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.     

Negative dependence and the Simes inequality

It is shown that the Simes inequality is reversed for a broad class of negatively dependent distributions.     

Cost-cautious designs for confirmatory bioassay

Confirmatory bioassay experiments take place in late stages of the drug discovery process when a small number of compounds have to be compared with respect to their properties. As the cost of the observations may differ considerably, the design problem is well specified by the cost of compound used rather than by the number of observations. We show that cost-efficient designs can be constructed using useful properties of the minimum support designs. These designs are particularly suited for studies where the parameters of the model to be estimated are known with high accuracy prior to the experiment, although they prove to be robust against typical inaccuracies of these values. When the parameters of the model can only be specified with ranges of values or by a probability distribution, we use a Bayesian criterion of optimality to construct the required designs. Typically, the number of their support points depends on the prior knowledge for the model parameters. In all cases we recommend identifying a set of designs with good statistical properties but different potential costs to choose from.     

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.     

An optimality criterion for supersaturated designs with quantitative factors

A supersaturated design (SSD) is a factorial design in which the degrees of freedom for all its main effects exceed the total number of distinct factorial level-combinations (runs) of the design. Designs with quantitative factors, in which level permutation within one or more factors could result in different geometrical structures, are very different from designs with nominal ones which have been treated as traditional designs. In this paper, a new criterion is proposed for SSDs with quantitative factors. Comparison and analysis for this new criterion are made. It is shown that the proposed criterion has a high efficiency in discriminating geometrically nonisomorphic designs and an advantage in computation.     

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.     

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.     

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/.     

How robust are tests for two independent samples?

Non-parametric procedures are sometimes in use even in cases where the corresponding parametric procedure is preferable. This is mainly due to the fact that in practical applications of statistical methods too much attention is paid to any violation of the normality assumption–normal distribution is, however, primarily supposed in order to easily derive the exact distribution of the statistic used within parametric approaches.     

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.     

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.     

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.     

Prediction intervals for general balanced linear random models

The main interest of prediction intervals lies in the results of a future sample from a previously sampled population. In this article, we develop procedures for the prediction intervals which contain all of a fixed number of future observations for general balanced linear random models. Two methods based on the concept of a generalized pivotal quantity (GPQ) and one based on ANOVA estimators are presented. A simulation study using the balanced one-way random model is conducted to evaluate the proposed methods. It is shown that one of the two GPQ-based and the ANOVA-based methods are computationally more efficient and they also successfully maintain the simulated coverage probabilities close to the nominal confidence level. Hence, they are recommended for practical use. In addition, one example is given to illustrate the applicability of the recommended methods.     

A constrained optimum experimental design problem for model discrimination with a continuously varying factor

We present the theoretical background and the numerical procedure for calculating optimum experimental designs for non-linear model discrimination in the presence of constraints. The design support points consist of two kinds of factors: a continuous function of time and discrete levels of other quantitative factors. That is, some of the experimental conditions are allowed to continually vary during the experimental run. We implement the theory in a chemical kinetic model discrimination problem.     

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.