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.     

Confidence intervals in regression utilizing prior information

We consider a linear regression model with regression parameter β=(_β1,…,_βp)$\beta =({\beta }_1,\dots ,{\beta }_p)$ and independent and identically N(0,σ²)$N(0,{\sigma }^2)$ distributed errors. Suppose that the parameter of interest is θ=a^Tβ$\theta =a^T\beta$ where a$a$ is a specified vector. Define the parameter τ=c^Tβ-t$\tau =c^T\beta -t$ where the vector c$c$ and the number t$t$ are specified and a$a$ and c$c$ are linearly independent. Also suppose that we have uncertain prior information that τ=0$\tau =0$. We present a new frequentist 1-α$1-\alpha$ confidence interval for θ$\theta$ that utilizes this prior information. We require this confidence interval to (a) have endpoints that are continuous functions of the data and (b) coincide with the standard 1-α$1-\alpha$ confidence interval when the data strongly contradict this prior information. This interval is optimal in the sense that it has minimum weighted average expected length where the largest weight is given to this expected length when τ=0$\tau =0$. This minimization leads to an interval that has the following desirable properties. This interval has expected length that (a) is relatively small when the prior information about τ$\tau$ is correct and (b) has a maximum value that is not too large. The following problem will be used to illustrate the application of this new confidence interval. Consider a 2×2$2\times 2$ factorial experiment with 20 replicates. Suppose that the parameter of interest θ$\theta$ is a specified simple   effect and that we have uncertain prior information that the two-factor interaction is zero. Our aim is to find a frequentist 0.95 confidence interval for θ$\theta$ that utilizes this prior information.     

Efficiency in estimation of memory

We study the efficiency of semiparametric estimates of memory parameter. We propose a class of shift invariant tapers of order (p,q). For a fixed p, the variance inflation factor of the new tapers approaches 1 as q   goes to infinity. We show that for d∈(−1/2,p+1/2)$d\in (-1/2,p+1/2)$, the proposed tapered Gaussian semiparametric estimator has the same limiting distribution as the nontapered version for d∈(−1/2,1/2)$d\in (-1/2,1/2)$. The new estimator is mean and polynomial trend invariant, and is computationally advantageous in comparison to the recently proposed exact local Whittle estimator. The simulation study shows that our estimator has comparable or better mean squared error in finite samples for a variety of models.     

Exact D-optimal designs for a linear log contrast model with mixture experiment for three and four ingredients

This study investigates the exact D-optimal designs of the linear log contrast model using the mixture experiment suggested by Aitchison and Bacon-Shone (1984) and the design space restricted by Lim (1987) and Chan (1988). Results show that for three ingredients, there are six extreme points that can be divided into two non-intersect sets S₁ and S₂. An exact N-point D  -optimal design for N=3p+q,p≥1,1≤q≤2$N=3p+q,p\ge 1,1\le q\le 2$ arranges equal weight n/N,0≤n≤p$n/N,0\le n\le p$ at the points of S₁ (S₂) and puts the remaining weight (N−3n)/N$(N-3n)/N$ on the points of S₂ (S₁) as evenly as possible. For four ingredients and N=6p+q,p≥1,1≤q≤5$N=6p+q,p\ge 1,1\le q\le 5$, an exact N-point design that distributes the weights as evenly as possible among the six supports of the approximate D-optimal design is exact D-optimal.     

Skew Dyck paths

In this paper we study the class S$\mathbb{S}$ of skew Dyck paths, i.e. of those lattice paths that are in the first quadrant, begin at the origin, end on the x-axis, consist of up steps  U=(1,1)$U=(1,1)$, down steps  D=(1,-1)$D=(1,-1)$, and left steps  L=(−1,-1)$L=(-1,-1)$, and such that up steps never overlap with left steps. In particular, we show that these paths are equinumerous with several other combinatorial objects, we describe some involutions on this class, and finally we consider several statistics on S$\mathbb{S}$.     

Skew Dyck paths,area, and superdiagonal bargraphs

Skew Dyck paths are a generalization of ordinary Dyck paths, defined as paths using up steps  U=(1,1)$U=(1,1)$, down steps  D=(1,-1)$D=(1,-1)$, and left steps  L=(−1,-1)$L=(-1,-1)$, starting and ending on the x-axis, never going below it, and so that up and left steps never overlap. In this paper we study the class of these paths according to their area, extending several results holding for Dyck paths. Then we study the class of superdiagonal bargraphs, which can be naturally defined starting from skew Dyck paths.     

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.     

Testing of a sub-hypothesis in linear regression models with long memory errors and deterministic design

This paper considers the problem of testing a sub-hypothesis in homoscedastic linear regression models where errors form long memory moving average processes and designs are non-random. Unlike in the random design case, asymptotic null distribution of the likelihood ratio type test based on the Whittle quadratic form is shown to be non-standard and non-chi-square. Moreover, the rate of consistency of the minimum Whittle dispersion estimator of the slope parameter vector is shown to be n^-(1-α)/2$n^{-(1-\alpha )/2}$, different from the rate n^-1/2$n^{-1/2}$ obtained in the random design case, where α$\alpha$ is the rate at which the error spectral density explodes at the origin. The proposed test is shown to be consistent against fixed alternatives and has non-trivial asymptotic power against local alternatives that converge to null hypothesis at the rate n^-(1-α)/2$n^{-(1-\alpha )/2}$.     

Distribution of order statistics from selected subsets of concomitants

For a random sample of size n from an absolutely continuous bivariate population (X, Y), let X_i:n be the i th X-order statistic and Y_[i:n] be its concomitant. We study the joint distribution of (V_s:m, W_t:n−m), where V_s:m is the s th order statistic of the upper subset {Y_[i:n], i=n−m+1,…,n}, and W_t:n−m is the t th order statistic of the lower subset {Y_[j:n], j=1,…,n−m  } of concomitants. When m=⌈_np0⌉$m=\lceil {\mathit{np}}_0\rceil$, s=⌈_mp1⌉$s=\lceil {\mathit{mp}}_1\rceil$, and t=⌈(n−m)_p2⌉$t=\lceil (n-m)p_2\rceil$, 0<_pi<1,i=0,1,2$0<p_i<1,i=0,1,2$, and n→∞$n\to \infty$, we show that the joint distribution is asymptotically bivariate normal and establish the rate of convergence. We propose second order approximations to the joint and marginal distributions with significantly better performance for the bivariate normal and Farlie–Gumbel bivariate exponential parents, even for moderate sample sizes. We discuss implications of our findings to data-snooping and selection problems.     

A note on the selection of optimal foldover plans for 16- and 32-run fractional factorial designs

The minimum aberration criterion has been advocated for ranking foldovers of 2^k−p$2^{k-p}$ fractional factorial designs (Li and Lin, 2003); however, a minimum aberration design may not maximize the number of clear low-order effects. We propose using foldover plans that sequentially maximize the number of clear low-order effects in the combined (initial plus foldover) design and investigate the extent to which these foldover plans differ from those that are optimal under the minimum aberration criterion. A small catalog is provided to summarize the results.     

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.     

The height of two types of generalised Motzkin paths

Consider Motzkin paths which are lattice paths in the plane starting at the origin, running weakly above the x-axis and after n   unit steps returning at the point (n,0)$(n,0)$. The allowed steps are the up and down steps (1,1)$(1,1)$ and (1,−1)$(1,-1)$ respectively and certain horizontal steps. We consider two types of horizontal steps that have attracted recent attention in the literature. First, we consider unit horizontal steps (1,0)$(1,0)$ coloured with k colours, secondly, we look at paths where the horizontal steps are of length k, for a non-negative integer k. Using generating functions, we study the sum of heights of such paths of size n. With the use of the Mellin transform, we find asymptotic expressions for the mean heights as n tends to infinity.     

Improving the estimators of the parameters of a probit regression model: A ridge regression approach

This paper considered the estimation of the regression parameters of a general probit regression model. Accordingly, we proposed five ridge regression (RR) estimators for the probit regression models for estimating the parameters (β)$(\beta )$ when the weighted design matrix is ill-conditioned and it is suspected that the parameter β$\beta$ may belong to a linear subspace defined by Hβ=h$H\beta =h$. Asymptotic properties of the estimators are studied with respect to quadratic biases, MSE matrices and quadratic risks. The regions of optimality of the proposed estimators are determined based on the quadratic risks. Some relative efficiency tables and risk graphs are provided to illustrate the numerical comparison of the estimators. We conclude that when q≥3$q\ge 3$, one would uses PRRRE; otherwise one uses PTRRE with some optimum size α$\alpha$. We also discuss the performance of the proposed estimators compare to the alternative ridge regression method due to Liu (1993).     

Optimal and efficient semi-Latin squares

An (n×n)/k$(n\times n)/k$semi-Latin square is an n×n square array in which nk distinct symbols (representing treatments) are placed in such a way that there are exactly k   symbols in each cell (row–column intersection) and each symbol occurs once in each row and once in each column. Semi-Latin squares form a class of row–column designs generalising Latin squares, and have applications in areas including the design of agricultural experiments, consumer testing, and via their duals, human–machine interaction. In the present paper, new theoretical and computational methods are developed to determine optimal or efficient (n×n)/k$(n\times n)/k$ semi-Latin squares for values of n and k for which such semi-Latin squares were previously unknown. The concept of subsquares of uniform semi-Latin squares is studied, new applications of the DESIGN package for GAP are developed, and exact algebraic computational techniques for comparing efficiency measures of binary equireplicate block designs are described. Applications include the complete enumeration of the (4×4)/k$(4\times 4)/k$ semi-Latin squares for k=2,…,10$k=2,\dots ,10$, and the determination of those that are A-, D-, and E-optimal, the construction of efficient (6×6)/k$(6\times 6)/k$ semi-Latin squares for k=4,5,6$k=4,5,6$, and counterexamples to a long-standing conjecture of R.A. Bailey and to a similar conjecture of D. Bedford and R.M. Whitaker.     

Quasi-minimax estimation in the general linear regression model

This paper is mainly concerned with minimax estimation in the general linear regression model y=Xβ+ε$y=X\beta +\epsilon$ under ellipsoidal restrictions on the parameter space and quadratic loss function. We confine ourselves to estimators that are linear in the response vector y  . The minimax estimators of the regression coefficient β$\beta$ are derived under homogeneous condition and heterogeneous condition, respectively. Furthermore, these obtained estimators are the ridge-type estimators and mean dispersion error (MDE) superior to the best linear unbiased estimator b=(X^′W^-1X)^-1X^′W^-1y$b=(X^{\prime }{W}^{-1}X{)}^{-1}{X}^{\prime }{W}^{-1}y$ under some conditions.