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

I studied the dispersion of winter wrens in 100-year-old, second-growth, coastal western hemlock forest at the University of British Columbia Research Forest in southwestern British Columbia from 1978-81. Male winter wrens were territorial on non-overlapping territories at an average density of 60 per km2. An average of 8% were polygamous. Females occupied generally non-overlapping home ranges at least during the breeding period, but were not shown territorial, although this possibility could not be excluded. Winter wrens were not uniformly distributed but showed differential use of various individual ecosystems (as mapped by Klinka 1976) and ecosystems grouped by forest floor habitats. Surplus, potentially territorial males were available during the breeding period which could have occupied the empty or sparsely occupied areas. Invertebrate food was more abundant in habitats used by winter wrens as compared to avoided habitats, and, food is proposed as a factor in habitat selection by winter wrens. I propose that winter wrens are spaced by territoriality and clumped by suitable habitat, and suggest that these two factors influence the patterns of dispersion of winter wrens in coastal western hemlock forest, and perhaps elsewhere as well. Klinka's ecosystems and grouped ecosystems were proposed as indicative of the distribution of winter wrens, and perhaps of other wildlife species generally. If so, ecosystems can arid should be used as the base for the study and management of wildlife in the province of British Columbia, and perhaps elsewhere as well. / Forestry, Faculty of / Graduate

It is a common belief that there are invariant acoustic patterns in speech signals, which can be related to their phonetic description. These patterns are expected to remain invariant, independent of the language, speaker, phonetic context, etc. Although many investigations based on short-time spectral analysis have established the feasibility of extracting invariant cues in certain contexts, they could not provide a set of invariant cues in any given phonetic context. In this thesis, the Wigner distribution (WD) was used to analyze speech signals for the first time, to investigate acoustic invariance. The WD, like the spectrogram, provides a time-frequency description of the signal. Unlike the spectrogram, it provides correct marginals in the time and frequency domains, but it is not a positive distribution. It is demonstrated here that the partially smoothed WD, in which both the properties of positivity and correct marginals are sacrificed to some extent, provides a better time-frequency resolution than short-time spectral analyses methods. An implementation and an interpretation of the partially smoothed WD are presented. The choice of smoothing parameters and the nature of cross-term suppression in a partially smoothed WD are discussed in detail. It is shown that the cross-terms in a partially smoothed WD do not mask the underlying nature of a signal in the time-frequency plane. A partially smoothed WD was used to investigate acoustic invariance in voiceless, unaspirated stop consonants spoken by native speakers of English, Telugu and French. Contrary to reports in the literature, it was shown that the features "diffuse-rising" and "compact" spectral shapes were not unique to alveolar and velar places of articulation, respectively, but depended on the vowel context. The resulting ambiguities when specifying the place of articulation were resolved using Formant Onset Duration (time taken for the steady state formants to occur in the vocal tract after the consonantal release) and F₂ of the following vowel. The place of articulation was specified correctly for 86% of the tokens. Unlike in other investigations, the errors in specifying the place of articulation were uniformly distributed over all vowel contexts. / Applied Science, Faculty of / Electrical and Computer Engineering, Department of / Graduate

If spacing behaviour of snowshoe hares limits juvenile survival and recruitment during summer, removal of this behaviour should produce an increase in these parameters. During the summers of 1980 and 1981 I removed all adults from an 8 ha trapping grid and all first litter juveniles from another. Experiments were conducted in the southwestern Yukon during a period when hare populations were at peak densities. The experimental removals did not increase survival, but recruitment relative to control areas was higher to the adult removal grid in 1980 and to both the adult removal and juvenile removal grids in 1981. To determine whether juveniles trapped for the first time were residents or immigrants, I implanted adult females with, calcium-45. This was passed to nursing young and could be detected by scintillation counting of a sample of bone tissue taken from new recruits. Any juvenile without radioactive calcium was classed as an immigrant. The increase in recruitment on the removal areas was due to increased immigration. The number of resident recruits was equal on all study areas. Results support the hypothesis that spacing behaviour limits juvenile immigration but not survival. However, immigration to control areas was also high with immigrants making up 70% of the total number of juveniles present on the areas in October. If food limits snowshoe hare numbers, addition of food should lead to increased numbers through higher survival and immigration. If food supply influences spacing behaviour of hares, home range size should decrease with food addition. I supplied peak (1980) and declining (1981) hare populations on 8 ha grids (one in 1980 and 2 in 1981) with laboratory rabbit chow for 1 - 4 months during March through June. Population size was determined by live-trapping and movements of animals were monitored by radio telemetry. Food addition decreased weight loss and improved survival of hares in both years. Onset of breeding was advanced in males but not females. In 1980, the number of males on the food addition area was 1.4 times higher than those on the control area while the number of females did not differ. In 1981, numbers of males and females were up to 3.6 and 3.2 times higher respectively on the food addition area as compared to those on ;the control area. The differences were due mainly to increased immigration. Residents responded to food addition by decreasing home range size in 1980 but not in 1981. Movement of immigrants, as monitored by telemetry, to the food addition area indicated that some established home ranges there while others returned to their old home ranges. Results support the hypothesis that hare densities are limited by winter food supply during the early decline phase of the cycle and possibly during the peak phase as well. A decrease in home range size was not necessary for immigration to occur. To examine the relationship of dispersal to changes in snowshoe hare numbers, I monitored dispersal of hares during a population increase, peak, and early decline (1978-1982). Two methods were used: 1) a conventional removal grid in which all animals caught each trapping session were removed and 2) telemetry monitoring of radio-collared individuals. The number of animals caught on the removal area was correlated with density on the control area but per capita dispersal rate was not. Both the number of dispersers and the per capita dispersal rate were highest during the period of peak densities on the control area. Dispersal, as measured by the removal grid, was not density dependent. Only 23 of 265 radio-collared animals dispersed during the study. Dispersal accounted for an average of 11% of the losses of radio-collared animals during the population decline. Results from both telemetry and the removal grid indicated that the decline in hare numbers was not due to dispersal. The amount of dispersal as determined by the removal grid was much higher than that determined by telemetry. The difference was more pronounced during the population peak and early decline. This was due to the removal grid over-estimating the average amount of dispersal that was occurring because it attracted animals to it. These results point to the need to be more critical of the underlying assumptions of the removal grid method as a way of monitoring dispersal. / Science, Faculty of / Zoology, Department of / Graduate

As a classic parameter originated from the binomial distribution, the binomial pro- portion has been well studied in the literature due to its wide range of applications. In contrast, the reciprocal of the binomial proportion, also known as the inverse proportion, is often overlooked, although it plays an important role in sampling designs and clinical studies. To estimate the inverse proportion, a simple method is to apply the maximum likelihood estimation (MLE). This estimator is, however, not a valid estimator because it suffers from the zero-event problem, which occurs when there is no successful event in the trials. At first, we review a number of methods proposed to overcome the zero-event problem and discuss whether they are feasible to estimate the inverse proportion. Inspired by the Wilson (1927) and Agresti and Coull (1998), in this thesis, we focus on a family of shrinkage estimators of the inverse proportion and propose to derive the optimal estimator within this family. The shrinkage estimator overcomes the zero-event problem by including a positive shrinkage parameter, which is intrinsically related to the expected value of the resulting estimator. To find the best shrinkage parameter, the relationship between the shrinkage parameter and the estimation bias of the shrinkage estimator is investigated systematically. Note that the explicit expression of the expected value function of the estimator and the best shrinkage parameter are quite complicated to compute when the number of trials is large. Hence, we review three methods in the literature which were proposed to approximate the expected value function. And after being inspired, we propose a new approximate formula for the expected value function and derive an approximate solution of the optimal shrinkage parameter by the Taylor expansion. Because there still exist an unknown binomial proportion in the optimal shrinkage parameter, we suggest a plug-in estimator for the unknown proportion with an adaptive threshold. Finally, simulation studies are conducted to evaluate the performance of our new estimator. As baselines for comparison, we also include the Fattorini estimator, the Haldane estimator and a piecewise estimator in the simulations. According to the simulation results, the new estimator is able to achieve a better or equally good performance compared with the Fattorini estimators in most settings. Hence, our new estimator can be a reliable estimator for the inverse proportion in most practical cases

The biogeography of the 50 currently identified Oregon species is reported from analyses of data from 2220 Berlese samples collected in a stratified, extensive sampling procedure, and from collection data on several hundred specimens from major private and public collections of pseudoscorpions. Individual species accounts including maps and data relating to geographical and seasonal distribution and habitats are provided, distributional areas and habitat types are categorized, dispersal mechanisms including examples of phoresy are presented and an illustrated key to Oregon species is given.

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