Moments of the decisive sample number and of the average time to termination of sequential tests

The moments of the decisive sample number and of the average time to termination of sequential tests are derived. All moments depend only on the probability of continuation of the sequential test at each step, or at each instant of time for time processes. Two theorems are proved. Although the second theorem may be proved by quoting a result used in Barlow and Proschan (1965), p. 33, or both theorems from a lemma in Govindarajulu (1975), pp. 518-519, the author’s direct proofs illuminate the results.     

A monte carlo investigation of a statistic for a bivariate missing data problem

Testing the equal means hypothesis of a bivariate normal distribution with homoscedastic varlates when the data are incomplete is considered. If the correlational parameter, ρ, is known, the well-known theory of the general linear model is easily employed to construct the likelihood ratio test for the two sided alternative. A statistic, T, for the case of ρ unknown is proposed by direct analogy to the likelihood ratio statistic when ρ is known. The null and nonnull distribution of T is investigated by Monte Carlo techniques. It is concluded that T may be compared to the conventional t distribution for testing the null hypothesis and that this procedure results in a substantial increase in power-efficiency over the procedure based on the paired t test which ignores the incomplete data. A Monte Carlo comparison to two statistics proposed by Lin and Stivers (1974) suggests that the test based on T is more conservative than either of their statistics.     

Two stage multiple comparison procedures based on the studentized range

Stein’s (1945) two sample approach and Tukey’s T-Method of multiple comparisons (see e.g. Miller, 1966, Ch. 2) are combined to obtain fixed width simultaneous confidence intervals and simultaneous test procedures of predetermined Type I and Type II error levels, for all contrasts, in a one way layout. The necessary constants for implementing the two stage procedure are obtained under a least favorable configuration of the parameters. This provides the required protection of the null and alternative hypotheses under any configuration of parameters. A table is provided for some selected designs and error levels and an example is given to illustrate certain features of the new procedure.     

On adaptive statistical inferences

k?i=1 T_l T_k     

Tables of the cramér-von mises distributions

Tables of the one- and two-sample unweighted Cramer-von Mises statistics are given, and compared with the limiting distribution. The two-sample statistic may be useful in (for example) clinical trials when a proportional hazards assumption (which leads to the use of the log-rank test) is unjustified: see, for example, Schumacher (1984). It is often possible to stop clinical trials early if the Cramer-von Mises test (rather than say, the log-rank test) is employed.     

Combining Univariate Tests for Multivariate Location Problem

In this article, three methods of combining dependent univariate tests are studied. The Bahadur approximate efficiencies are derived under the asymptotic normal assumption. These procedures are applied to the multivariate location problem and compared with two Hotelling-type tests. A Monte Carlo study indicates that in certain cases the powers of the combination methods are much better than Hotelling's T ² and other multivariate nonparametric tests.     

Admissibility of Test Procedures for Linear Hypotheses in General Linear Model

This article shows that an F-test procedure is admissible for testing a linear hypothesis concerning one of the split mean vectors in a general linear model and an F-test procedure is also admissible for testing a linear hypothesis concerning another of the split mean vectors in the same model. These results are proved by showing that the critical functions of the tests are unique Bayes procedures with respect to proper prior distributions set in common for the null hypotheses and for the alternative ones, respectively.     

The Use of Noncentral F Approximations for Calculation of Power and Sample Size

Approximations to the noncentral F distribution yield surprisingly accurate results for power and sample size problems arising from linear hypotheses about normal random variables. The approximations are easy to use with a desk (or hand-held) calculator that computes cumulative F probabilities. These approximations are particularly advantageous for testing the hypothesis that differences among the means are small against the alternative that the differences are large.     

Use of the Correlation Coefficient with Normal Probability Plots

The use of the correlation coefficient is suggested as a technique for summarizing and objectively evaluating the information contained in probability plots. Goodness-of-fit tests are constructed using this technique for several commonly used plotting positions for the normal distribution. Empirical sampling methods are used to construct the null distribution for these tests, which are then compared on the basis of power against certain nonnormal alternatives. Commonly used regression tests of fit are also included in the comparisons. The results indicate that use of the plotting position p_i = (i - .375)/(n + .25) yields a competitive regression test of fit for normality.     

General treatment of goodness-of-fit tests based on Kullback–Leibler information

This paper introduces a general goodness-of-fit test based on the estimated Kullback–Leibler information. The test uses the Vasicek entropy estimate. Two special cases of the test for location–scale and shape families are discussed. The results are used to introduce goodness-of-fit tests for the uniform, Laplace, Weibull and beta distributions. The critical values and powers for some alternatives are obtained by simulation.