A surprising property of uniformly best linear affine estimation in linear regression when prior information is fuzzy

It is already shown in Arnold and Stahlecker (2009) that, in linear regression, a uniformly best estimator exists in the class of all Γ-compatible$\Gamma \text{-compatible}$ linear affine estimators. Here, prior information is given by a fuzzy set Γ$\Gamma$ defined by its ellipsoidal α-cuts$\alpha \text{-cuts}$. Surprisingly, such a uniformly best linear affine estimator is uniformly best not only in the class of all Γ-compatible$\Gamma \text{-compatible}$ linear affine estimators but also in the class of all estimators satisfying a very weak and sensible condition. This property of a uniformly best linear affine estimator is shown in the present paper. Furthermore, two illustrative special cases are mentioned, where a generalized least squares estimator on the one hand and a general ridge or Kuks–Olman estimator on the other hand turn out to be uniformly best.     

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.     

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.     

Re-weighted Nadaraya–Watson estimation of second-order jump-diffusion model

We propose a re-weighted Nadaraya–Watson estimator for the infinitesimal conditional expectation of the second-order jump-diffusion model. Consistency and asymptotic normality of the estimator require certain regular conditions.     

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.     

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.     

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).     

New 3-level response surface designs constructed from incomplete block designs

Box and Behnken [1958. Some new three level second-order designs for surface fitting. Statistical Technical Research Group Technical Report No. 26. Princeton University, Princeton, NJ; 1960. Some new three level designs for the study of quantitative variables. Technometrics 2, 455–475.] introduced a class of 3-level second-order designs for fitting the second-order response surface model. These 17 Box–Behnken designs (BB designs) are available for 3–12 and 16 factors. Although BB designs were developed nearly 50 years ago, they and the central-composite designs of Box and Wilson [1951. On the experimental attainment of optimum conditions. J. Royal Statist. Soc., Ser. B 13, 1–45.] are still the most often recommended response surface designs. Of the 17 aforementioned BB designs, 10 were constructed from balanced incomplete block designs (BIBDs) and seven were constructed from partially BIBDs (PBIBDs). In this paper we show that these seven BB designs constructed from PBIBDs can be improved in terms of rotatability as well as average prediction variance, D$D$- and G$G$-efficiency. In addition, we also report new orthogonally blocked solutions for 5, 8, 9, 11 and 13 factors. Note that an 11-factor BB design is available but cannot be orthogonally blocked. All new designs can be found at http://www.math.montana.edu/∼$\sim$jobo/bbd/.     

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.     

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.     

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 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.     

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.     

Bayesian variable selection using an adaptive powered correlation prior

The problem of selecting the correct subset of predictors within a linear model has received much attention in recent literature. Within the Bayesian framework, a popular choice of prior has been Zellner's g$g$-prior which is based on the inverse of empirical covariance matrix of the predictors. An extension of the Zellner's prior is proposed in this article which allow for a power parameter on the empirical covariance of the predictors. The power parameter helps control the degree to which correlated predictors are smoothed towards or away from one another. In addition, the empirical covariance of the predictors is used to obtain suitable priors over model space. In this manner, the power parameter also helps to determine whether models containing highly collinear predictors are preferred or avoided. The proposed power parameter can be chosen via an empirical Bayes method which leads to a data adaptive choice of prior. Simulation studies and a real data example are presented to show how the power parameter is well determined from the degree of cross-correlation within predictors. The proposed modification compares favorably to the standard use of Zellner's prior and an intrinsic prior in these examples.