Analysis of one-way layout of count data in the presence of over or under dispersion

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)$C(\alpha )$ 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)$C(\alpha )$ statistics compared to the C(α)$C(\alpha )$ 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)$C(\alpha )$ 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)$C(\alpha )$ statistic also shows robustness for moderate heterogeneity due to dispersion. Finally, applications to ecological, toxicological and biological data are given.     

On the distributions of multivariate sample skewness

In this paper, we consider the multivariate normality test based on measure of multivariate sample skewness defined by Srivastava (1984). Srivastava derived asymptotic expectation up to the order N⁻¹ for the multivariate sample skewness and approximate χ²${\chi }^2$ test statistic, where N   is sample size. Under normality, we derive another expectation and variance for Srivastava's multivariate sample skewness in order to obtain a better test statistic. From this result, improved approximate χ²${\chi }^2$ test statistic using the multivariate sample skewness is also given for assessing multivariate normality. Finally, the numerical result by Monte Carlo simulation is shown in order to evaluate accuracy of the obtained expectation, variance and improved approximate χ²${\chi }^2$ test statistic. Furthermore, upper and lower percentiles of χ²${\chi }^2$ test statistic derived in this paper are compared with those of χ²${\chi }^2$ test statistic derived by Mardia (1974) which is used multivariate sample skewness defined by Mardia (1970).     

Two sets of isotones for comparing tests of exponentiality

Isotones   are a deterministic graphical device introduced by Mudholkar et al. [1991. A graphical procedure for comparing goodness-of-fit tests. J. Roy. Statist. Soc. B 53, 221–232], in the context of comparing some tests of normality. An isotone of a test is a contour of p$p$ values of the test applied to “ideal samples”, called profiles, from a two-shape-parameter family representing the null and the alternative distributions of the parameter space. The isotone is an adaptation of Tukey's sensitivity curves, a generalization of Prescott's stylized sensitivity contours, and an alternative to the isodynes   of Stephens. The purpose of this paper is two fold. One is to show that the isotones can provide useful qualitative information regarding the behavior of the tests of distributional assumptions other than normality. The other is to show that the qualitative conclusions remain the same from one two-parameter family of alternatives to another. Towards this end we construct and interpret the isotones of some tests of the composite hypothesis of exponentiality, using the profiles of two Weibull extensions, the generalized Weibull and the exponentiated Weibull families, which allow IFR, DFR, as well as unimodal and bathtub failure rate alternatives. Thus, as a by-product of the study, it is seen that a test due to Csörg? et al. [1975. Application of characterizations in the area of goodness-of-fit. In: Patil, G.P., Kotz, S., Ord, J.K. (Eds.), Statistical Distributions in Scientific Work, vol. 2. Reidel, Boston, pp. 79–90], and Gnedenko's Q(r)$Q(r)$ test [1969. Mathematical Methods of Reliability Theory. Academic Press, New York], are appropriate for detecting monotone failure rate alternatives, whereas a bivariate F$F$ test due to Lin and Mudholkar [1980. A test of exponentiality based on the bivariate F$F$ distribution. Technometrics 22, 79–82] and their entropy test [1984. On two applications of characterization theorems to goodness-of-fit. Colloq. Math. Soc. Janos Bolyai 45, 395–414] can detect all alternatives, but are especially suitable for nonmonotone failure rate alternatives.     

On batch queueing systems: A combinatorial approach

We consider batch queueing systems M/M^H/1$M/M^H/1$ and M^H/M/1$M^H/M/1$ with catastrophes. The transient probability functions of these queueing systems are obtained by a Lattice Path Combinatorics approach that utilizes randomization and dual processes. Steady state distributions are also determined. Generalization to systems having batches of different sizes are discussed.     

Optimal block designs for three treatments when observations are correlated

E$E$-optimal designs for comparing three treatments in blocks of size three are identified, where intrablock observations are correlated according to a first order autoregressive error process with parameter ρ∈(0,1)$\rho \in (0,1)$. For number of blocks b   of the form b=3n+1$b=3n+1$, there are two distinct optimal designs depending on the value of ρ$\rho$, with the best design being unequally replicated for large ρ$\rho$. For other values of b$b$, binary, equireplicate designs with specified within-block assignment patterns are best. In many cases, the stronger majorization optimality is established.     

Optimally robust credible sets with a class of priors

We determine a credible set A   that is the “best” with respect to the variation of the prior distribution in a neighborhood Γ$\Gamma$ of the starting prior _π0(θ)${\pi }_0(\theta )$. Among the class of sets with credibility γ$\gamma$ under _π0${\pi }_0$, the “optimally robust” set will be the one which maximizes the minimum probability of including θ$\theta$ as the prior varies over Γ$\Gamma$. This procedure is also Γ-minimax$\Gamma \text{-minimax}$ with respect to the risk function, probability of non-inclusion. We find the optimally robust credible set for three neighborhood classes Γ$\Gamma$, the ε-contamination$\epsilon \text{-contamination}$ class, the density ratio class and the density bounded class. A consequence of this investigation is that the maximum likelihood set is seen to be an optimal credible set from a robustness perspective.     

Kernel density estimators for random fields satisfying an interlaced mixing condition

For a sequence of strictly stationary random fields that are uniformly ρ′-mixing$\rho \prime \mathrm{-}\mathrm{mixing}$ and satisfy a Lindeberg condition, a central limit theorem is obtained for sequences of “rectangular” sums from the given random fields. The “Lindeberg CLT” is then used to prove a CLT for some kernel estimators of probability density for some strictly stationary random fields satisfying ρ′-mixing$\rho \prime \mathrm{-}\mathrm{mixing}$, and whose probability density and joint densities are absolutely continuous.     

Statistical analysis for rounded data

When random variables do not take discrete values, observed data are often the rounded values of continuous random variables. Errors caused by rounding of data are often neglected by classical statistical theories. While some pioneers have identified and made suggestions to rectify the problem, few suitable approaches were proposed. In this paper, we propose an approximate MLE (AMLE) procedure to estimate the parameters and discuss the consistency and asymptotic normality of the estimates. For our illustration, we shall consider the estimates of the parameters in AR(p)$\mathit{AR}(p)$ and MA(q)$\mathit{MA}(q)$ models for rounded data.     

Addition of runs to an s-level supersaturated design

The purpose of the present work is to extend the work of Gupta et al. (2010) to s  -level column balanced supersaturated designs. Addition of runs to an existing E(χ²)-optimal$E({\chi }^2)\mathrm{-}\mathrm{optimal}$ supersaturated design and to study the optimality of the resulting design is an important issue. This paper considers the study of the optimality of the resulting design. A lower bound to E(χ²)$E({\chi }^2)$ has been obtained for the extended supersaturated designs. Some examples and a small catalogue of E(χ²)-optimal$E({\chi }^2)\mathrm{-}\mathrm{optimal}$ supersaturated designs are also presented.     

Explicit formulas for generalized family-wise error rates and unimprovable step-down multiple testing procedures

Many multiple testing procedures (MTPs) are available today, and their number is growing. Also available are many type I error rates: the family-wise error rate (FWER), the false discovery rate, the proportion of false positives, and others. Most MTPs are designed to control a specific type I error rate, and it is hard to compare different procedures. We approach the problem by studying the exact level at which threshold step-down (TSD) procedures (an important class of MTPs exemplified by the classic Holm procedure) control the generalized FWER   defined as the probability of k$k$ or more false rejections. We find that level explicitly for any TSD procedure and any k$k$. No assumptions are made about the dependency structure of the p$p$-values of the individual tests. We derive from our formula a criterion for unimprovability   of a procedure in the class of TSD procedures controlling the generalized FWER at a given level. In turn, this criterion implies that for each k$k$ the number of such unimprovable procedures is finite and is greater than one if k>1$k>1$. Consequently, in this case the most rejective procedure in the above class does not exist.     

Using pilot study information to increase efficiency in clinical trials

It is often necessary to conduct a pilot study to determine the sample size required for a clinical trial. Due to differences in sampling environments, the pilot data are usually discarded after sample size calculation. This paper tries to use the pilot information to modify the subsequent testing procedure when a two-sided t$t$-test or a regression model is used to compare two treatments. The new test maintains the required significance level regardless of the dissimilarity between the pilot and the target populations, but increases the power when the two are similar. The test is constructed based on the posterior distribution of the parameters given the pilot study information, but its properties are investigated from a frequentist's viewpoint. Due to the small likelihood of an irrelevant pilot population, the new approach is a viable alternative to the current practice.     

Asymptotic normality of kernel type regression estimators for random fields

The asymptotic normality of the Nadaraya–Watson regression estimator is studied for α-mixing$\alpha \text{-}\mathrm{mixing}$ random fields. The infill-increasing setting is considered, that is when the locations of observations become dense in an increasing sequence of domains. This setting fills the gap between continuous and discrete models. In the infill-increasing case the asymptotic normality of the Nadaraya–Watson estimator holds, but with an unusual asymptotic covariance structure. It turns out that this covariance structure is a combination of the covariance structures that we observe in the discrete and in the continuous case.     

Statistical inference for partially time-varying coefficient errors-in-variables models

This paper studies the partially time-varying coefficient models where some covariates are measured with additive errors. In order to overcome the bias of the usual profile least squares estimation when measurement errors are ignored, we propose a modified profile least squares estimator of the regression parameter and construct estimators of the nonlinear coefficient function and error variance. The proposed three estimators are proved to be asymptotically normal under mild conditions. In addition, we introduce the profile likelihood ratio test and then demonstrate that it follows an asymptotically χ²${\chi }^2$ distribution under the null hypothesis. Finite sample behavior of the estimators is investigated via simulations too.