposed and bounds on the error probabilities are obtained. The expected sample sizes are investigated and a weak form of optimality is demonstrated under certain conditions.
TL;DR: In this paper, the authors define a countable family of probability measures and show that a member of such a family is "Sequentially Distinguishable" i f for any given ε > 0.
Abstract: Let {X,, n ~ l } be an iid (independent and identically distributed) stochastic sequence assumed to be governed by a member of a countable family of probability measures 4 = {P,: 8 e ~} where P, are defined on an appropriate probability space and ~ is countable. Observing sequent/ally the stochastic sequence {X,, n>=l} we want to stop at some finite stage and decide in favour of a member of the family 9 with a uniformly small probability of error. The family 9 is said to be "Sequentially Distinguishable" i f for any given ~ (O l and ~,(~l~___e for a given s (0<~<1) and V].
01 Feb 1974
TL;DR: In this paper, the use of Patnaik's (1949) Chi-square approximation to the noncentral Chi-squares distribution and the Wilson-Hilferty (1931) transformation of Chi-quare to approximate normality is explored as a simple, efficient means of finding one-or-two-order statistic confidence bounds on parameters of the one-and two-parameter negative exponential distributions.
Abstract: : The use of Patnaik's (1949) Chi-square approximation to the non-central Chi-square distribution and the Wilson-Hilferty (1931) transformation of Chi-square to approximate normality are explored herein as a simple, efficient means of finding one-or-two-order statistic confidence bounds on parameters of the one- and two-parameter negative exponential distributions. Such methods can be used when it is known that r of n sample items, r < or = n, have failed during a life test, but the times of some early failures are not known exactly. An important implication of the result applying to approximate confidence bounds on the mean of an exponential distribution from a single order statistic is that of obtaining simple-closed-form approximations to percentiles of the Beta distribution (with integer parameters). It is shown that a generalization of the approximation applies to Beta variates with noninteger parameters. (Modified author abstract)
01 Mar 1974
TL;DR: In this article, the size of the coefficients an of an exponential series f(x) = O ane Anx,x > 0, Re An> 0, to the function f was compared between weighted ip sums of the sequence (an) and weighted LP integrals of f on [O, W].
Abstract: Results are obtained which relate the size of the coefficients an of an exponential series f(x) = _n=O ane Anx,x > 0, Re An> 0, to the function f. These results involve comparisons between weighted ip sums of the sequence (an) and weighted LP integrals of f on [O, W).
TL;DR: In this article, the distribution of any linear combination of a finite number of truncated exponential variates from possibly n distinct populations is obtained by using the Laplace transform, and the distribution is demonstrated in a compact form which is quite suitable for computational purposes.
Abstract: The distribution of any linear combination of a finite number of truncated exponential variates from possibly n distinct populations is obtained by using the Laplace transform. The distribution is demonstrated in a compact form which is quite suitable for computational purposes. The results are exemplified. Finally, a brief remark on the distribution of the product of truncated exponential variates is also added.