Though some of the discrete distributions, for example the binomial, hypergeometric, Poisson, are well tabulated, often statisticians use the percentage points of approximating continuous distributions when analysing discrete data. In this thesis, the exact relationships between certain discrete and continuous distributions are established, and these relationships are used for setting confidence limits and significance testing of hypotheses. In Chapter 1, statements of all distributions and mathematical functions used in this thesis are made, and also some approximations are mentioned without proofs. In Chapter 2, exact relationships between discrete distributions (the binomial, negative-binomial, and Poisson) and continuous distributions (the F and χ²) are proved. In Chapter 3, use is made of the approximate and exact relationships between discrete and continuous distributions, for setting confidence limits on the parameters of the discrete distributions. Chapter 4 consists of the approximate and exact significance testing of hypotheses by using the approximate and exact relationships, given in Chapter 2. In Chapter 5, two-sample, exact and approximate, significance tests of hypotheses on the Poisson distribution are performed, in the case of fixed number of events experimentation and fixed time experimentation. / M.S.

This paper provides a systematic study of order statistics drawn from discrete parent distributions. New procedures are followed for the derivation of the distribution of the r^th order statistic x_(r) and of the joint distribution of x_(r), x_(s) (s > r), that is, we first derive the cumulative probability distribution, from which the probability distribution comes directly. This approach is easier than the usual method, moreover the formulae for the c.d.f. derived in this way can be easily computed. To get the moments of X_(r), we use convenient formulae involving the tails of the c.d.f. of X_(r) rather than the p.d.f. of X_(r). The moments are then readily derived from general results for discrete distributions. We show the analogy between the results in the continuous and discrete cases. Applications to three discrete distributions are given. We consider some results on uncorrelated statistics which were established in the continuous case and show that the same results hold also for the discrete case. Many recurrence relations between moments of order statistics are derived in the discrete case yielding the same results as previously given by Govindarajulu (1963) and Sillitto (1951, 1964) in the continuous case. / M.S.

A statistical test is called"robust" if it is insensitive to departures from the underlying assumptions, this term was introduced by Box (4). Theoretical study made by Gayen (13), (14) and (15) showed that"Student's" t-test and the closely related F-test of analysis of variance are insensitive to departures from normality. But the F-test on the equality of two variances is very sensitive to such departures. Empirical studies made by Norton (19) and Boneau (3) agree with Gayen's theoretical conclusions. Norton studied the effect of non-normality on the F-test of analysis of variance,and showed that the form of the sampled population had very little effect on this test. For example, for the case of three groups of sample sizes 3, for the 5% level, the percentages exceeding the theoretical limits were 7.83 and 4.77% respectively for sampling from a leptokurtic and an extremely skewed population. Such property of robustness to non-normality on the F-test of analysis of variance is also possessed by t-test. Boneau's empirical study on the effect of non-normality on the two-sample t-test showed that for two samples of size 5, the significance level is respectively 3.1 and 5.1% for the empirical distribution of t’s from the exponential and uniform distribution compare to the nominal 5% value. The discrepancy is decreased when sample size is increased. / Master of Science

This thesis is concerned with an application of the principle of inclusion and exclusion and with related approximation techniques. These procedures are extensively employed for developing test criteria based on statistics expressible as maxima. Upper percentage points of a number of such statistics have been tabulated by various special methods. However, the application of the principle of inclusion and exclusion, coupled with the Bonferroni inequalities, is often useful in providing good approximations. An extensive review of this method is presented in this report. This procedure allows one to establish upper and lower limits to upper percentage points, say λ_α, of statistics expressible as maxima. The upper bound approximation to λ_α requires only the knowledge of the distribution(s) of the variates under consideration. The lower bound, however, requires also the joint distribution(s) of pairs of the variates. Since the joint distribution is often difficult to calculate, an approximation technique may be necessary. A detailed discussion of such an approximation with guidelines for its applicability to statistics other than those discussed is presented. Two alternative methods for the determination of upper percentage points for statistics expressed as maxima are discussed: Whittle's lower bound approximation and the assumption of independence. It is pointed out that Whittle's lower bound is stronger than that of Bonferroni only under certain conditions. The assumption of independence leads to approximately the same result as Bonferroni. / Master of Science

This thesis gives known theorems on which the concept and construction of confidence and two types of tolerance limits for normal distributions are based. Procedures are presented for computing two-sided conference and tolerance limits for means and simple linear regression data (simultaneous and non-simultaneous limits for each type). A numerical simple linear regression example is presented showing the six types of limits. An additional bibliography is given for reference on confidence and tolerance limits when information other than what is given in the thesis is desired. / M.S.

When some population values are completely from observation, the distribution from which the observations came is said to be truncated. Estimation of the parameters from truncated distributions has been an open field for research. This thesis examines the developments which have taken place in this area, giving the major writers and the methods used by them to obtain estimators. A. C. Cohen is responsible for much work involving the maximum likelihood procedure. Using the method of moments and several methods which they have developed, Rider, Plackett, Samford, Moore, Des Raj, and Halperin have made significant contributions. The Poisson, Normal, Binomial, Negative Binomial, and Gamma distributions are included in the investigation and along with the estimators, in some cases, asymptotic variances are given. Though much work has been done, there are many things left to be investigated. Only a small number of distributions have been dealt with, with all multivariate distributions other than the normal lacking any investigation. It is not known how the estimators are affected by small sample sizes, and with the aid of the computer variances can be examined. A new problem arises when the points of truncation are not clearly defined and complicated equations often make estimators difficult to fine. / M.S.

Orthogonal parameters for a distribution, f(x,8_i), (i=l,2,...,n), are defined such that E (- ∂² log f/∂β_i∂β_j=0 for all i, j, (i≠j). It has been pointed out that the problem of estimating the parameters of a distribution by maximum likelihood procedures, when the likelihood equations require iterative schemes for their solution, may be simplified by the use of orthogonal parameters. Distribution: having maximum likelihood equations where iteration is required include a number of two-parameter contagious distributions, like the Neyman type A and the Poisson-Binomial. The general method for finding orthogonal parameters is examined, and is seen to be inappropriate for the contagious distributions. An alternate method is developed by which orthogonal parameters are obtained for the Neyman type A, Poisson-Binomial, Binomial-Poisson, Geometric-Poisson, Logarithmic-Binomial, Logarithmic-Poisson and Normal-Poisson distributions, as well as for three Gram-Charlier type distributions. Some characteristics of the class of distributions to which the alternate method is applicable are discussed. The limitations of the general and the alternate methods are examined, and an example given where neither is of any use. It is also pointed out that, in many cases where orthogonal parameters are determined, simple transformations by which one can write the distribution in terms of the orthogonal parameters may not exist. It is concluded that the methods for determining the parameters are somewhat limited in scope; and that although the characteristics of the orthogonal parameters may be useful, the disadvantages associated with them may restrict their application. / Master of Science

Hill and Monticino (1998) introduced a constructive method for generating random probability measures with a prescribed mean or distribution on the mean. The method involves sequentially generating an array of barycenters that uniquely defines a probability measure. This work analyzes statistical properties of the measures generated by sequential barycenter array constructions. Specifically, this work addresses how changing the base measures of the construction affects the statististics of measures generated by the SBA construction. A relationship between statistics associated with a finite level version of the SBA construction and the full construction is developed. Monte Carlo statistical experiments are used to simulate the effect changing base measures has on the statistics associated with the finite level construction.

<p>An Archimedean lattice is a graph of a regular tiling of the plane, such that all corners are equivalent. A tiling is regular if all tiles are regular polygons: equilateral triangles, squares, et cetera. There exist exactly 11 Archimedean lattices. Being planar graphs, the Archimedean lattices have duals, 3 of which are Archimedean, the other 8 are called Laves lattices.</p><p>In the thesis, three measures of connectivity of these 19 graphs are studied: the connective constant for self-avoiding walks, and bond and site percolation critical probabilities. The connective constant measures connectivity by the number of walks in which all visited vertices are unique. The critical probabilities quantify the proportion of edges or vertices that can be removed, so that the produced subgraph has a large connected component.</p><p>A common issue for these measures is that they, although intensely studied by both mathematicians and scientists from other fields, have been calculated only for very few graphs. With the goal of comparing the induced orders of the Archimedean and Laves lattices under the three measures, the thesis gives improved bounds and estimates for many graphs. </p><p>A large part of the thesis focuses on the problem of deciding whether a given graph is a subgraph of another graph. This, surprisingly difficult problem, is considered for the set of Archimedean and Laves lattices, and for the set of matching Archimedean and Laves lattices.</p>

An Archimedean lattice is a graph of a regular tiling of the plane, such that all corners are equivalent. A tiling is regular if all tiles are regular polygons: equilateral triangles, squares, et cetera. There exist exactly 11 Archimedean lattices. Being planar graphs, the Archimedean lattices have duals, 3 of which are Archimedean, the other 8 are called Laves lattices. In the thesis, three measures of connectivity of these 19 graphs are studied: the connective constant for self-avoiding walks, and bond and site percolation critical probabilities. The connective constant measures connectivity by the number of walks in which all visited vertices are unique. The critical probabilities quantify the proportion of edges or vertices that can be removed, so that the produced subgraph has a large connected component. A common issue for these measures is that they, although intensely studied by both mathematicians and scientists from other fields, have been calculated only for very few graphs. With the goal of comparing the induced orders of the Archimedean and Laves lattices under the three measures, the thesis gives improved bounds and estimates for many graphs. A large part of the thesis focuses on the problem of deciding whether a given graph is a subgraph of another graph. This, surprisingly difficult problem, is considered for the set of Archimedean and Laves lattices, and for the set of matching Archimedean and Laves lattices.