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Two k-sample versions of an Anderson-Darling rank statistic are proposed for testing the homogeneity of samples. Their asymptotic null distributions are derived for the continuous as well as the discrete case. In the continuous case the asymptotic distributions coincide with the (k - 1)-fold convolution of the asymptotic distribution for the Anderson-Darling one-sample statistic. The quality of this large sample approximation is investigated for small samples through Monte Carlo simulation. This is done for both versions of the statistic under various degrees of data rounding and sample size imbalances. Tables for carrying out these tests are provided, and their usage in combining independent one- or k-sample Anderson-Darling tests is pointed out. The test statistics are essentially based on a doubly weighted sum of integrated squared differences between the empirical distribution functions of the individual samples and that of the pooled sample. One weighting adjusts for the possibly different sample sizes, and the other is inside the integration placing more weight on tail differences of the compared distributions. The two versions differ mainly in the definition of the empirical distribution function. These tests are consistent against all alternatives. The use of these tests is two-fold: (a) in a one-way analysis of variance to establish differences in the sampled populations without making any restrictive parametric assumptions or (b) to justify the pooling of separate samples for increased sample size and power in further analyses. Exact finite sample mean and variance formulas for one of the two statistics are derived in the continuous case. It appears that the asymptotic standardized percentiles serve well as approximate critical points of the appropriately standardized statistics for individual sample sizes as low as 5. The application of the tests... [ABSTRACT FROM AUTHOR]

Large-sample covariance matrices for the analysis of variance (ANOVA), minimum norm quadratic unbiased estimator (MINQUE), restricted maximum likelihood (REML), and maximum likelihood (ML) estimates of variance components are presented for the unbalanced one-way model when the underlying distributions are not necessarily normal. The limiting variances depend on the design sequence, on the actual values of the variance components, and on the kurtosis parameters of the underlying distributions. (The skewness parameters and other moments do not affect the limiting distributions.) Because all estimates are consistent and asymptotically normal, it is reasonable to compare the estimates using their asymptotic variances. Thus the efficiency of one estimate relative to another is defined as the ratio of their asymptotic variances, and these efficiencies are evaluated numerically and analytically for a variety of nonnormal situations. Various authors, including Hocking and Kutner (1975), Corbeil and Searle (1976), and Swallow and Monahan (1984), have compared the variances and/or mean squared errors of variance component estimates in a variety of finite-sample models assuming normally distributed effects. The purpose of this article is to extend this line of research to incorporate nonnormal distributions and large samples. Some special cases of MINQUE considered in this article are called MINQUE(0), MINOUE(1), MINOUE(Infinity), and MIVQUE (minimum variance quadratic unbiased estimator); these are defined by setting the a priori variance ratio to 0, 1, infinity (in a sense to be explained), and to the actual variance ratio, respectively. Because the actual variance ratio is usually unknown, the MIVQUE estimates cannot be computed for unbalanced designs. (In balanced designs MIVQUE and ANOVA correspond.) It turns out, however, that ML... [ABSTRACT FROM AUTHOR]

The usual ratio estimates for a population mean use information on an auxiliary variable to improve the estimation. The standard randomization analysis, however, does not take enough account of the observations on the auxiliary variable, so estimates and estimated variates can be conditionally biased unless the sample is balanced with respect to the auxiliary variable. I propose a method of adjustment for the bias of the estimate and its variance based only on the assumption of simple random sampling, which uses an approximate conditional distribution of the estimate given the mean of the auxiliary variable. Royall and Cumberland (1981) illustrated empirically that the usual random sampling estimates and estimated variances could be badly conditionally biased when samples were obtained that were unbalanced in the auxiliary variate. So if the sample mean and population mean, both of which are known at the time of analysis, are not close, it is known that the results are biased, but the usual random sampling formulas, which are purely unconditional, do not take account of this. They suggested that prediction models were necessary to adjust these. If the joint distribution of the sample means of the variate and covariate are considered together, then using the fact that these are asymptotically bivariate normal, it is possible to adjust approximately for the linear regression of the variate mean on the mean of the auxiliary variate. In this way a bias-adjusted estimate is suggested and formulas for the conditional variances of the estimates are obtained. A simulation study of two populations, one where the model on which the ratio estimate is based holds and one where it does not, is carried out to illustrate the improvements given by the results and to compare them with results based on the formulas... [ABSTRACT FROM AUTHOR]

The article is a technical comment on the negative correlation observed between the data-determined bandwidth and a value that minimizes ASE. This effect may also be demonstrated in a much simpler setting. This phenomenon is related to a negative correlation between ASE(h) and author's unbiased risk estimate, which was observed in a related situation. The negative correlation between unbiased estimates of bandwidths and their true values found here by author occurs widely and appears worthy of further study.