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A theorem by Shah and Khatri [A.M.S. (1961) 321883-887] is extended to give the distribution of Q/X² , where Q is a positive definite quadratic form involving non-central normal variates and X² is an independently distributed chi-square variate. Conditions are given under which the distribution of this ratio reduces to that of non-central F. Kramer’s method for analysis of variance of a two-way classification with disproportionate subclass numbers is reviewed and shown to satisfy these conditions. Various functional forms of the non-centrality parameter for evaluating the power of his method are given. Additional algebraic and numerical results are obtained to compare the power of Kramer's method and the method of fitting constants (least squares) outlined by Yates [J.A.S.A. (1934) 24151-66]. A mnemonic rule, based on 310 randomly generated two-way classifications, is given for discriminating against use of Kramer’s method in situations where his method potentially may be very deficient in power as compared to the method of fitting constants.

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.

A brief introduction to the concepts and terminology of spectral analysis and a review of the standard methods for cross-spectral estimation, based on discrete time history data, are incorporated in Chapter 1. Co-spectral and quadrature-spectral estimators which are characterized by non-negative spectral windows are developed in Chapter 2. While the spectral windows for the co-spectral estimators are non-negative for all relevant values of the assignable constants, certain restrictions on these constants are necessary to assure the non-negativity of the quadrature-spectral window. The properties of these estimators are considered in detail. In Chapter 3 randomized co-spectral and quadrature spectral estimators are presented. These estimators depend on the random selection of sets of time differences, as opposed to the systematic evaluation of all possible time differences for the standard estimators. By suitable choices of probability distributions for the time differences and of weight functions, the expectations of the randomized estimators can be made equivalent to the expectations of the standard estimators or the estimators of Chapter 2. Since the randomized estimator is much simpler to use than the standard estimator, these estimators are compared in terms of their variances, given that they have equal expectations. The choice of probability distributions to yield minimum variance, given that the expectation is specified, is considered. Extremely simple co-spectral and quadrature-spectral estimators, for the case where the coefficients of the Fourier series expansions of realizations of the processes over a finite time interval can be obtained by means of suitable analog equipment, are developed in Chapter 4. The expectations, variances and covariances of these estimators are derived.

This paper considers the problem of placing fiducial limits on an unknown parameter of a population, on the basis of a sample drawn from the population. The concept of fiducial limits is generalized to that of finding a fiducial distribution for the parameter. Necessary conditions for application of fiducial theory are considered. Examples are given to illustrate the methods used. Particular attention is given to the meaning which should be associated with a fiducial probability statement. It is shown that, in many cases, fiducial probability has a meaning related to frequency of events. An example is given to illustrate a case where fiducial probability cannot be given such an interpretation. The relationship of the fiducial distribution to a Bayesian posterior distribution is considered. The use of fiducial theory is shown by applying it to solve two important problems in statistical estimation.

The female of Spalangia drosophilae, Ashm, attack the puparia of Drosophila melanogaster, laying eggs between the puparial case and the pupa. The procedure is as follows: the parasite makes a careful examination of the puparia, tapping it with its antennae; then it stings the pupa to immobility and lays an egg. In a series of observations carried at the Imperial Institute of Entomology, Canada, using Spalangia as parasite and Drosophila as host, it was noted that the ovipositing female seemed to avoid superparasitism -- that is, laying eggs in already parasitized puparia. Naturally, if the number of non-parasitized insects still available was relatively small, the restraint exercised by the female tended to break down, and several eggs would be laid in some of the hosts. In order to determine the number of times the female Spalangia could exercise restraint before she was compelled to lay an egg, a series of three hundred and sixty separate experiments were performed. These involved the exposure of groups of five, ten and twenty-five Drosophila puparia to different numbers of parasites for various lengths of time, so that considerable variation in the number of eggs laid occurred. The observational results were expressed in terms of the number of puparia containing zero, one, two, etc., eggs. As the probability theory involved in this type of situation does not seem to be available, it is proposed to develop it in such a manner that experimental results may be analyzed to estimate the number of times the parasite could withhold an egg. The aspect of the experimental data which is of primary interest is the number of cases of superparasitism occurring for any given number of eggs laid and hosts exposed.

This dissertation is a study of certain aspects of restricted ranking, a method intended for use by a panel of m judges evaluating the relative merits of N subjects, candidates tor scholarships, awards, etc. Each judge divides the N subjects into R classes, so that n₁ individuals receive a grade i (i = 1, 2, R; Σnᵢ = N) where the R numbers nᵢ are close to N/R (nᵢ = N/R when N is divisible by R) and are preassigned and the same for all judges. When this method is used, all subjects are treated alike, the grading system is the same for all judges and the grades of each judge are given equal weight. Equally important, the meaning of a particular grade is clear to each judge and the same for each judge. Under the null hypothesis that all nR = N subjects are of equal merit, tests of significance are developed to determine whether (1) a particular individual is superior or inferior to the rest of the subjects; (2) two particular subjects are of equal merit; (3) the individuals with the highest and lowest scores are respectively superior and interior to the rest of the subjects and (4) the nR subjects form a homogeneous group. The critical values of the test statistics for (1), (2) and (3) are tabled for small to moderate values of m, an approximation based on the asymptotic normality of the appropriate test statistic proving suitable for large m. The test of homogeneity (4) employs a sum of squares of subjects’ scores which is shown to be asymptotically distributed for m→∞ as chi-square with nR-1 degrees of freedom. For the special case of complete ranking (R=N), this statistic is identical to one proposed by Friedman (1937) form rankings. The behavior of two of these tests is theoretically investigated for the non-null case of nR-1 subjects having equal merit and one "outlying" subject whose merit exceeds the others. The assumption is made that each judge j assigns a grade to every subject i on the basis of a "subjective random variable" xᵢⱼ with mean equal to the "true"

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.

Elmer Roberts reports that German advisors to the Emperor believe the Japanese navy is superior to the American navy, should the two nations go to war. Roberts believes that this is unbiased information because the Germans despise Japan and are generally favorable to the United States.

A model is presented for the reliability growth of a system during a test program. Parameters of the model are assumed to be random variables with appropriate prior density functions . Expressions are then derived that enable estimates (in the form of expectations) and precision statements (in the form of variances) to be made of . projected system reliability at time t after the start of the test program . system reliability after the observation of failure data Numerical examples are presented, and extension to multi-mode failures is mentioned., This task was supported by: Special Projects, Code Sp-114, http://archive.org/details/bayesianreliabil08poll, NA

2 v. diagrs. 20 cm., First published in Latin in 1713., 1. Bd. 1. Th. Abhandlungen über die bei glücksspielen möglichen berechnungen, von C. Huyghens, mit anmerkungen von J. Bernoulli. 2. Th. Permutations- und combinationslehre.--2. Bd. 3 Th. Anwendungen der combinationslehre auf verschiedene glücks- und würfelspiele. 4. Th. Anwendung der vorhergehenden lehre auf bürgerliche, sittliche und wirthschaftliche verhältnisse. Anhang: Brief an einem freund über das ballspiel., The University of Michigan Historical Mathematics Collection, (dlps) ABZ9501.0001.001, (lccallno) QA273.B529 Zg 1899, https://www.lib.umich.edu/about-us/policies/copyright-policy

46 p., In Pamphlets. Philosophy, v. 8, no. 1., math: Math copy bound separately, The University of Michigan Historical Mathematics Collection, (dlps) ACB4604.0001.001, (lccallno) B29 .P18 v.8, https://www.lib.umich.edu/about-us/policies/copyright-policy

2 v. diagrs. 16 cm., "Literatur": v.1, p. [5]-6; v.2, p. [5], The University of Michigan Historical Mathematics Collection, (dlps) ACL9494.0001.001, (lccallno) QA273 .K72, https://www.lib.umich.edu/about-us/policies/copyright-policy

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.