J. C. W. Rayner

Welch's approximate solution for the Behrens-Fisher problem

The Wald, likelihood ratio, and score statistics have been calculated for the Behrens–Fisher problem. These statistics provide asymptotically equivalent and weakly optimal tests. The test based on the Wald statistic was found to be very similar to the V test discussed by Welch (1937); hence the V test was compared with the likelihood ratio and score tests in a power study. Since the powers were very similar and the V test was more convenient to use from several points of view, we recommend this test. Sizes of the V test, as well as a power comparison with the usual pooled t test, were also obtained.

Interpreting the skewness coefficient

We examine and extend the argument of Horswell and Looney (1993), who claimed that the test for normality based on the sample skewness coefficient does not reliably discriminate between skewed and symmetric distributions. Theoretical and simulation evidence is given.

Smooth extensions of Pearsons's product moment correlation and Spearman's rho

A smooth model for doubly ordered two-way contingency tables with no fixed marginals is given and the score test of the hypothesis of independence derived. For the saturated model the score statistic is the familiar Pearons Xp2, and the first component is simply related to Pearsons product moment correlation. The higher-order components provide the promised extensions. They provide powerful direction tests and are easy to use and interpret, assessing if the bivariate moments of the data are consistent with what might be expected under the independence model. If ranks are used the score statistic is still Xp2, and the first component is simply related to Spearmans rho. The higher-order components again provide the promised extensions. In both cases the components permit an informative and close scrutiny of the data.

Nonparametric Analysis for Doubly Ordered Two-Way Contingency Tables

Tsujitani (1992, Biometrics 48, 267-269) and Kateri (1993, Biometrics 49, 950-951) considered analyses for two-way contingency tables. In this reader reaction a nonparametric analysis using mnidrank scores is given. This analysis can be interpreted in terms of bivariate moments, does not make model assumptions, takes into account the ordinal nature of the data, and allows Monte Carlo P values to be given.

The asymptotically optimal tests

This paper reviews the asymptotically optimal tests: the likelihood ratio, score and Wald tests, with emphasis on the score and Wald tests. These tests are also given after a linear transformation of the parameter space. Use of this approach often simplifies the calculations involved, especially when multiple tests are to be derived.

Robustness of the sample correlation - the bivariate lognormal case

The sample correlation coefficient R is almost universally used to estimate the population correlation coefficient ρ. If the pair (X,Y) has a bivariate normal distribution, this would not cause any trouble. However, if the marginals are nonnormal, particularly if they have high skewness and kurtosis, the estimated value from a sample may be quite different from the population correlation coefficient ρ.

Multivariate geometric distributions

Families of multivariate geometric distributions with flexible correlations can be constructed by applying inverse sampling to a sequence of multinomial trials, and counting outcomes in possibly overlapping categories. Further multivariate families can be obtained by considering other stopping rules, with the possibility of different stopping roles for different counts, A simple characterisation is given for stopping rules which produce joint distributions with marginals having the same form as that of the number of trials. The inverse sampling approach provides a unified treatment of diverse results presented by earlier authors, including Goldberg (1934), Bates and Meyman (1952), Edwards and Gurland (1961), Hawkes (1972), Paulson and Uppulori (1972) and Griffiths and Milne (1987). It also provides a basis for investigating the range of possible correlations for a given set of marginal parameters. In the case of more than two joint geometric or negative binomial variables, a convenient matrix formulation ...

Tests of Fit for the Geometric Distribution

Abstract This article gives power comparisons of some tests of fit for the Geometric distribution. These tests include a Chernoff–Lehmann X 2 test, some smooth tests, a Kolmogorov–Smirnov test, and an Anderson–Darling test. This article suggests that a good test of fit analysis is provided by a data dependent Chernoff–Lehmann X 2 test with class expectations greater than unity, and its components. These data dependent statistics involve arithmetically simple parameter estimation, convenient approximate distributions, and provide a fairly complete assessment of how well the data agrees with a Geometric distribution. The power comparisons indicate also that the best performed single statistic is the Anderson–Darling statistic.

Goodness of fit for the Poisson distribution

Two problems with the usual X2 test of fit for the Poisson distribution are how to pool the data and how much power is lost by this pooling. Smooth tests of fit as outlined in Rayner and Best (1989) avoid the pooling problems and provide weakly optimal and therefore powerful tests. Power comparisons between X2, smooth tests and a modified Kolmogorov-Smirnov statistic are given.

A test for bivariate normality

A test of multivariate normality given by Koziol (1986, 1987) is examined in some detail for the bivariate case. The small sample null distribution is considered and power comparisons given. Examples are to illustrate the use of components of the test statistic.