Showing papers on "Natural exponential family published in 1971"

A Prediction Problem Concerning Samples From the Exponential Distribution, With Application in Life Testing

Abstract: The problem of predicting the rth ordered observation Xr in a sample of n from an exponential distribution, based on the observed values of the first k ordered observations from the sample (k < r ≤ n), is discussed. It is shown how to find an interval estimate for Xr , and the result is used to predict the remaining elapsed time in certain life test experiments involving items whose life times follow an exponential distribution.

Approximate Fiducial Bounds on Reliability for the Two Parameter Negative Exponential Distribution

Abstract: We suggest herein a procedure for finding approximate fiducial bounds on the true reliability for the two parameter negative exponential distribution, which may be used as approximate “confidence” limits on reliability. These limits are simple and easy to compute. Also, they reduce to the exact confidence limits for the case of the single parameter negative exponential distribution.

Two-variate Exponential Distribution and Its Numerical Table for Engineering Application

Abstract: This study aims to develop the fundamental theory of a two-variate gamma distribution, especially of a two-variate exponential distribution for engineering application. In outline, the study is as follows : (1) Methods of estimating the parameters included in the probability density fuction of the distribution, the shape parameter in the marginal distribution of which is the same in each, are developed by using the techniques of maximum likelihood and moments. The results show that the estimator for the correlation parameter by the latter is coincident with the ordinary Pearsonian definition of correlation coefficient, but that by the former is not. (2) The characteristics of the two-variate exponential distribution, which is a special type of gamma distribution, especially the characteristics of a correlation surface and locus of the mode of the conditional probability density function are clarified theoretically and numerically in relation to the correlation parameters. (3) For convenience of engineering application of two-variate exponential distribution, numerical values of the conditional probability function are provided in a table, That is, for the fixed values of one variate, the computational values of the other variate are prepared under various conditional probabilities and correlation parameters.

A Monte Carlo Investigation of the Robustness of T 2

Abstract: The robustness of T2 for samples of size 5, 10 and 20 from several bivariate distributions is investigated. Samples are presented from bivariate normal, uniform, exponential, gamma, lognormal and double exponential distributions. Related observations on the one sample t and paired t are also given. Highly skewed distributions resulted in too many extreme values of T2 Other distributions gave conservative results. The use of the t-test and non-simultaneous techniques gave large overall levels of significance.

Reliability Estimation for the Generalized Gamma Distribution and Robustness of the Weibull Model

Abstract: Two test statistics are suggested for discriminating between the exponential model and the more general Weibull or gamma models, and these are compared to some previously used test statistics by Monte Carlo methods. The results of estimating reliability under an exponential assumption when the true model is Weibull is also investigated. These results as well as the tests mentioned above indicate that the exponential model is often not adequate when the more general models hold. In contrast to this result it was found that the Weibull model was quite robust relative to the generalized gamma distribution with regard to reliability estimation. Some general pivotal function properties are presented for the maximum likelihood estimator of reliability for the generalized gamma distribution and similar results also hold for the Weibull procedure under a generalized gamma assumption. These results made a Monte Carlo study of this problem feasible. Since the maximum likelihood estimators are apparently ill-behaved...

Note on a Characterizing Property of the Exponential Distribution

Abstract: In this note we discuss a characteristic property of the exponential distribution which is closely related to the "lack-of-memory" characterization.

Small Sample Power Functions for Nonparametric Tests of Location in the Double Exponential Family

Abstract: Eight nonparametric tests of location are examined in a small sample setting. Power functions are presented for samples drawn from the double exponential distribution. The results provide an example where the asymptotically most powerful rank test (the Mood median test) performs poorly for alternatives which are not very close to the null hypothesis.

On entire functions of exponential type

Abstract: Le t J be a n en tire fun c ti o n a nd le t p. \"\" 1 and 1(1 , r)= {f:~ 1 f 11)(re iO) I \"dO} 1/\". for a U s uffic ie ntly la rge r, the n J is of ex pone nti a l type not exceeding. {2 log (l-t. ~) + 1 + log (2N) !} .. If thi s co ndition is re place d by re lated co nditi ons, th e n a lso is of expo ne nti a l t ype. An e ntire fun c tion f(z) is said to be of bounde d ind ex if and only if th ere exi sts a non-negative integer N (ind e pe nde nt of z) s uch thatO\"'j \",N j!-k! (1.1) for all k and all z, and the smallest s uc h integer N is calle d the index off(z) ([1], [4] , [5]).1 It is known that a function of bounde d inde x N is of ex ponenti al type not exceedin g N+ 1 [6] but that a function of expon e ntial type need not be of bounde d inde x. In fac t any e ntire fun c tion havin g ze ros of arbitrarily large multipli city is not of bound e d index and th ere exist fun c tion s with simple zeros and of exponential type whic h are not of bounded index [8]. In a recent paper [2] Fred Gross considers interesting variations of condition (1.1) and proves the following THEOREM A: Let f be entire and C a positive constant. If there exists a positive integer N such that for k=O, 1,. . , N, f satisfies one ofthefollowing,for all z with l z I sufficiently large:

Simplified Estimation of Parameters in a Double Exponential Distribution

Abstract: Simplified estimators of the location and scale parameters of a double exponential distribution are given. Complete and symmetrically censored samples are considered. The high efficiency of these estimators relative to the best linear unbiased estimator (BLUE) make them useful in practice.

Approximate Confidence Limits for Complex Systems with Exponential Component Lives

Abstract: The asymptotic distribution of the log-likelihood ratio is shown to provide a method of determining approximate confidence bounds for the reliability function of any coherent system when each component has an exponential life with unknown failure rate and component performance data are provided in the form: number of failures (minimum of one) and total operating time. Thus the method applies under all general types of censoring. This extends the results of the authors, Ann. Math. Statist. (1968), on confidence limits for coherent structures with binomial data on the component's reliability. Methods similar to those previously utilized are combined with some special properties of the exponential distribution to obtain the results.

Maximum Likelihood Estimation in the Truncated, Single Parameter, Discrete Exponential Family

Abstract: A number of papers appear in the literature in which estimates are derived for parameters from truncated discrete probability density functions. A partial list is given in the references. Almost all of these papers deal with either the truncated binomial, negative binomial or poisson cases. These probability density functions are members of the single parameter, discrete exponential family (regular case). In the following, the maximum likelihood estimation equation is derived for all truncated densities which belong to this family. This more general derivation should help unify the work done on maximum likelihood estimation which is scattered throughout the literature.

The Effect of the Two Extremes in Estimating the Best Linear Unbiased Estimates (BLUE) of the Parameters of the Exponential Distribution

Abstract: Several authors, [1], [2], [3], [4], [5], [6], [7], have considered linear estimation of the parameters of the exponential distribution from censored samples or by using selected order statistics. It was shown in [7] that missing the smallest observation in small samples from the two-parameter single exponential distribution greatly affects the efficiency of the BLUE of the location parameter. Moreover, this loss in efficiency depends upon the sample size. Insofar as the BLUE of the scale parameter is concerned, missing the largest observation in small samples results in a considerable loss of its efficiency. These empirical results were of considerable interest but the expressions for the loss of efficiencies due to missing one or both extremes in the oneand twoparameter exponential distributions were not derived. It is important to obtain such expressions in order to investigate such losses in relation to sample size. A comparison of the loss of information caused by missing the smallest and/or the largest observation will complete that part of the picture regarding estimation.

On models for business failure data

Abstract: It is pointed out in this paper that Lomax's hyperbolic function is a special case of both Compound Gamma and Compound Weibull distributions, and both of these distributions provide better models for Lomax's business failure data than his hyperbolic and exponential functions. Since his exponential function fails to yield a valid distribution function, a necessary condition is established to remedy this drawback. In the light of this result, his exponential function is modified in several ways. It is further shown that a natural complement of Lomax's exponential function does not suffer from this drawback.

A method for nonlinear exponential regression analysis

Abstract: A computer-oriented technique is presented for performing a nonlinear exponential regression analysis on decay-type experimental data. The technique involves the least squares procedure wherein the nonlinear problem is linearized by expansion in a Taylor series. A linear curve fitting procedure for determining the initial nominal estimates for the unknown exponential model parameters is included as an integral part of the technique. A correction matrix was derived and then applied to the nominal estimate to produce an improved set of model parameters. The solution cycle is repeated until some predetermined criterion is satisfied.