Andrew P. Soms

The use of the tetrachoric series for evaluating multivariate normal probabilities

The tetrachoric series is a technique for evaluating multivariate normal probabilities frequently cited in the statistical literature. In this paper we have examined the convergence properties of the tetrachoric series and have established the following. For orthant probabilities, the tetrachoric series converges if ;[varrho]ij; 1/(k - 1) or k is odd and [varrho]ij > 1/(k - 2), 1 = 2 and all [varrho]ij such that the correlation matrix is positive definite is false.

An Algorithm for the Discrete Fisher's Permutation Test

Abstract An algorithm, especially suitable for a computer, is presented for carrying out Fishers two sample permutation test for integer-valued data with ties. It is shown that the same method can be applied to carry out the permutation Wilcoxon test in the presence of ties using average ranks. Some numerical examples are given.

Exact Confidence Intervals, Based on the Z Statistic, for the Difference Between Two Proportions

We show that the confidence interval version of the extended exact unconditional Z test of Suissa and Shuster (1985) for testing the equality of two binomial proportions is due to general results of Buehler (1957), Sudakov and references cited there (1974), and Harris and Soms (1984). We apply these results to obtain exact unconditional confidence intervals for the difference between two proportions, deriving an explicit solution for the “best” outcome, make some comments on Buehlers (1957) method and give a numerical example. The Appendix contains a listing of the necessary FORTRAN programs.

Bounds for the t-tail area

The bounds of Birnbaum (1942) and Sampford (1953) for the upper tail area of the normal distribution are extended to the upper tail of the t-distribution. Numerical and theoretical comparisons are made with the bounds of Peizer and Pratt (1968), Wallace (1959) and Soms (1977).

Rational Bounds for the t-Tail Area

Abstract The bounds of Boyd (1959) for the tail area of the normal distribution are extended to the t distribution. The adequacy of the approximation is discussed and numerical examples provided.

Applications of schur theory to reliability inference for k of n systems

The results of Hoeffding (1956), Pledger and Proschan (1971), Gleser (1975) and Boland and Proschan (1983) are used to obtain Buehler (1957) 1-α lower confidence limits for the reliability of k of n systems of independent components when the subsystem data have equal sample sizes and the observed failures satisfy certain conditions. To the best of our knowledge, for k ≠ 1 or n, this is the first time the exact optimal lower confidence limits for system reliability have been given. The observed failure vectors are a generalization of key test results for k of n systems, k ≠ n (Soms (1984) and Winterbottom (1974)). Two examples applying the above theory are also given.

The Lindstrom-Madden Method for Series Systems with Repeated Components.

The Lindstrom-Madden method of approximating lower confidence limits for series systems with unlike components is extended to series systems with repeated components utilizing the methods of Buehler, Sudakov and of Harris and Sons. An exact solution is given for no failures and key test results, together with an approximation for the general case. Numerical examples are also provided.

Some recent results for series system reliability

Asymptotic properties of key test results ate obtained. Some further results on the Lindstrom-Madden method are given, including a binomial-Poisson inequality, together with some numerical examples. Simplified proofs of some results of Pledger and Proschan (1971) and Nevius, Proschan and Sethuraman (1977) are provided and the listing of a FORTRAN program to calculate a lower bound to the 1-α lower confidence limit on the system reliability is given.

Permuiation tesis for k-sample binomial data with comparisons of exact and approximate P-levels

Exact ksample permutation tests for binary data for three commonly encountered hypotheses tests are presented,, The tests are derived both under the population and randomization models . The generating function for the number of cases in the null distribution is obtained, The asymptotic distributions of the test statistics are derived . Actual significance levels are computed for the asymptotic test versions , Random sampling of the null distribution is suggested as a superior alternative to the asymptotics and an efficient computer technique for implementing the random sampling is described., finally, some numerical examples are presented and sample size guidelines given for computer implementation of the exact tests.