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In stratified otolaryngologic (or ophthalmologic) studies, the misleading results may be obtained when ignoring the confounding effect and the correlation between responses from two ears. Score statistic and Wald-type statistic are presented to test equality in a stratified bilateral-sample design, and their corresponding sample size formulae are given. Score statistic for testing homogeneity of difference between two proportions and score confidence interval of the common difference of two proportions in a stratified bilateral-sample design are derived. Empirical results show that (1) score statistic and Wald-type statistic based on dependence model assumption outperform other statistics in terms of the type I error rates; (2) score confidence interval demonstrates reasonably good coverage property; (3) sample size formula via Wald-type statistic under dependence model assumption is rather accurate. A real example is used to illustrate the proposed methodologies.

In this paper, we extend the concept of near order statistic observation by considering observations that fall into a random region determined by a given order statistic and a Borel set. We study asymptotic properties of numbers of such observations as the sample size tends to infinity and the order statistic is a central one. We show that then proportions of these numbers converge in probability to some population probabilities. We also prove that these numbers can be centered and normalized to yield normal limit law. First, we derive results for one order statistic; next we give extensions to the multivariate case of two or more order statistics.

The Wilcoxon–Mann–Whitney statistic is commonly used for a distribution-free comparison of two groups. One requirement for its use is that the sample sizes of the two groups are fixed. This is violated in some of the applications such as medical imaging studies and diagnostic marker studies; in the former, the violation occurs since the number of correctly localized abnormal images is random, while in the latter the violation is due to some subjects not having observable measurements. For this reason, we propose here a random-sum Wilcoxon statistic for comparing two groups in the presence of ties, and derive its variance as well as its asymptotic distribution for large sample sizes. The proposed statistic includes the regular Wilcoxon rank-sum statistic. Finally, we apply the proposed statistic for summarizing location response operating characteristic data from a liver computed tomography study, and also for summarizing diagnostic accuracy of biomarker data.

The paper gives an asymptotic distribution of a test statistic for detecting a change in a mean of random vectors with dependent components. The studied test statistic has a form of a maximum of a square Euclidean norms of vectors with components being standardized partial cumulative sums of deviations from means. The limit distribution was obtained using a result of Piterbarg [1994. High deviations for multidimensional stationary Gaussian processes with independent components. In: Zolotarev, V.M. (Ed.), Stability Problems for Stochastic Models, pp. 197–210].

We discuss the nature of ancillary information in the context of the continuous uniform distribution. In the one-sample problem, the existence of sufficient statistics mitigates conditioning on the ancillary configuration. In the two-sample problem, additional ancillary information becomes available when the ratio of scale parameters is known. We give exact results for conditional inferences about the common scale parameter and for the difference in location parameters of two uniform distributions. The ancillary information affects the precision of the latter through a comparison of the sample value of the ratio of scale parameters with the known population value. A limited conditional simulation compares the Type I errors and power of these exact results with approximate results using the robust pooled t-statistic.

This article develops test statistics for the homogeneity of the means of several treatment groups of count data in the presence of over-dispersion or under-dispersion when there is no likelihood available. The C ( α ) or score type tests based on the models that are specified by only the first two moments of the counts are obtained using quasi-likelihood, extended quasi-likelihood, and double extended quasi-likelihood. Monte Carlo simulations are then used to study the comparative behavior of these C ( α ) statistics compared to the C ( α ) statistic based on a parametric model, namely, the negative binomial model, in terms of the following: size; power; robustness for departures from the data distribution as well as dispersion homogeneity. These simulations demonstrate that the C ( α ) statistic based on the double extended quasi-likelihood holds the nominal size at the 5% level well in all data situations, and it shows some edge in power over the other statistics, and, in particular, it performs much better than the commonly used statistic based on the quasi-likelihood. This C ( α ) statistic also shows robustness for moderate heterogeneity due to dispersion. Finally, applications to ecological, toxicological and biological data are given.

In a two-sample (say, X-sample and Y-sample) non-parametric testing problem, two joint null distributions of a statistic similar to the Galton statistic and either the number of Y-runs or the maximum length of Y-runs are derived by using two well-known combinatorial results which are associated with lattice paths.

Anderson (Biometrics 15 (1959) 582) proposed a χ2-type statistic for the nonparametric analysis of a randomized blocks design with no ties in the data. In this paper, we propose an Anderson statistic that allows for ties in the data. We show that the asymptotic distribution of the statistic under the null hypothesis of no treatment effect is a χ2 distribution. Under weak assumptions on the tie structure it is shown that the degrees of freedom for the asymptotic distribution is unchanged compared to the untied case. An extended analysis based on a partition of the statistic into independent components is suggested. The first component is shown to equal the Friedman rank statistic corrected for ties. The subsequent components allow for the detection of dispersion effects, higher order effects and differences in distribution. A simulation study is given and the new analysis is applied to a sensory evaluation data set.

We propose a goodness-of-fit statistic for testing exponentiality based on Hoeffding's maximum correlation. We study its small and large sample properties, we obtain its exact distribution, tables of exact and asymptotic critical values, and some power curves. We compare this statistic with the Gini statistic, the Shapiro–Wilk statistic for exponentiality and the Stephens’ modification of the latter.

Theory of variable selection for structural models that do not have clear dependent variables is developed. Theory is derived within the framework of the curved exponential family of distributions for observed variables. The idea of Rao's score test was taken to construct a test statistic for variable selection, and its statistical properties are examined. In particular, the test statistic is shown to have asymptotic central chi-square distribution under a kind of alternative hypothesis. This fact will provide an evidence for excellent performance of the score statistic for real data sets.

We define a chi-squared statistic for p-dimensional data as follows. First, we transform the data to remove the correlations between the p variables. Then, we discretize each variable into groups of equal size and compute the cell counts in the resulting p-way contingency table. Our statistic is just the usual chi-squared statistic for testing independence in a contingency table. Because the cells have been chosen in a data-dependent manner, this statistic does not have the usual limiting distribution. We derive the limiting joint distribution of the cell counts and the limiting distribution of the chi-squared statistic when the data is sampled from a multivariate normal distribution. The chi-squared statistic is useful in detecting hidden structure in raw data or residuals. It can also be used as a test for multivariate normality.

In applied statistics, the coefficient of variation is widely calculated and interpreted even when the sample size of the data set is very small. However, confidence intervals for the coefficient of variation are rarely reported. One of the reasons is the exact confidence interval for the coefficient of variation, which is given in Lehmann (Testing Statistical Hypotheses, 2nd Edition, Wiley, New York, 1996), is very difficult to calculate. Various asymptotic methods have been proposed in literature. These methods, in general, require the sample size to be large. In this article, we will apply a recently developed small sample asymptotic method to obtain approximate confidence intervals for the coefficient of variation for both normal and nonnormal models. These small sample asymptotic methods are very accurate even for very small sample size. Numerical examples are given to illustrate the accuracy of the proposed method.

This paper develops second-order asymptotic theory for score tests in proper dispersion models without imposing known dispersion. Our results can be used to provide an Edgeworth expansion for the test statistic or to obtain a Bartlett-corrected statistic. The latter is a modified version of the original statistic distributed as chi-squared with an error of order O (n −3/2 ) , n being the sample size, which is an improvement over Rao's original score statistic which has a chi-squared distribution with error of order O (n −1 ) . We also show that the formulae we obtain generalize a number of previously published results.

The confidence level of an inference table is defined as a weighted truth probability of the inference when sampling the reference set. The reference set is recognized by conditioning on the values of maximal partially ancillary statistics. In the sampling experiment values of incidental parameters are sampled from fiducial distributions. The concepts in this paper generalize ideas of Fisher's ‘Sampling the reference set’ (1961).

In this paper we propose test statistics for a general hypothesis concerning the adequacy of multivariate random-effects covariance structures in a multivariate growth curve model with differing numbers of random effects (Lange, N., N.M. Laird, J. Amer. Statist. Assoc. 84 (1989) 241–247). Since the exact likelihood ratio (LR) statistic for the hypothesis is complicated, it is suggested to use a modified LR statistic. An asymptotic expansion of the null distribution of the statistic is obtained. The exact LR statistic is also discussed.

The problem of prediction is concerned with predicting an unobserved random variable using a data dependent statistic. We extend the Rao-Blackwell theorem of Johansson ( Scand. J. Statist . (1990) 17, 135–145) in the prediction context to an arbitrary convex loss function. Two situations in which the problem of obtaining an unbiased predictor with minimum mean squared error of prediction can be reduced to UMVU estimation of an appropriate parametric function are described. The inadequacy of Rao-Blackwellization of an unbiased predictor when the prediction sufficient statistic is not complete is illustrated with two examples.

Generalized definitions of the invariance, (partial) sufficiency and ancillarity principles are given in a measure-theoretic context. Data reduction in statistical inference is described in terms of these principles.

We will show that an ancillary paradox, which is similar to Brown (Ann. Statist. 18, 1990), can occur in the estimation of the means vector in a situation involving random sample sizes. Suppose that some population has been partitioned into k strata. The mi observations will be drawn independently from the ith stratum. Assume that the vector of sample means has a k-variate normal distribution with mean vector θ. If mi is fixed, δ0, the admissible and minimax estimator of θ under ordinary squared error loss function, is given in Berger (Ann. Statist. 4, 1976). If mi is an ancillary statistic, we prove that δ0 is not admissible. Two new estimators for k⩾4 are investigated where one of them was proposed by Baranchik (Ann. Statist. 1, 1973). Some comparisons among all these estimators are performed.

This paper is concerned with testing the equality of several scale parameters in the presence of a common shape parameter β for samples drawn from Weibull distributions, or, equivalently and more conveniently we deal with testing the equality of several location parameters in presence of a common scale parameter b (= β -1 ) for samples drawn from extreme value distributions. In particular we derive a C (α) statistic (Neyman, 1959) for type II censored data. The performance of the C (α) statistic and several others, such as the likelihood ratio statistic LRu (Lawless, 1982), the quadratic statistic Qu (Nelson, 1982) and the shape parameter ratio statistic SP (McCool, 1979), is then compared in terms of size and power by using extensive Monte Carlo simulations. The statistics LRu and Qu are in general very liberal. The statistic SP does not have a known asymptotic distribution and in some situations it is beaten by some other statistics in terms of power. The statistic CMLu is conservative for small samples and for censored samples situations. However, it performs best in terms of power. So we developed an adjusted CMLu which maintains nomial level and hence will perform best in terms of power. Further, we derive a C (α) statistic for testing the assumption of a common b . The performance of this statistic and three others considered by Lawless and Mann (1976) and McCool (1979) are compared in terms of size and power by using some limited simulation experiments. Based on the simulation results we recommend the marginal likelihood ratio statistic (Lawles and Mann, 1976).

The results by Godambe (1980, 1984) and Liang (1983) concerning ancillarity of statistic T for parameter of interest θ1 in the presence of parameter θ2, and the loss of information in using the conditional distribution for inference concerning θ1, are extended to the general multiparametric case of θ1 and θ2. Also, relationships are examined concerning the appropriate information matrices.

We propose a test statistic for discrimination between alternative univariate binary response models which is asymptotically equivalent to the likelihood ratio statistic and Pearson's goodness of fit statistic. We propose an optimal design procedure. Under certain conditions we prove that the maximum value of the power can be obtained when the degrees of freedom of the test statistic is one. Several mathematical properties of the incomplete gamma function ratio and the non-central chi-squared distribution are required in the discussion and these are established.

We propose a test statistic for incomplete block designs in which the treatment effects are fixed and the block effect is random. The statistic uses combined intra-block and inter-block treatment estimates, with weights derived from the data. A Monte Carlo study produced two interesting results. First, the statistic is, in general, more powerful than the usual F -statistic based on intra-block estimates alone. The increase in power depends upon the eigenvalues of the incidence matrix, as well as the block and unit variances. Second, the proposed test statistics, under the null hypothesis of no treatment differences, behaves very much like an F -variate. Both results holt true for a wide variety of designs, for a range of variance values, and for several different variance estimators.

An L1 nonuniform central limit bound of order O(n − 1 2 ) is obtained under alternatives as well as under hypothesis for the generalized rank statistic (including the simple linear rank statistic, signed linear rank statistic, L-statistic, and a rank combinatorial statistic as special cases).