Thesis--Copenhagen. / Translated by Poul Reppien. Errata slip inserted. Bibliography: p. [96]-99.

Thesis--University of Wisconsin--Madison. / Typescript. Vita. Description based on print version record. Includes bibliographical references (leaves 74-75).

Thesis (Ph. D.)--University of Wisconsin--Madison, 1975. / Typescript. Vita. Description based on print version record. Includes bibliographical references (leaves 238-242).

Thesis (Ph. D.)--University of Wisconsin--Madison, 1968. / Typescript. Vita. eContent provider-neutral record in process. Description based on print version record. Includes bibliography.

In this thesis we present two approximations to the distribution function of the sum of n independent random variables. They are obtained from generalizations of asymptotic expansions derived by Rubin and Zidek who considered the case of chi random variables. These expansions are obtained from Gurland's inversion formula for the distribution function by using an adaptation of Laplace's method for integrals. By means of numerical results obtained for a variety of common distributions and small values of n these approximations arc compared to the classical methods of Edgeworth and Cramer. Finally, the method is used to obtain approximations to the non-central chi-square distribution and to the doubly non-central F-distribution for various cases defined in terms of its parameters. / Science, Faculty of / Mathematics, Department of / Graduate

This research study had shown various probable rational methods of quantifying subjective information in a probability distribution with particular reference to the evaluation of economic projects by computer simulation. Computer simulation to give all the possible outcomes of a capital project using the Monte Carlo technique (method of statistical trials) provides a strong practical appeal for the evaluation of a risky project. However, a practical problem in the application of computer simulation to the evaluation of capital expenditures is the numerical quantification of uncertainty in the input variables in a probability distribution. One serious shortcoming in the use of subjective probabilities is that subjective probability distributions are not in a reproducible or mathematical form. They do not, therefore, allow for validation of their general suitability in particular cases to characterize input variables by independent means. At the same time the practical derivation of subjective probability distributions is by no means considered an easy or exact task. The present study was an attempt to suggest a simplification to the problem of deriving a probability distribution by the usual method of direct listing of subjective probabilities. The study examined the possible applicability of four theoretical probability distributions (lognormal, Weibull, normal and triangular) to the evaluation of capital projects by computer simulation. Both theory and procedures were developed for employing the four theoretical probability distributions to quantify the probability of occurrence of input variables in a simulation model. The procedure established for fitting the lognormal probability function to three-level estimates of probabilistic information was the principal contribution from this study to research in the search for improved techniques for the analysis of risky projects. A priori considerations for studying the lognormal function were discussed. Procedures were also shown on how to apply the triangular probability function and the normal approximation to simulate the outcomes of a capital project. The technique of fitting the Weibull probability function to three-level estimates of forecasts was adopted from a paper by William D. Lamb. The four theoretical probability functions wore applied to a case problem which was analyzed using subjective probabilities by David B. Hertz and reported in the Harvard Business Review. The proposal considered was a $10/-million extension to a chemical processing plant for a medium-sized industrial chemical producer. The investigations of the present study disclosed that the log-normal function showed considerable promise as a suitable probability distribution to quantify the uncertainties surrounding project variables. The normal distribution was also found to hold promise of being an appropriate distribution to use in simulation studies. The Weibull probability function did not show up too favourably by the results obtained when it was applied to the case problem under study. The triangular probability function was found to be either an inexact or unsuitable approximation to use in simulation studies as shown by the results obtained on this case problem. Secondary investigations were conducted to test the sensitivity of Monte Carlo simulation outputs to (l) number of statistical trials; (2) assumptions made on tail probabilities and (3) errors in the three-level estimates. / Business, Sauder School of / Graduate

During the past three decades or so there has been much work done concerning contagious probability distributions in an attempt to explain the behavior of certain types of biological populations. The distributions most widely discussed have been the Poisson-binomial, the Poisson Pascal or Poisson-negative binomial, and the Poisson-Poisson or Neyman Type A. Many generalizations of the above distributions have also been discussed. The purpose of this work is to discuss the multivariate analogues of the above three distributions, i.e. the Poisson-multinomial, Poisson-negative multinomial, and Poisson-multivariate Poisson, respectively. In chapter one the first of these distributions is discussed. Initially a biological model is suggested which leads us to a probability generating function. From this a recursion formula for the probabilities is found. Parameter estimation by the methods of moments and maximum likelihood is discussed in some detail and an approximation for the asymptotic efficiency of the former method is found. The latter method is asymptotically efficient. Finally sample zero and unit sample frequency estimators are briefly discussed. In chapter two, exactly the same procedure is followed for the Poisson-negative multinomial distribution. Many close similarities are obvious between the two distributions. The last chapter is devoted to a particular common limiting case of the first two distributions. This is the Poisson-multivariate Poisson. In this case the desired results are obtained by carefully considering appropriate limits in either of the previous two cases. / Science, Faculty of / Mathematics, Department of / Graduate

Let ℱ = {F} be a family of n-variate cumulative distribution functions (c.d.f.'s). If F₁...,F(k) belong to ℱ and P₁,...,P(k) are positive numbers that sum to 1, then the convex combination M(x₁ ,. . . , x(n)) = [formula omitted](x₁,...,x(n)) is called a finite mixture generated by ℱ. The F₁,…,F(k) are called the components of the mixture and the P₁,…,P(k) are called their weights, respectively. The mixture M(x₁,...,x(n)) is said to be identifiable with respect to ℱ if no other convex combination of a finite number of c.d.f.'s from ℱ will generate M(x₁,...,x(n)). We establish the identiflability of mixtures consisting of at most k components when the components belong to a family of univariate c.d.f.'s that have the following properties: (a) no two c.d.f.'s have the same mean; (b) each c.d.f. has the same r(th) central moment for r = 1,...,2k-1; and (c) the first 2k-1 central moments are finite. If the mixture and the 2k-1 central moments are known, a solution for the weights and means of the components is given. If a random sample is taken from the mixture, then asymptotically normal estimates of the weights and means are given, providing the 2k-1 central moments are known. Matrix mixtures are introduced and are found to be of use in estimating the density functions and c.d.f.'s of the components. In the .case of the above family, the estimates of the density functions are shown to have an asymptotically normal distribution. Consistent and least squares estimates are obtained for the component c.d.f.'s. We show that for multivariate mixtures identifiability of any one of the marginal mixtures implies the identifiability of the multivariate mixture, but not conversely. Finally, the univariate results are generalized to the multivariate case, and an example of the use of matrix mixtures is given. / Science, Faculty of / Mathematics, Department of / Graduate

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