Optimal designs for covariates’ models

A linear model with one treatment at V levels and first order regression on K continuous covariates with values on a K-cube is considered. The D-criterion is used to judge the ‘goodness’ of any design for estimating the parameters of this model. Since this criterion is based on the determinant of the information matrix M(d) of a design d, upper bounds for |M(d)| yield lower bounds for the D-efficiency of any design d in estimating the vector of parameters in the model. We consider here only classes of designs d for which the number N of observations to be taken is a multiple of V, that is, there exists R≥2 such that N=V×R.Under these conditions, we determine the maximum of |M(d)|, and conditions under which the maximum is attained. These conditions include R being even, each treatment level being observed the same number of times, that is, R times, and N being a multiple of four. For the other cases of congruence of N (modulo 4) we further determine upper bounds on |M (d)| for equireplicated designs, i.e. for designs with equal number of observations per treatment level. These upper bounds are shown to depend also on the congruence of V (modulo 4). For some triples (N,V,K), the upper bounds determined are shown to be attained.Construction methods yielding families of designs which attain the upper bounds of |M(d)| are presented, for each of the sixteen cases of congruence of N and V.We also determine the upper bound for D-optimal designs for estimating only the treatment parameters, when first order regression on one continuous covariate is present.     

Linear discrimination for multi-level multivariate data with separable means and jointly equicorrelated covariance structure

In this article we study a linear discriminant function of multiple m-variate observations at u-sites and over v-time points under the assumption of multivariate normality. We assume that the m-variate observations have a separable mean vector structure and a “jointly equicorrelated covariance” structure. The new discriminant function is very effective in discriminating individuals in a small sample scenario. No closed-form expression exists for the maximum likelihood estimates of the unknown population parameters, and their direct computation is nontrivial. An iterative algorithm is proposed to calculate the maximum likelihood estimates of these unknown parameters. A discriminant function is also developed for unstructured mean vectors. The new discriminant functions are applied to simulated data sets as well as to a real data set. Results illustrating the benefits of the new classification methods over the traditional one are presented.     

Model-robust supersaturated and partially supersaturated designs

Supersaturated designs are an increasingly popular tool for screening factors in the presence of effect sparsity. The advantage of this class of designs over resolution III factorial designs or Plackett–Burman designs is that n, the number of runs, can be substantially smaller than the number of factors, m. A limitation associated with most supersaturated designs produced thus far is that the capability of these designs for estimating g active effects has not been discussed. In addition to exploring this capability, we develop a new class of model-robust supersaturated designs that, for a given n and m, maximizes the number g   of active effects that can be estimated simultaneously. The capabilities of model-robust supersaturated designs for model discrimination are assessed using a model-discrimination criterion, the subspace angle. Finally, we introduce the class of partially supersaturated designs, intended for use when we require a specific subset of _m1$m_1$ core factors to be estimable, and the sparsity of effects principle applies to the remaining (m-_m1$m-m_1$) factors.     

Nonparametric location-scale models for censored successive survival times

Let (T₁,T₂) be gap times corresponding to two consecutive events, which are observed subject to (univariate) random right-censoring. The censoring variable corresponding to the second gap time T₂ will in general depend on this gap time. Suppose the vector (T₁,T₂) satisfies the nonparametric location-scale regression model T₂=m(T₁)+σ(T₁)?, where the functions m and σ are ‘smooth’, and ? is independent of T₁. The aim of this paper is twofold. First, we propose a nonparametric estimator of the distribution of the error variable under this model. This problem differs from others considered in the recent related literature in that the censoring acts not only on the response but also on the covariate, having no obvious solution. On the basis of the idea of transfer of tail information (Van Keilegom and Akritas, 1999), we then use the proposed estimator of the error distribution to introduce nonparametric estimators for important targets such as: (a) the conditional distribution of T₂ given T₁; (b) the bivariate distribution of the gap times; and (c) the so-called transition probabilities. The asymptotic properties of these estimators are obtained. We also illustrate through simulations, that the new estimators based on the location-scale model may behave much better than existing ones.     

A geometrically motivated parametric model in manifold estimation

The general aim of manifold estimation is reconstructing, by statistical methods, an m-dimensional compact manifold S on ^d (with m≤d) or estimating some relevant quantities related to the geometric properties of S. Focussing on the cases d=2 and d=3, with m=d or m=d?1, we will assume that the data are given by the distances to S from points randomly chosen on a band surrounding S. The aim of this paper is to show that, if S belongs to a wide class of compact sets (which we call sets with polynomial volume), the proposed statistical model leads to a relatively simple parametric formulation. In this setup, standard methodologies (method of moments, maximum likelihood) can be used to estimate some interesting geometric parameters, including curvatures and Euler characteristic. We will particularly focus on the estimation of the (d?1)-dimensional boundary measure (in Minkowski's sense) of S. It turns out, however, that the estimation problem is not straightforward since the standard estimators show a remarkably pathological behaviour: while they are consistent and asymptotically normal, their expectations are infinite. The theoretical and practical consequences of this fact are discussed in some detail.     

On block-transitive designs with affine automorphism group

Detailed necessary and sufficient conditions for a k-subset of AG(d, 3) to generate the block set of a block-transitive t-design with automorphism group AGL(d, 3) are derived for t = 3, 4, 5. Similar necessary conditions are found for the existence of a block-transitive design with automorphism group AGL(d, p) when p is an arbitrary odd prime. A search was carried out to find feasible parameter sets satisfying the implied divisibility conditions. The only ‘small’ feasible parameter sets found with k or v − k not exceeding 1000 were for t = 4 and (d, p) = (7, 3), (8, 3), and (3, 7). Examples of block-transitive 4-designs admitting AGL(7, 3) are found for each of the values k = 115, 116, 230, 437, and 552.     

Cyclic designs for a covariate model

A linear model with one treatment at V levels and first order regression on K continuous covariates with values on a K-cube is considered. We restrict our attention to classes of designs d for which the number of observations N to be taken is a multiple of V, i.e. N = V × R with R ≥2, and each treatment level is observed R times. Among these designs, called here equireplicated, there is a subclass characterized by the following: the allocation matrix of each treatment level (for short, allocation matrix) is obtained through cyclic permutation of the columns of the allocation matrix of the first treatment level. We call these designs cyclic. Besides having easy representation, the most efficient cyclic designs are often D-optimal in the class of equireplicated designs. A known upper bound for the determinant of the information matrix M(d) of a design, in the class of equireplicated ones, depends on the congruences of N and V modulo 4. For some combinations of parameter moduli, we give here methods of constructing families of D-optimal cyclic designs. Moreover, for some sets of parameters (N, V,K = V), where the upper bound on ∣M(d)∣ (for that specific combination of moduli) is not attainable, it is also possible to construct highly D-efficient cyclic designs. Finally, for N≤24 and V≤6, computer search was used to determine the most efficient design in the class of cyclic ones. They are presented, together with their respective efficiency in the class of equireplicated designs.     

Concerning the vector v(m,t)

Let q = mt + 1 be a prime power, and let v(m, t) be the (m + 1)-vector (b₁, b₂, …, b_{m + 1}) of elements of GF(q) such that for each k, 1 ⩽ k ⩽ m + 1, the set {b_i−b_j:i∈{1,2,…m+1} − {m + 2 − k}, j  i + k(mod m + 2) and 1⩽j⩽m+1} forms a system of representatives for the cyclotomic classes of index m in GF(q). In this paper, we investigate the existence of such vectors. An upper bound on t for the existence of a v(m, t) is given for each fixed m unless both m and t are even, in which case there is no such a vector. Some special cases are also considered.     

Bayesian Estimation of Change Point in Inverse Weibull Sequence

A sequence of independent lifetimes X ₁,…, X _m , X _m+1,…, X _n were observed from inverse Weibull distribution with mean stress θ₁ and reliability R ₁(t ₀) at time t ₀ but later it was found that there was a change in the system at some point of time m and it is reflected in the sequence after X _m by change in mean stress θ₁ and in reliability R ₂(t ₀) at time t ₀. The Bayes estimators of m, R ₁(t ₀) and R ₂(t ₀) are derived when a poor and a more detailed prior information is introduced into the inferential procedure. The effects of correct and wrong prior information on the Bayes estimators are studied.     

Optimal U-type design for Bayesian nonparametric multiresponse prediction

This paper presents an extension of the work of Yue and Chatterjee (2010) about U-type designs for Bayesian nonparametric response prediction. We consider nonparametric Bayesian regression model with p responses. We use U-type designs with n runs, m factors and q levels for the nonparametric multiresponse prediction based on the asymptotic Bayesian criterion. A lower bound for the proposed criterion is established, and some optimal and nearly optimal designs for the illustrative models are given.     

Quantile inference for heteroscedastic regression models

Consider the nonparametric heteroscedastic regression model Y=m(X)+σ(X)?, where m(·) is an unknown conditional mean function and σ(·) is an unknown conditional scale function. In this paper, the limit distribution of the quantile estimate for the scale function σ(X) is derived. Since the limit distribution depends on the unknown density of the errors, an empirical likelihood ratio statistic based on quantile estimator is proposed. This statistics is used to construct confidence intervals for the variance function. Under certain regularity conditions, it is shown that the quantile estimate of the scale function converges to a Brownian motion and the empirical likelihood ratio statistic converges to a chi-squared random variable. Simulation results demonstrate the superiority of the proposed method over the least squares procedure when the underlying errors have heavy tails.     

Search designs for 2m factorials derived from balanced arrays of strength 2(l+1) and ad-optimal search designs

We consider a search design for the 2^m type such that at most knonnegative effects can be searched among (l+1)-factor interactions and estimated along with the effects up to l- factor interactions, provided (l+1)-factor and higher order interactions are negligible except for the k effects. We investigate some properties of a search design which is yielded by a balanced 2^m design of resolution 2l+1 derived from a balanced array of strength 2(l+1). A necessary and sufficient condition for the balanced design of resolution 2l+1 to be a search design for k=1 is given. Optimal search designs for k=1 in the class of the balanced 2^m designs of resolution V (l=2), with respect to the AD-optimality criterion given by Srivastava (1977), with N assemblies are also presented, where the range of (m,N) is (m=6; 28≤N≤41), (m=7; 35≤N≤63) and (m=8; 44≤N≤74).     

Efficient crossover designs in the presence of interactions between direct and carry-over treatment effects

Crossover designs, or repeated measurements designs, are used for experiments in which t treatments are applied to each of n experimental units successively over p time periods. Such experiments are widely used in areas such as clinical trials, experimental psychology and agricultural field trials. In addition to the direct effect on the response of the treatment in the period of application, there is also the possible presence of a residual, or carry-over, effect of a treatment from one or more previous periods. We use a model in which the residual effect from a treatment depends upon the treatment applied in the succeeding period; that is, a model which includes interactions between the treatment direct and residual effects. We assume that residual effects do not persist further than one succeeding period.A particular class of strongly balanced repeated measurements designs with n=t² units and which are uniform on the periods is examined. A lower bound for the A-efficiency of the designs for estimating the direct effects is derived and it is shown that such designs are highly efficient for any number of periods p=2,…,2t.     

Estimation of a standard measuring error by repeated measurements and sortings

A sorting-and-measuring machine (SMM) measures and sorts (classifies) on-line produced items into several groups according to their size. The measuring devices of the SMM perceive the actual item size with a random error ε and classify the item as being smaller than b iff z+ε<b. Here ε is a normal zero-mean r.v. with unknown standard deviation σ which is the main parameter characterizing the precision and technical condition of an SMM. The paper gives the following method of estimating σ. N₀ items are measured and N₁ of them are recognized by the SMM as belonging to the group a<z≤b. These N₁ items are sorted again and N₂ of them return to this group, these are sorted again, and so on. The estimation of σ is based on the statistics N_m/N_n. Moments of the ratio statistics N_m/N_n and their distributional properties are investigated. It turns out that the expected value of N_m/N_n depends almost linearly on σ which allows us to construct ‘almost’ unbiased estimators of type $\text{σ}\text{?}{}_{\mathrm{mn}}\text{=A}\textcolor{red}{}\text{N}{}_{\mathrm{m}}\text{/N}{}_{\mathrm{n}}\text{+B}$ with good propert including robustness with respect to the distribution of item size. Convex combinations of $\text{σ}\text{?}{}_{\mathrm{mn}}$ statistics are considered to obtain an estimator with minimal variance.     

On the spectrum of [v,k, t] trades

A [v, k, t] trade of volume m consists of two disjoint collections T₁ and T₂, each of m k-subsets of a v-set V, such that each t-subset of V is contained in the same number of blocks of T₁ and T₂, and each element of V is contained in at least one block of T₁. We study [v, k, t] trades, and investigate their spectrum (i.e., the collections of allowable volumes), using both theoretical techniques and computer-based searches.     

Two new data-dependent choices of m when applying the m-out-of-n bootstrap to hypothesis testing

The traditional non-parametric bootstrap (referred to as the n-out-of-n bootstrap) is a widely applicable and powerful tool for statistical inference, but in important situations it can fail. It is well known that by using a bootstrap sample of size m, different from n, the resulting m-out-of-n bootstrap provides a method for rectifying the traditional bootstrap inconsistency. Moreover, recent studies have shown that interesting cases exist where it is better to use the m-out-of-n bootstrap in spite of the fact that the n-out-of-n bootstrap works. In this paper, we discuss another case by considering its application to hypothesis testing. Two new data-based choices of m are proposed in this set-up. The results of simulation studies are presented to provide empirical comparisons between the performance of the traditional bootstrap and the m-out-of-n bootstrap, based on the two data-dependent choices of m, as well as on an existing method in the literature for choosing m. These results show that the m-out-of-n bootstrap, based on our choice of m, generally outperforms the traditional bootstrap procedure as well as the procedure based on the choice of m proposed in the literature.     

An extension of the ranked set sampling theory

This paper is concerned with ranked set sampling theory which is useful to estimate the population mean when the order of a sample of small size can be found without measurements or with rough methods. Consider n sets of elements each set having size m. All elements of each set are ranked but only one is selected and quantified. The average of the quantified elements is adopted as the estimator. In this paper we introduce the notion of selective probability which is a generalization of a notion from Yanagawa and Shirahata (1976). Uniformly optimal unbiased procedures are found for some (n,m). Furthermore, procedures which are unbiased for all distributions and are good for symmetric distributions are studied for (n,m) which do not allow uniformly optimal unbiased procedures.     

Model based labeling for mixture models

Label switching is one of the fundamental problems for Bayesian mixture model analysis. Due to the permutation invariance of the mixture posterior, we can consider that the posterior of a m-component mixture model is a mixture distribution with m! symmetric components and therefore the object of labeling is to recover one of the components. In order to do labeling, we propose to first fit a symmetric m!-component mixture model to the Markov chain Monte Carlo (MCMC) samples and then choose the label for each sample by maximizing the corresponding classification probabilities, which are the probabilities of all possible labels for each sample. Both parametric and semi-parametric ways are proposed to fit the symmetric mixture model for the posterior. Compared to the existing labeling methods, our proposed method aims to approximate the posterior directly and provides the labeling probabilities for all possible labels and thus has a model explanation and theoretical support. In addition, we introduce a situation in which the “ideally” labeled samples are available and thus can be used to compare different labeling methods. We demonstrate the success of our new method in dealing with the label switching problem using two examples.     

A statistical method for synthesizing mediation analyses using the product of coefficient approach across multiple trials

Mediation analysis often requires larger sample sizes than main effect analysis to achieve the same statistical power. Combining results across similar trials may be the only practical option for increasing statistical power for mediation analysis in some situations. In this paper, we propose a method to estimate: (1) marginal means for mediation path a, the relation of the independent variable to the mediator; (2) marginal means for path b, the relation of the mediator to the outcome, across multiple trials; and (3) the between-trial level variance–covariance matrix based on a bivariate normal distribution. We present the statistical theory and an R computer program to combine regression coefficients from multiple trials to estimate a combined mediated effect and confidence interval under a random effects model. Values of coefficients a and b, along with their standard errors from each trial are the input for the method. This marginal likelihood based approach with Monte Carlo confidence intervals provides more accurate inference than the standard meta-analytic approach. We discuss computational issues, apply the method to two real-data examples and make recommendations for the use of the method in different settings.     

Particle swarm optimization based Liu-type estimator

In this study, a new method for the estimation of the shrinkage and biasing parameters of Liu-type estimator is proposed. Because k is kept constant and d is optimized in Liu’s method, a (k, d) pair is not guaranteed to be the optimal point in terms of the mean square error of the parameters. The optimum (k, d) pair that minimizes the mean square error, which is a function of the parameters k and d, should be estimated through a simultaneous optimization process rather than through a two-stage process. In this study, by utilizing a different objective function, the parameters k and d are optimized simultaneously with the particle swarm optimization technique.