Tools for constructing optimal two-level factorial designs for a linear model containing main effects and one two-factor interaction

In Hedayat and Pesotan [1992, Two-level factorial designs for main effects and selected two-factor interactions. Statist. Sinica 2, 453–464.] the concepts of a g(n,e)$g(n,e)$-design and a g(n,e)$g(n,e)$-matrix are introduced to study designs of n$n$ factor two-level experiments which can unbiasedly estimate the mean, the n$n$ main effects and e$e$ specified two-factor interactions appearing in an orthogonal polynomial model and it is observed that the construction of a g-design is equivalent to the construction of a g  -matrix. This paper deals with the construction of D-optimal g(n,1)$g(n,1)$-matrices. A standard form for a g(n,1)$g(n,1)$-matrix is introduced and some lower and upper bounds on the absolute determinant value of a D-optimal g(n,1)$g(n,1)$-matrix in the class of all g(n,1)$g(n,1)$-matrices are obtained and an approach to construct D-optimal g(n,1)$g(n,1)$-matrices is given for 2?n?8$2?n?8$. For two specific subclasses, namely a certain class of g(n,1)$g(n,1)$-matrices within the class of g(n,1)$g(n,1)$-matrices of index one and the class C(H)$C(H)$ of g(8t+2,1)$g(8t+2,1)$-matrices constructed from a normalized Hadamard matrix H   of order 8t+4(t?1)$8t+4(t?1)$ two techniques for the construction of the restricted D-optimal matrices are given.     

On the maximum of periodic integer-valued sequences with exponential type tails via max-semistable laws

In this work we study the limiting distribution of the maximum term of periodic integer-valued sequences with marginal distribution belonging to a particular class where the tail decays exponentially. This class does not belong to the domain of attraction of any max-stable distribution. Nevertheless, we prove that the limiting distribution is max-semistable when we consider the maximum of the first k_n observations, for a suitable sequence {_kn}$\{k_n\}$ increasing to infinity. We obtain an expression for calculating the extremal index of sequences satisfying certain local conditions similar to conditions D^(m)(_un)$D^{(m)}(u_n)$, m∈N$m\in \mathbb{N}$, defined by Chernick et al. (1991). We apply the results to a class of max-autoregressive sequences and a class of moving average models. The results generalize the ones obtained for the stationary case.     

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.     

Representing a function of two variables as a superposition over angles of ridge functions with a given shape via linear programming

A ridge function with shape function g   in the horizontal direction is a function of the form g(x)h(y,0)$g(x)h(y,0)$. Along each horizontal line it has the shape g(x)$g(x)$, multiplied by a function h(y,0)$h(y,0)$ which depends on the y-value of the horizontal line. Similarly a ridge function with shape function g   in the vertical direction has the form g(y)h(x,π/2)$g(y)h(x,\pi /2)$. For a given shape function g it may or may not be possible to represent an arbitrary   function f(x,y)$f(x,y)$ as a superposition over all angles of a ridge function with shape g   in each direction, where h=_hf=_hf,g$h=h_f=h_{f,g}$ depends on the functions f and g   and also on the direction, θ:h=_hf,g(·,θ)$\theta :h=h_{f,g}(·,\theta )$. We show that if g   is Gaussian centered at zero then this is always possible and we give the function _hf,g$h_{f,g}$ for a given f(x,y)$f(x,y)$. For highpass or for odd shapes g  , we show it is impossible to represent an arbitrary f(x,y)$f(x,y)$, i.e. in general there is no _hf,g$h_{f,g}$. Note that our problem is similar to tomography, where the problem is to invert the Radon transform, except that the use of the word inversion is here somewhat “inverted”: in tomography f(x,y)$f(x,y)$ is unknown and we find it by inverting the projections of f  ; here, f(x,y)$f(x,y)$ is known, g(z)$g(z)$ is known, and _hf(·,θ)=_hf,g(·,θ)$h_f(·,\theta )=h_{f,g}(·,\theta )$ is the unknown.     

Estimation of smooth regression functions in monotone response models

We consider the estimation of smooth regression functions in a class of conditionally parametric co-variate-response models. Independent and identically distributed observations are available from the distribution of (Z,X)$(Z,X)$, where Z is a real-valued co-variate with some unknown distribution, and the response X conditional on Z   is distributed according to the density p(·,ψ(Z))$p(·,\psi (Z))$, where p(·,θ)$p(·,\theta )$ is a one-parameter exponential family. The function ψ$\psi$ is a smooth monotone function. Under this formulation, the regression function E(X|Z)$E(X|Z)$ is monotone in the co-variate Z   (and can be expressed as a one–one function of ψ$\psi$); hence the term “monotone response model”. Using a penalized least squares approach that incorporates both monotonicity and smoothness, we develop a scheme for producing smooth monotone estimates of the regression function and also the function ψ$\psi$ across this entire class of models. Point-wise asymptotic normality of this estimator is established, with the rate of convergence depending on the smoothing parameter. This enables construction of Wald-type (point-wise) as well as pivotal confidence sets for ψ$\psi$ and also the regression function. The methodology is extended to the general heteroscedastic model, and its asymptotic properties are discussed.     

Combinatorially composing Chebyshev polynomials

We present a combinatorial proof of two fundamental composition identities associated with Chebyshev polynomials. Namely, for all m,n?0$m,n?0$, _Tm(_Tn(x))=_Tmn(x)$T_m(T_n(x))=T_{\mathit{mn}}(x)$ and _Um−1(_Tn(x))_Un−1(x)=_Umn−1(x).$U_{m-1}(T_n(x))U_{n-1}(x)=U_{\mathit{mn}-1}(x).$     

Limit laws for the cumulative number of ties for the maximum in a random sequence

Let {_Xn,n?1}$\{X_n,n?1\}$ be a sequence of independent identically distributed random variables, taking nonnegative integer values. An observation _Xn$X_n$ is a tie for the maximum if _Xn=max{_X1,…,_Xn-1}$X_n=\max \{X_1,\dots ,X_{n-1}\}$. In this paper, we obtain weak and strong laws of large numbers and central limit theorems for the cumulative number of ties for the maximum among the first n$n$ observations.     

Prediction in moving average processes

For the stationary invertible moving average process of order one with unknown innovation distribution F, we construct root-n   consistent plug-in estimators of conditional expectations E(h(_Xn+1)|_X1,…,_Xn)$E(h(X_{n+1})|X_1,\dots ,X_n)$. More specifically, we give weak conditions under which such estimators admit Bahadur-type representations, assuming some smoothness of h or of F. For fixed h it suffices that h   is locally of bounded variation and locally Lipschitz in _L2(F)$L_2(F)$, and that the convolution of h and F   is continuously differentiable. A uniform representation for the plug-in estimator of the conditional distribution function P(_Xn+1?·|_X1,…,_Xn)$P(X_{n+1}?·|X_1,\dots ,X_n)$ holds if F has a uniformly continuous density. For a smoothed version of our estimator, the Bahadur representation holds uniformly over each class of functions h that have an appropriate envelope and whose shifts are F-Donsker, assuming some smoothness of F. The proofs use empirical process arguments.     

Minimum Hellinger distance estimation in a nonparametric mixture model

In this paper, we investigate the estimation problem of the mixture proportion λ$\lambda$ in a nonparametric mixture model of the form λF(x)+(1-λ)G(x)$\lambda F(x)+(1-\lambda )G(x)$ using the minimum Hellinger distance approach, where F and G are two unknown distributions. We assume that data from the distributions F and G   as well as from the mixture distribution λF+(1-λ)G$\lambda F+(1-\lambda )G$ are available. We construct a minimum Hellinger distance estimator of λ$\lambda$ and study its asymptotic properties. The proposed estimator is chosen to minimize the Hellinger distance between a parametric mixture model and a nonparametric density estimator. We also develop a maximum likelihood estimator of λ$\lambda$. Theoretical properties such as the existence, strong consistency, asymptotic normality and asymptotic efficiency of the proposed estimators are investigated. Robustness properties of the proposed estimator are studied using a Monte Carlo study. Two real data examples are also analyzed.     

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.