Construction of blocked two-level regular designs with general minimum lower order confounding

Zhang et al. (2008) proposed a general minimum lower order confounding (GMC for short) criterion, which aims to select optimal factorial designs in a more elaborate and explicit manner. By extending the GMC criterion to the case of blocked designs, Wei et al. (submitted for publication) proposed a B¹-GMC criterion. The present paper gives a construction theory and obtains the B¹-GMC 2^n−m:2^r$2^{n-m}:2^r$ designs with n≥5N/16+1$n\ge 5N/16+1$, where 2^n−m:2^r$2^{n-m}:2^r$ denotes a two-level regular blocked design with N=2^n−m$N=2^{n-m}$ runs, n   treatment factors, and 2^r$2^r$ blocks. The construction result is simple. Up to isomorphism, the B¹-GMC 2^n−m:2^r$2^{n-m}:2^r$ designs can be constructed as follows: the n   treatment factors and the 2^r−1$2^r-1$ block effects are, respectively, assigned to the last n   columns and specific 2^r−1$2^r-1$ columns of the saturated 2^{(N−1)−(N−1−n+m)}$2^{(N-1)-(N-1-n+m)}$ design with Yates order. With such a simple structure, the B¹-GMC designs can be conveniently used in practice. Examples are included to illustrate the theory.     

General minimum lower order confounding designs: An overview and a construction theory

For fractional factorial (FF) designs, Zhang et al. (2008) introduced a new pattern for assessing regular designs, called aliased effect-number pattern (AENP), and based on the AENP, proposed a general minimum lower order confounding (denoted by GMC for short) criterion for selecting design. In this paper, we first have an overview of the existing optimality criteria of FF designs, and then propose a construction theory for 2^n−m$2^{n-m}$ GMC designs with 33N/128≤n≤5N/16$33N/128\le n\le 5N/16$, where N=2^n−m$N=2^{n-m}$ is the run size and n is the number of factors, for all N's and n  's, via the doubling theory and SOS resolution IV designs. The doubling theory is extended with a new approach. By introducing a notion of rechanged (RC) Yates order for the regular saturated design, the construction result turns out to be quite transparent: every GMC 2^n−m$2^{n-m}$ design simply consists of the last n columns of the saturated design with a specific RC Yates order. This can be very conveniently applied in practice.     

Optimal block sequences for blocked fractional factorial split-plot designs

It is known that for blocked 2^n-k$2^{n-k}$ designs a judicious sequencing of blocks may allow one to obtain early and insightful results regarding influential parameters in the experiment. Such findings may justify the early termination of the experiment thereby producing cost and time savings. This paper introduces an approach for selecting the optimal sequence of blocks for regular two-level blocked fractional factorial split-plot screening experiments. An optimality criterion is developed so as to give priority to the early estimation of low-order factorial effects. This criterion is then applied to the minimum aberration blocked fractional factorial split-plot designs tabled in McLeod and Brewster [2004. The design of blocked fractional factorial split-plot experiments. Technometrics 46, 135–146]. We provide a catalog of optimal block sequences for 16 and 32-run minimum aberration blocked fractional factorial split-plot designs run in either 4 or 8 blocks.     

Local asymptotic minimax theory for block-decreasing densities

In this paper, we study Lebesgue densities on (0,∞)^d$(0,\infty {)}^d$ that are non-increasing in each coordinate, while keeping all other coordinates fixed, from the perspective of local asymptotic minimax lower bound theory. In particular, we establish a local optimal rate of convergence of the order n^−1/(d+2)$n^{-1/(d+2)}$.     

On E(s)-optimal supersaturated designs

A popular measure to assess 2-level supersaturated designs is the E(s²)$E(s^2)$ criterion. In this paper, improved lower bounds on E(s²)$E(s^2)$ are obtained. The same improvement has recently been established by Ryan and Bulutoglu [2007. E(s²)$E(s^2)$-optimal supersaturated designs with good minimax properties. J. Statist. Plann. Inference 137, 2250–2262]. However, our analysis provides more details on precisely when an improvement is possible, which is lacking in Ryan and Bulutoglu [2007. E(s²)$E(s^2)$-optimal supersaturated designs with good minimax properties. J. Statist. Plann. Inference 137, 2250–2262]. The equivalence of the bounds obtained by Butler et al. [2001. A general method of constructing E(s²)$E(s^2)$-optimal supersaturated designs. J. Roy. Statist. Soc. B 63, 621–632] (in the cases where their result applies) and those obtained by Bulutoglu and Cheng [2004. Construction of E(s²)$E(s^2)$-optimal supersaturated designs. Ann. Statist. 32, 1662–1678] is established. We also give two simple methods of constructing E(s²)$E(s^2)$-optimal designs.     

Markov chain Monte Carlo tests for designed experiments

We consider conditional exact tests of factor effects in designed experiments for discrete response variables. Similarly to the analysis of contingency tables, a Markov chain Monte Carlo method can be used for performing exact tests, when large-sample approximations are poor and the enumeration of the conditional sample space is infeasible. For designed experiments with a single observation for each run, we formulate log-linear or logistic models and consider a connected Markov chain over an appropriate sample space. In particular, we investigate fractional factorial designs with 2^p-q$2^{p-q}$ runs, noting correspondences to the models for 2^p-q$2^{p-q}$ contingency tables.     

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.     

Truncated linear estimation of a bounded multivariate normal mean

We consider the problem of estimating the mean θ$\theta$ of an _Np(θ,_Ip)${\mathrm{N}}_{\mathrm{p}}(\theta ,I_p)$ distribution with squared error loss ∥δ−θ∥²$\parallel \delta -\theta {\parallel }^2$ and under the constraint ∥θ∥≤m$\parallel \theta \parallel \le m$, for some constant m>0$m>0$. Using Stein's identity to obtain unbiased estimates of risk, Karlin's sign change arguments, and conditional risk analysis, we compare the risk performance of truncated linear estimators with that of the maximum likelihood estimator _δmle${\delta }_{\mathrm{mle}}$. We obtain for fixed (m,p)$(m,p)$ sufficient conditions for dominance. An asymptotic framework is developed, where we demonstrate that the truncated linear minimax estimator dominates _δmle${\delta }_{\mathrm{mle}}$, and where we obtain simple and accurate measures of relative improvement in risk. Numerical evaluations illustrate the effectiveness of the asymptotic framework for approximating the risks for moderate or large values of p.     

Constrained Bayes and empirical Bayes estimation under random effects normal ANOVA model with balanced loss function

The paper develops constrained Bayes and empirical Bayes estimators in the random effects ANOVA model under balanced loss functions. In the balanced normal–normal model, estimators of the Bayes risks of the constrained Bayes and constrained empirical Bayes estimators are provided which are correct asymptotically up to O(m^-1)$\mathrm{O}(m^{-1})$, that is the remainder term is o(m^-1)$\mathrm{o}(m^{-1})$, m$m$ denoting the number of strata.     

Empirical Bayes regression analysis with many regressors but fewer observations

In this paper, we consider the prediction problem in multiple linear regression model in which the number of predictor variables, p, is extremely large compared to the number of available observations, n  . The least-squares predictor based on a generalized inverse is not efficient. We propose six empirical Bayes estimators of the regression parameters. Three of them are shown to have uniformly lower prediction error than the least-squares predictors when the vector of regressor variables are assumed to be random with mean vector zero and the covariance matrix (1/n)X^tX$(1/n){\mathit{X}}^{\mathrm{t}}\mathit{X}$ where X^t=(_x1,…,_xn)${\mathit{X}}^{\mathrm{t}}=({\mathit{x}}_1,\dots ,{\mathit{x}}_n)$ is the p×n$p\times n$ matrix of observations on the regressor vector centered from their sample means. For other estimators, we use simulation to show its superiority over the least-squares predictor.     

Rates of convergence for the k-nearest neighbor estimators with smoother regression functions

Let (X, Y  ) be a R^d×R-valued${\mathbb{R}}^d\times \mathbb{R}\mathrm{-}\mathrm{valued}$ random vector. In regression analysis one wants to estimate the regression function m(x)?E(Y|X=x)$m(x)?\mathbf{E}(Y|X=x)$ from a data set. In this paper we consider the rate of convergence for the k-nearest neighbor estimators in case that X   is uniformly distributed on [0,1]^d$[\mathrm{0,1}{]}^d$, Var(Y|X=x)$\mathbf{Var}(Y|X=x)$ is bounded, and m is (p, C)-smooth. It is an open problem whether the optimal rate can be achieved by a k  -nearest neighbor estimator for 1<p≤1.5$1<p\le 1.5$. We solve the problem affirmatively. This is the main result of this paper. Throughout this paper, we assume that the data is independent and identically distributed and as an error criterion we use the expected L₂ error.     

Bayesian D-optimal supersaturated designs

We introduce a new class of supersaturated designs using Bayesian D-optimality. The designs generated using this approach can have arbitrary sample sizes, can have any number of blocks of any size, and can incorporate categorical factors with more than two levels. In side by side diagnostic comparisons based on the E(s²)$E(s^2)$ criterion for two-level experiments having even sample size, our designs either match or out-perform the best designs published to date. The generality of the method is illustrated with quality improvement experiment with 15 runs and 20 factors in 3 blocks.     

Regularization and variable selection for infinite variance autoregressive models

Autoregressive models with infinite variance are of great importance in modeling heavy-tailed time series and have been well studied. In this paper, we propose a penalized method to conduct model selection for autoregressive models with innovations having Pareto-like distributions with index α∈(0,2)$\alpha \in (\mathrm{0,2})$. By combining the least absolute deviation loss function and the adaptive lasso penalty, the proposed method is able to consistently identify the true model and at the same time produce efficient estimators with a convergence rate of n^−1/α$n^{-1/\alpha }$. In addition, our approach provides a unified way to conduct variable selection for autoregressive models with finite or infinite variance. A simulation study and a real data analysis are conducted to illustrate the effectiveness of our method.     

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.     

An identity for the Fisher information and Mahalanobis distance

Consider a mixture problem consisting of k classes. Suppose we observe an s-dimensional random vector X   whose distribution is specified by the relations P(X∈A|Y=i)=_Pi(A)$P(X\in A|Y=i)=P_i(A)$, where Y   is an unobserved class identifier defined on {1,…,k}$\{1,\dots ,k\}$, having distribution P(Y=i)=_pi$P(Y=i)=p_i$. Assuming the distributions _Pi$P_i$ having a common covariance matrix, elegant identities are presented that connect the matrix of Fisher information in Y   on the parameters _p1,…,_pk$p_1,\dots ,p_k$, the matrix of linear information in X, and the Mahalanobis distances between the pairs of P  's. Since the parameters are not free, the information matrices are singular and the technique of generalized inverses is used. A matrix extension of the Mahalanobis distance and its invariant forms are introduced that are of interest in their own right. In terms of parameter estimation, the results provide an independent of the parameter upper bound for the loss of accuracy by esimating _p1,…,_pk$p_1,\dots ,p_k$ from a sample of X^′$X^{\prime }$s, as compared with the ideal estimator based on a random sample of Y^′$Y^{\prime }$s.     

On empirical likelihood for linear models with missing responses

Suppose that we have a linear regression model Y=X^′β+_ν0(X)ε$Y=X^{\prime }\beta +{\nu }_0(X)\epsilon$ with random error ε$\epsilon$, where X is a random design variable and is observed completely, and Y is the response variable and some Y-values are missing at random (MAR). In this paper, based on the ‘complete’ data set for Y after inverse probability weighted imputation, we construct empirical likelihood statistics on EY   and β$\beta$ which have the χ²${\chi }^2$-type limiting distributions under some new conditions compared with Xue (2009). Our results broaden the applicable scope of the approach combined with Xue (2009).