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.     

A marked point process perspective in fitting spatial point process models

This paper discusses a new perspective in fitting spatial point process models. Specifically the spatial point process of interest is treated as a marked point process where at each observed event x$\mathbf{x}$ a stochastic process M(x;t)$M(\mathbf{x};t)$, 0<t<r$0<t<r$, is defined. Each mark process M(x;t)$M(\mathbf{x};t)$ is compared with its expected value, say F(t;θ)$F(t;\theta )$, to produce a discrepancy measure at x$\mathbf{x}$, where θ$\theta$ is a set of unknown parameters. All individual discrepancy measures are combined to define an overall measure which will then be minimized to estimate the unknown parameters. The proposed approach can be easily applied to data with sample size commonly encountered in practice. Simulations and an application to a real data example demonstrate the efficacy of the proposed approach.     

Bayesian population estimation for small sample capture-recapture data using noninformative priors

One critical issue in the Bayesian approach is choosing the priors when there is not enough prior information to specify hyperparameters. Several improper noninformative priors for capture-recapture models were proposed in the literature. It is known that the Bayesian estimate can be sensitive to the choice of priors, especially when sample size is small to moderate. Yet, how to choose a noninformative prior for a given model remains a question. In this paper, as the first step, we consider the problem of estimating the population size for _Mt$M_{\mathrm{t}}$ model using noninformative priors. The _Mt$M_{\mathrm{t}}$ model has prodigious application in wildlife management, ecology, software liability, epidemiological study, census under-count, and other research areas. Four commonly used noninformative priors are considered. We find that the choice of noninformative priors depends on the number of sampling occasions only. The guidelines on the choice of noninformative priors are provided based on the simulation results. Propriety of applying improper noninformative prior is discussed. Simulation studies are developed to inspect the frequentist performance of Bayesian point and interval estimates with different noninformative priors under various population sizes, capture probabilities, and the number of sampling occasions. The simulation results show that the Bayesian approach can provide more accurate estimates of the population size than the MLE for small samples. Two real-data examples are given to illustrate the method.     

Improving density estimators of discretely observed processes by interpolation

We consider density estimation for a smooth stationary process _Xt$X_t$, t∈R$t\in \mathbf{R}$, based on a discrete sample _Yi=_XΔi$Y_i=X_{\mathrm{\Delta }i}$, i=0,…,n=T/Δ$i=0,\dots ,n=T/\Delta$. By a suitable interpolation scheme of order p  , we augment data to form an approximation _Xp,t$X_{p,t}$, t∈[0,T]$t\in [0,T]$, of the continuous-time process and base our density estimate on the augmented sample path. Our results show that this can improve the rate of convergence (measured in terms of n) of the density estimate. Among other things, this implies that recording n   observations using a small Δ$\Delta$ can be more efficient than recording n independent observations.     

Regression models for functional data by reproducing kernel Hilbert spaces methods

Non-parametric regression models are developed when the predictor is a function-valued random variable X={_Xt}t∈T$X=\{X_t{\}}_{t\in T}$. Based on a representation of the regression function f(X)$f(X)$ in a reproducing kernel Hilbert space such models generalize the classical setting used in statistical learning theory. Two applications corresponding to scalar and categorical response random variable are performed on stock-exchange and medical data. The results of different regression models are compared.     

Optimal simple step stress accelerated life test design for reliability prediction

A step stress accelerated life testing model is presented to obtain the optimal hold time at which the stress level is changed. The experimental test is designed to minimize the asymptotic variance of reliability estimate at time ζ$\zeta$. A Weibull distribution is assumed for the failure time at any constant stress level. The scale parameter of the Weibull failure time distribution at constant stress levels is assumed to be a log-linear function of the stress level. The maximum likelihood function is given for the step stress accelerated life testing model with Type I censoring, from which the asymptotic variance and the Fisher information matrix are obtained. An optimal test plan with the minimum asymptotic variance of reliability estimate at time ζ$\zeta$ is determined.     

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 the E-optimality of complete block designs under a mixed interference model

In experiments in which the response to a treatment can be affected by other treatments, the interference model with neighbor effects is usually used. It is known that circular neighbor balanced designs (CNBDs) are universally optimal under such a model if the neighbor effects are fixed (Druilhet, 1999) or random (4 and 7). However, such designs cannot exist for every combination of design parameters. In the class of block designs with the same number of treatments as experimental units per block, a CNBD cannot exist if the number of blocks, b  , is equal to p(t−1)±1$p(t-1)\pm 1$, where p is a positive integer and t is the number of treatments. Filipiak et al. (2008) gave the structure of the left-neighboring matrix of E-optimal complete block designs with p  =1 under the model with fixed neighbor effects. The purpose of this paper is to generalize E-optimality results for designs with p∈N$p\in \mathbb{N}$ assuming random neighbor effects.     

On asymptotic properties of the parameter estimator for a type of SPDE

In this paper, we study a random field _U?(t,x)$U_{?}(t,x)$ governed by some type of stochastic partial differential equations with an unknown parameter θ$\theta$ and a small noise ?$?$. We construct an estimator of θ$\theta$ based on the continuous observation of N   Fourier coefficients of _U?(t,x)$U_{?}(t,x)$, and prove the strong convergence and asymptotic normality of the estimator when the noise ?$?$ tends to zero.     

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.     

Improved WLP and GWP lower bounds based on exact integer programming

By using exact integer programming (IP) (integer programming in infinite precision) bounds on the word-length patterns (WLPs) and generalized word-length patterns (GWPs) for fractional factorial designs are improved. In the literature, bounds on WLPs are formulated as linear programming (LP) problems. Although the solutions to such problems must be integral, the optimization is performed without the integrality constraints. Two examples of this approach are bounds on the number of words of length four for resolution IV regular designs, and a lower bound for the GWP of two-level orthogonal arrays. We reformulate these optimization problems as IP problems with additional valid constraints in the literature and improve the bounds in many cases. We compare the improved bound to the enumeration results in the literature to find many cases for which our bounds are achieved. By using the constraints in our integer programs we prove that f(16λ,2,4)?9$f(16\lambda ,2,4)?9$ if λ$\lambda$ is odd where f(2^tλ,2,t)$f(2^t\lambda ,2,t)$ is the maximum n   for which an OA(N,n,2,t)$\mathrm{OA}(N,n,2,t)$ exists. We also present a theorem for constructing GMA OA(N,N/2-u,2,3)$\mathrm{OA}(N,N/2-u,2,3)$ for u=1,…,5$u=1,\dots ,5$.     

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.     

Optimality of the quasi-score estimator in a mean–variance model with applications to measurement error models

We consider a regression of y$y$ on x$x$ given by a pair of mean and variance functions with a parameter vector θ$\theta$ to be estimated that also appears in the distribution of the regressor variable x$x$. The estimation of θ$\theta$ is based on an extended quasi-score (QS) function. We show that the QS estimator is optimal within a wide class of estimators based on linear-in-y$y$ unbiased estimating functions. Of special interest is the case where the distribution of x$x$ depends only on a subvector α$\alpha$ of θ$\theta$, which may be considered a nuisance parameter. In general, α$\alpha$ must be estimated simultaneously together with the rest of θ$\theta$, but there are cases where α$\alpha$ can be pre-estimated. A major application of this model is the classical measurement error model, where the corrected score (CS) estimator is an alternative to the QS estimator. We derive conditions under which the QS estimator is strictly more efficient than the CS estimator.