Robust Linear Calibration

We regard the simple linear calibration problem where only the response y of the regression line y = β₀ + β₁ t is observed with errors. The experimental conditions t are observed without error. For the errors of the observations y we assume that there may be some gross errors providing outlying observations. This situation can be modeled by a conditionally contaminated regression model. In this model the classical calibration estimator based on the least squares estimator has an unbounded asymptotic bias. Therefore we introduce calibration estimators based on robust one-step-M-estimators which have a bounded asymptotic bias. For this class of estimators we discuss two problems: The optimal estimators and their corresponding optimal designs. We derive the locally optimal solutions and show that the maximin efficient designs for non-robust estimation and robust estimation coincide.     

Robust designs and optimality of least squares for regression problems

Unbiased linear estimators are considered for the model

$\text{Y(}\text{x}{}_{\mathrm{i}}\text{)=θ}{}_0\text{+}\text{∑k}\text{j=1}\text{θ}{}_{\mathrm{j}}\text{x}{}_{\mathrm{ij}}\text{+ψ(}\text{x}{}_{\mathrm{i}}\text{)+ε}{}_{\mathrm{i}}\text{, i=1,2,…,n,}$

where ψ(x) is an unknown contamination. It is assumed that |ψ(x)|?φ(6x6) where φ is a convex function. Minimax analogues of Φ_p-optimality criteria are introduced. It is shown that, under certain (sufficient) conditions, the least squares estimators and corresponding designs are optimal in the class of all unbiased linear estimators and designs. It is also shown that, in the case when least squares estimators with symmetric design do not lead to an optimal solution, the relative efficiency of optimal least squares is not diminishing and has a uniform lower bound.     

Implications of survey design for generalized regression estimation of linear functions

The paper presents a general randomization theory approach to point and interval estimation of Q linear functions T_q = Σ^N₁c_kqY_k(q = 1,…,Q), where Y₁,…,Y_N are values of a variable of interest Y in a finite population. Such linear functions include population and domain means and totals, population regression coefficients, etc. We assume that some auxiliary information can be exploited. This suggests the generalized regression technique based on the fit of a linear model, whereby is created approximately design unbiased estimators T?_q. The paper focuses on estimation of the variance-covariance matrix of the T?_q for single stage and two stage designs. Two techniques based on Taylor expansions are compared. Results of Monte-Carlo experiments (not reported here) show that the coverage properties are good of normal-theory confidence intervals flowing from one or the other variance estimate.     

Efficient experimental designs for sigmoidal growth models

For the Weibull- and Richards-regression model robust designs are determined by maximizing a minimum of D  - or _D1$D_1$-efficiencies, taken over a certain range of the non-linear parameters. It is demonstrated that the derived designs yield a satisfactory solution of the optimal design problem for this type of model in the sense that these designs are efficient and robust with respect to misspecification of the unknown parameters. Moreover, the designs can also be used for testing the postulated form of the regression model against a simplified sub-model.     

Equivalence theorem for Schur optimality of experimental designs

An experimental design is said to be Schur optimal, if it is optimal with respect to the class of all Schur isotonic criteria, which includes Kiefer's criteria of _Φp${\Phi }_p$-optimality, distance optimality criteria and many others. In the paper we formulate an easily verifiable necessary and sufficient condition for Schur optimality in the set of all approximate designs of a linear regression experiment with uncorrelated errors. We also show that several common models admit a Schur optimal design, for example the trigonometric model, the first-degree model on the Euclidean ball, and the Berman's model.     

On the stability of the jackknife variance estimator in ratio estimation

The stability of a slightly modified version of the usual jackknife variance estimator is evaluated exactly in small samples under a suitable linear regression model and compared with that of two different linearization variance estimators. Depending on the degree of heteroscedasticity of the error variance in the model, the stability of the jackknife variance estimator is found to be somewhat comparable to that of one or the other of the linearization variance estimators under conditions especially favorable to ratio estimation (i.e., regression approximately through the origin with a relatively small coefficient of variation in the x population). When these conditions do not hold, however, the jackknife variance estimator is found to be less stable than either of the linearization variance estimators.     

Model robust extrapolation designs

We seek designs which are optimal in some sense for extrapolation when the true regression function is in a certain class of regression functions. More precisely, the class is defined to be the collection of regression functions such that its (h + 1)-th derivative is bounded. The class can be viewed as representing possible departures from an ‘ideal’ model and thus describes a model robust setting. The estimates are restricted to be linear and the designs are restricted to be with minimal number of points. The design and estimate sought is minimax for mean square error. The optimal designs for cases X = [0, ∞] and X = [-1, 1], where X is the place where observations can be taken, are discussed.     

On representing maximin efficient designs by Taylor series

This paper considers exponential and rational regression models that are nonlinear in some parameters. Recently, locally D-optimal designs for such models were investigated in [Melas, V. B., 2005. On the functional approach to optimal designs for nonlinear models. J. Statist. Plann. Inference 132, 93–116] based upon a functional approach. In this article a similar method is applied to construct maximin efficient D-optimal designs. This approach allows one to represent the support points of the designs by Taylor series, which gives us the opportunity to construct the designs by hand using tables of the coefficients of the series. Such tables are provided here for models with two nonlinear parameters. Furthermore, the recurrent formulas for constructing the tables for arbitrary numbers of parameters are introduced.     

Saturated designs for multivariate cubic regression

This paper deals with the problem of finding saturated designs for multivariate cubic regression on a cube which are nearly D-optimal. A finite class of designs is presented for the k dimensional cube having the property that the sequence of the best designs in this class for each k is asymptotically efficient as k increases. A method for constructing good designs in this class is discussed and the construction is carried out for 1?k?8. These numerical results are presented in the last section of the paper.     

Inferences from biased samples with a memory effect

Biased sampling occurs often in observational studies. With one biased sample, the problem of nonparametrically estimating both a target density function and a selection bias function is unidentifiable. This paper studies the nonparametric estimation problem when there are two biased samples that have some overlapping observations (i.e. recaptures) from a finite population. Since an intelligent subject sampled previously may experience a memory effect if sampled again, two general 2-stage models that incorporate both a selection bias and a possible memory effect are proposed. Nonparametric estimators of the target density, selection bias, and memory functions, as well as the population size are developed. Asymptotic properties of these estimators are studied and confidence bands for the selection function and memory function are provided. Our procedures are compared with those ignoring the memory effect or the selection bias in finite sample situations. A nonparametric model selection procedure is also given for choosing a model from the two 2-stage models and a mixture of these two models. Our procedures work well with or without a memory effect, and with or without a selection bias. The paper concludes with an application to a real survey data set.     

Families of Hadamard group divisible designs

By a family of designs we mean a set of designs whose parameters can be represented as functions of an auxiliary variable t where the design will exist for infinitely many values of t. The best known family is probably the family of finite projective planes with υ = b = t² + t + 1, r = k = t + 1, and λ = 1. In some instances, notably coding theory, the existence of families is essential to provide the degree of precision required which can well vary from one coding problem to another. A natural vehicle for developing binary codes is the class of Hadamard matrices. Bush (1977) introduced the idea of constructing semi-regular designs using Hadamard matrices whereas the present study is concerned mostly with construction of regular designs using Hadamard matrices. While codes constructed from these designs are not optimal in the usual sense, it is possible that they may still have substantial value since, with different values of λ₁ and λ₂, there are different error correcting capabilities.     

Semiparametric multiple kernel estimators and model diagnostics for count regression functions

Abstract

This study concerns semiparametric approaches to estimate discrete multivariate count regression functions. The semiparametric approaches investigated consist of combining discrete multivariate nonparametric kernel and parametric estimations such that (i) a prior knowledge of the conditional distribution of model response may be incorporated and (ii) the bias of the traditional nonparametric kernel regression estimator of Nadaraya-Watson may be reduced. We are precisely interested in combination of the two estimations approaches with some asymptotic properties of the resulting estimators. Asymptotic normality results were showed for nonparametric correction terms of parametric start function of the estimators. The performance of discrete semiparametric multivariate kernel estimators studied is illustrated using simulations and real count data. In addition, diagnostic checks are performed to test the adequacy of the parametric start model to the true discrete regression model. Finally, using discrete semiparametric multivariate kernel estimators provides a bias reduction when the parametric multivariate regression model used as start regression function belongs to a neighborhood of the true regression model.     

Optimality of random allocation design for the control of accidental bias in sequential experiments

In comparing two treatments, suppose the suitable subjects arrive sequentially and must be treated at once. Known or unknown to the experimenter there may be nuisance factors systematically affecting the subjects. Accidental bias is a measure of the influence of these factors in the analysis of data. We show in this paper that the random allocation design minimizes the accidental bias among all designs that allocate n, out of 2n, subjects to each treatment and do not prefer either treatment in the assignment. When the final imbalance is allowed to be nonzero, optimal and efficient designs are given. In particular the random allocation design is shown to be very efficient in this broader setup.     

Optimum designs in complete two-way layouts

In the usual two-way layout of ANOVA (interactions are admitted) let nij ? 1 be the number of observations for the factor-level combination(i, j). For testing the hypothesis that all main effects of the first factor vanish numbers $\text{n}{}^1\text{ij}$ are given such that the power function of the F-test is uniformly maximized (U-optimality), if one considers only designs (nij) for which the row-sums ni are prescribed. Furthermore, in the (larger) set of all designs for which the total number of observations is given, all D-optimum designs are constructed.     

Parametric inference based on judgment post stratified samples

In this paper, we consider a judgment post stratified (JPS) sample of set size $H$ from a location and scale family of distributions. In a JPS sample, ranks of measured units are random variables. By conditioning on these ranks, we derive the maximum likelihood (MLEs) and best linear unbiased estimators (BLUEs) of the location and scale parameters. Since ranks are random variables, by considering the conditional distributions of ranks given the measured observations we construct Rao-Blackwellized version of MLEs and BLUEs. We show that Rao-Blackwellized estimators always have smaller mean squared errors than MLEs and BLUEs in a JPS sample. In addition, the paper provides empirical evidence for the efficiency of the proposed estimators through a series of Monte Carlo simulations.     

On the simple linear regression model with correlated measurement errors

The simple linear regression model with measurement error has been subject to much research. In this work we will focus on this model when the error in the explanatory variable is correlated with the error in the regression equation. Specifically, we are interested in the comparison between the ordinary errors-in-variables estimator of the regression coefficient β$\beta$ and the estimator that takes account of the correlation between the errors. Based on large sample approximations, we compare the estimators and find that the estimator that takes account of the correlation should be preferred in most situations. We also compare the estimators in small sample situations. This is done by stochastic simulation. The results show that the estimators behave quite similarly in most of the simulated situations, but that the ordinary errors-in-variables estimator performs considerably worse than the estimator that takes account of the correlation for certain parameter combinations. In addition, we look briefly into the bias introduced by ignoring correlated errors when computing sample correlations, and in predictions.     

On optimality properties of simple block designs in the approximate design theory

Optimality properties of approximate block designs are studied under variations of (1) the class of competing designs, (2) the optimality criterion, (3) the parametric function of interest, and (4) the statistical model. The designs which are optimal turn out to be the product of their treatment and block marginals, and uniform designs when the support is specified in advance. Optimality here means uniform, universal, and simultaneous j_p-optimality. The classical balanced incomplete block designs are embedded into this approach, and shown to be simultaneously j_p-optimal for a maximal system of identifiable parameters. A geometric account of universal optimality is given which applies beyond the context of block designs.     

Considerations on group-wise identical designs for linear mixed models

We consider a general class of mixed models, where the individual parameter vector is composed of a linear function of the population parameter vector plus an individual random effects vector. The linear function can vary for the different individuals. We show that the search for optimal designs for the estimation of the population parameter vector can be restricted to the class of group-wise identical designs, i.e., for each of the groups defined by the different linear functions only one individual elementary design has to be optimized. A way to apply the result to non-linear mixed models is described.     

Robust prediction and extrapolation designs for censored data

In this paper we present the construction of robust designs for a possibly misspecified generalized linear regression model when the data are censored. The minimax designs and unbiased designs are found for maximum likelihood estimation in the context of both prediction and extrapolation problems. This paper extends preceding work of robust designs for complete data by incorporating censoring and maximum likelihood estimation. It also broadens former work of robust designs for censored data from others by considering both nonlinearity and much more arbitrary uncertainty in the fitted regression response and by dropping all restrictions on the structure of the regressors. Solutions are derived by a nonsmooth optimization technique analytically and given in full generality. A typical example in accelerated life testing is also demonstrated. We also investigate implementation schemes which are utilized to approximate a robust design having a density. Some exact designs are obtained using an optimal implementation scheme.     

The superharmonic condition for simultaneous estimation of means in exponential familles

The mean vector associated with several independent variates from the exponential subclass of Hudson (1978) is estimated under weighted squared error loss. In particular, the formal Bayes and “Stein-like” estimators of the mean vector are given. Conditions are also given under which these estimators dominate any of the “natural estimators”. Our conditions for dominance are motivated by a result of Stein (1981), who treated the N_p (θ, I) case with p ≥ 3. Stein showed that formal Bayes estimators dominate the usual estimator if the marginal density of the data is superharmonic. Our present exponential class generalization entails an elliptic differential inequality in some natural variables. Actually, we assume that each component of the data vector has a probability density function which satisfies a certain differential equation. While the densities of Hudson (1978) are particular solutions of this equation, other solutions are not of the exponential class if certain parameters are unknown. Our approach allows for the possibility of extending the parametric Stein-theory to useful nonexponential cases, but the problem of nuisance parameters is not treated here.