Lattice and Schröder paths with periodic boundaries

We consider paths in the plane with (1,0$1,0$), (0,1$0,1$), and (a,b$a,b$)-steps that start at the origin, end at height n$n$, and stay strictly to the left of a given non-decreasing right boundary. We show that if the boundary is periodic and has slope at most b/a$b/a$, then the ordinary generating function for the number of such paths ending at height n   is algebraic. Our argument is in two parts. We use a simple combinatorial decomposition to obtain an Appell relation or “umbral” generating function, in which the power zⁿ$z^n$ is replaced by a power series of the form zⁿ_φn(z),$z^n{\phi }_n(z),$ where _φn(0)=1.${\phi }_n(0)=1.$ Then we convert (in an explicit way) the umbral generating function to an ordinary generating function by solving a system of linear equations and a polynomial equation. This conversion implies that the ordinary generating function is algebraic. We give several concrete examples, including an alternative way to solve the tennis ball problem.     

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.     

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).     

Linear sufficiency and completeness in the context of estimating the parametric function in the general Gauss–Markov model

In this paper we consider linear sufficiency and linear completeness in the context of estimating the estimable parametric function K^′β${\mathbf{K}}^{\prime }\mathit{\beta }$ under the general Gauss–Markov model {y,Xβ,σ²V}$\{\mathbf{y},\mathbf{X}\mathit{\beta },{\sigma }^2\mathbf{V}\}$. We give new characterizations for linear sufficiency, and define and characterize linear completeness in a case of estimation of K^′β${\mathbf{K}}^{\prime }\mathit{\beta }$. Also, we consider a predictive approach for obtaining the best linear unbiased estimator of K^′β${\mathbf{K}}^{\prime }\mathit{\beta }$, and subsequently, we give the linear analogues of the Rao–Blackwell and Lehmann–Scheffé Theorems in the context of estimating K^′β${\mathbf{K}}^{\prime }\mathit{\beta }$.     

Distribution theory of quadratic forms for matrix multivariate elliptical distribution

This paper proposes the density and characteristic functions of a general matrix quadratic form X^(?)AX${\mathbf{X}}^{(?)}\mathbf{AX}$, when A=A^(?)$\mathbf{A}={\mathbf{A}}^{(?)}$ is a positive semidefinite matrix, X$\mathbf{X}$ has a matrix multivariate elliptical distribution and X^(?)${\mathbf{X}}^{(?)}$ denotes the usual conjugate transpose of X$\mathbf{X}$. These results are obtained for real normed division algebras. With particular cases we obtained the density and characteristic functions of matrix quadratic forms for matrix multivariate normal, Pearson type VII, t and Cauchy distributions.     

Optimal designs based on exact confidence regions for parameter estimation of a nonlinear regression model

This paper is concerned with the proposal of optimality criteria, referred to as X  - and X^′$X^{\prime }$-optimality criteria, and the construction of X  - and X^′$X^{\prime }$-optimal designs, for nonlinear regression models. These optimal designs aim at improving the estimation of parameters of this class of models. The principle of these criteria is the minimization, with respect to the design, of the expected volume of a particular exact parametric confidence region. In this paper we give detailed definitions, properties, and computation methods of X  - and X^′$X^{\prime }$-optimal designs. We also compare these designs with the classic local D-optimal designs, with regard to robustness and efficiency, for two very well-known academic models (Box–Lucas and Michaelis–Menten models).     

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.     

Tabu search for covering arrays using permutation vectors

A covering array  CA(N;t,k,v)$\mathsf{CA}(N;t,k,v)$ is an N×k$N\times k$ array, in which in every N×t$N\times t$ subarray, each of the v^t$v^t$ possible t  -tuples over v$v$ symbols occurs at least once. The parameter t is the strength   of the array. Covering arrays have a wide range of applications for experimental screening designs, particularly for software interaction testing. A compact representation of certain covering arrays employs “permutation vectors” to encode v^t×1$v^t\times 1$ subarrays of the covering array so that a covering perfect hash family whose entries correspond to permutation vectors yields a covering array. We introduce a method for effective search for covering arrays of this type using tabu search. Using this technique, improved covering arrays of strength 3, 4 and 5 have been found, as well as the first arrays of strength 6 and 7 found by computational search.     

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.     

A sequential design for a clinical trial with a linear prognostic factor

We study a randomized adaptive design to assign one of the L$L$ treatments to patients who arrive sequentially by means of an urn model. At each stage n$n$, a reward is distributed between treatments. The treatment applied is rewarded according to its response, 0?_Yn?1$0?Y_n?1$, and 1-_Yn$1-Y_n$ is distributed among the other treatments according to their performance until stage n-1$n-1$. Patients can be classified in K+1$K+1$ levels and we assume that the effect of this level in the response to the treatments is linear. We study the asymptotic behavior of the design when the ordinary least square estimators are used as a measure of performance until stage n-1$n-1$.     

Asymptotic properties of the EPMC for modified linear discriminant analysis when sample size and dimension are both large

We deal with the problem of classifying a new observation vector into one of two known multivariate normal distributions when the dimension p and training sample size N   are both large with p<N$p<N$. Modified linear discriminant analysis (MLDA) was suggested by Xu et al. [10]. Error rate of MLDA is smaller than the one of LDA. However, if p and N   are moderately large, error rate of MLDA is close to the one of LDA. These results are conditional ones, so we should investigate whether they hold unconditionally. In this paper, we give two types of asymptotic approximations of expected probability of misclassification (EPMC) for MLDA as n→∞$n\to \infty$ with p=O(n^δ)$p=O(n^{\delta })$, 0<δ<1$0<\delta <1$. The one of two is the same as the asymptotic approximation of LDA, and the other is corrected version of the approximation. Simulation reveals that the modified version of approximation has good accuracy for the case in which p and N are moderately large.     

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.     

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.     

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$n^{-(1-\alpha )/2}$.     

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.     

Distribution of concomitants of order statistics and their order statistics

For a random sample of size n$n$ from an absolutely continuous random vector (X,Y)$(X,Y)$, let _Yi:n$Y_{i:n}$ be i$i$th Y$Y$-order statistic and _Y[j:n]$Y_{[j:n]}$ be the Y$Y$-concomitant of _Xj:n$X_{j:n}$. We determine the joint pdf of _Yi:n$Y_{i:n}$ and _Y[j:n]$Y_{[j:n]}$ for all i,j=1$i,j=1$ to n$n$, and establish some symmetry properties of the joint distribution for symmetric populations. We discuss the uses of the joint distribution in the computation of moments and probabilities of various ranks for _Y[j:n]$Y_{[j:n]}$. We also show how our results can be used to determine the expected cost of mismatch in broken bivariate samples and approximate the first two moments of the ratios of linear functions of _Yi:n$Y_{i:n}$ and _Y[j:n]$Y_{[j:n]}$. For the bivariate normal case, we compute the expectations of the product of _Yi:n$Y_{i:n}$ and _Y[i:n]$Y_{[i:n]}$ for n=2$n=2$ to 8 for selected values of the correlation coefficient and illustrate their uses.