Optimal prediction designs in finite discrete spectrum linear regression models

In the paper, we solve the n-point optimal prediction design problem for the simplest nontrivial finite discrete spectrum linear regression models with correlated observations. We show that for all the models in consideration, there exists an optimal prediction design supported on at most three distinct points, which can be computed using one-dimensional optimization. In some cases, an optimal prediction design can be found explicitly.     

Distance optimality design criterion in linear models

Properties of the most familiar optimality criteria, for example A-, D- and E-optimality, are well known, but the distance optimality criterion has not drawn much attention to date. In this paper properties of the distance optimality criterion for the parameter vector of the classical linear model under normally distributed errors are investigated. DS-optimal designs are derived for first-order polynomial fit models. The matter of how the distance optimality criterion is related to traditional D- and E-optimality criteria is also addressed. Received: June 1999     

A stochastic characterization of Loewner optimality design criterion in linear models

In this paper we present a new stochastic characterization of the Loewner optimality design criterion. The result is obtained by proving a generalization to the well known corollary of Anderson's theorem. Certain connections between the Loewner optimality and the stochastic distance optimality design criterion are showed. We also present applications and generalizations of the main result. Received: 9 August 2000     

Testing fuzzy linear hypotheses in linear regression models

In this paper statistical tests with fuzzily formulated hypotheses are discussed, i.e., hypotheses H₀ and H₁ are fuzzy sets. The classical criteria of the errors of type I and type II are generalized, and this approach is applied to the linear hypothesis in the linear regression model. A sufficient condition to control both generalized criteria simultaneously is presented even in case of testing H₀ against the omnibus alternative H₁:¬ H₀. This is completely different from the classical case of testing crisp complementary hypotheses.     

Optimal designs for linear mixture models

Summary In a recent paper S nee and M arquardt [8] considered designs for linear mixture models, where the components are subject to individual lower and/or upper bounds. When the number of components is large their algorithm XVERT yields designs far too extensive for practical purposes.
The purpose of this paper is to describe a numerical procedure resulting in a design of fixed size N , which is approximately D -optimal, and where the components may be subject to linear constraints (f.e. upper or lower bounds). The proposed method is more generally applicable for models linear in the independent variables and the parameters and the convex hull of the experimental region is a polyhedron whose vertices are known.     

Estimation under linear regression models

Summary Royall and Herson considered balanced samples for ensuring robustness of standard ratio estimator under polynomial superpopulation models. Here we formulate a post-sample estimator of Royall type which remains robust (in respect of bias) under a wide class of polynomial regression models.     

Optimal and robust designs for trigonometric regression models

This article presents discussions on the optimal and robust designs for trigonometric regression models under different optimality criteria. First, we investigate the classical Q-optimal designs for estimating the response function in a full trigonometric regression model with a given order. The equivalencies of Q-, A-, and G-optimal designs for trigonometric regression in general are also articulated. Second, we study minimax designs and their implementation in the case of trigonometric approximation under Q-, A-, and D-optimality. Then, We indicate the existence of the symmetric designs that are D-optimal minimax designs for general trigonometric regression models, and prove the existence of the symmetric designs that are Q- or A-optimal minimax designs for two particular trigonometric regression models under certain conditions.     

Estimation and optimal designs for linear Haar-wavelet models

This paper gives an analytical expression for the best linear unbiased estimator (BLUE) of the unknown parameters in the linear Haar-wavelet model. From the analytical expression, we solve for the eigenvalues of the covariance matrix of the BLUE in analytical form. Further, we use these eigenvalues to construct some conventional discrete optimal designs for the model. The equivalences among these optimal designs are demonstrated and some examples are also given.      

A distribution-free least squares estimator for censored linear regression models

This paper describes a method for estimating simultaneously the parameter vector of the systematic component and the distribution function of the random component of a censored linear regression model. The estimator is obtained by minimizing the sum of the squares of the differences between the observed values of the dependent variable and the corresponding expected values of this variable according to the estimated parameter vector and distribution function. The resulting least squares parameter estimator incorporates information on the distribution of the random component of the regression model that is available from the estimation sample. Hence, it may often be more efficient than are parameter estimators that do not use such information. The results of numerical experiments with the least squares estimator tend to support this hypothesis.     

Empirical likelihood for high-dimensional linear regression models

High-dimensional data are becoming prevalent, and many new methodologies and accompanying theories for high-dimensional data analysis have emerged in response. Empirical likelihood, as a classical nonparametric method of statistical inference, has proved to possess many good features. In this paper, our focus is to investigate the asymptotic behavior of empirical likelihood for regression coefficients in high-dimensional linear models. We give regularity conditions under which the standard normal calibration of empirical likelihood is valid in high dimensions. Both random and fixed designs are considered. Simulation studies are conducted to check the finite sample performance.     

Optimal strategies in linear multisector models: Value function and optimality conditions

In this paper we study an optimal control problem with mixed constraints related to a multisector linear model with endogenous growth. The main aim is to establish a set of necessary and a set of sufficient conditions which are the basis for studying the qualitative properties of optimal trajectories. The presence of possibly degenerate mixed constraints, the unboundedness and non-strict convexity of the Hamiltonian, make the problem difficult to deal with. We develop first the dynamic programming approach, proving that the value function is a bilateral viscosity solution to the associated Hamilton–Jacobi–Bellman (HJB) equation. Then, using our results, we give a set of sufficient and a set of necessary optimality conditions which involve so-called co-state inclusion: this can be interpreted as the existence of a dual path of prices supporting the optimal path.     

Bootstrap J tests of nonnested linear regression models

The J test for nonnested regression models often overrejects very severely as an asymptotic test. We provide a theoretical analysis which explains why and when it performs badly. This analysis implies that, except in certain extreme cases, the J test will perform very well when bootstrapped. Using several methods to speed up the simulations, we obtain extremely accurate Monte Carlo results on the finite-sample performance of the bootstrapped J test. These results fully support the predictions of our theoretical analysis, even in contexts where the analysis is not strictly applicable.     

Optimal designs for generalized linear models with biased response

This paper is concerned with the search for locally optimal designs when the observations of the response variable arise from a weighted distribution in the exponential family. Locally optimal designs are derived for regression models in which the response follows a weighted version of Normal, Gamma, Inverse Gaussian, Poisson or Binomial distributions. Some conditions are given under which the optimal designs for the weighted and original (non-weighted) distributions are the same. An efficiency study is performed to find out the behavior of the D-optimal designs for the original distribution when they are used to estimate models with weighted distributions.     

Contamination in linear regression models and its influence on estimators

The consequences of the omission of possibly contaminated observations in a linear regression model for the performance of the ordinary least squares ( LS- ) estimator are discussed. We compare the ordinary L Sestimator with the corresponding 'never pooled' LS -estimator with respect to the matrix-valued mean squared error. Necessary and sufficient conditions are derived for the superiority of an estimator to another one and tests are proposed to check these conditions. Finally the resulting preliminary-test-estimators are investigated.     

Estimation of variances in orthogonal finite discrete spectrum linear regression models

The Invariant Quadratic Estimators, the Maximum Likelihood Estimator (MLE) and Restricted Maximum Likelihood Estimator (REML) of variances in an orthogonal Finite Discrete Spectrum Linear Regression Model (FDSLRM) are derived and the problems of unbiasedness and consistency of these estimators are investigated.Acknowledgement. The research was supported by the grants 1/0272/03, 1/0264/03 and 2/4026/04 of the Slovak Scientific Grant Agency VEGA.     

Asymptotic properties of M-estimators in linear and nonlinear multivariate regression models

We consider the (possibly nonlinear) regression model in \(\mathbb{R }^q\) with shift parameter \(\alpha \) in \(\mathbb{R }^q\) and other parameters \(\beta \) in \(\mathbb{R }^p\) . Residuals are assumed to be from an unknown distribution function (d.f.). Let \(\widehat{\phi }\) be a smooth \(M\) -estimator of \(\phi = {{\beta }\atopwithdelims (){\alpha }}\) and \(T(\phi )\) a smooth function. We obtain the asymptotic normality, covariance, bias and skewness of \(T(\widehat{\phi })\) and an estimator of \(T(\phi )\) with bias \(\sim n^{-2}\) requiring \(\sim n\) calculations. (In contrast, the jackknife and bootstrap estimators require \(\sim n^2\) calculations.) For a linear regression with random covariates of low skewness, if \(T(\phi ) = \nu \beta \) , then \(T(\widehat{\phi })\) has bias \(\sim n^{-2}\) (not \(n^{-1}\) ) and skewness \(\sim n^{-3}\) (not \(n^{-2}\) ), and the usual approximate one-sided confidence interval (CI) for \(T(\phi )\) has error \(\sim n^{-1}\) (not \(n^{-1/2}\) ). These results extend to random covariates.     

Locally D- and c-optimal designs for Poisson and negative binomial regression models

The research on optimal experimental designs for nonlinear regression models is of great interest because these models are used to characterize chemical, biological or agricultural phenomena. Much of them involve an exponential decay. In this paper, locally D- and c-optimal designs are derived analytically for Poisson and negative binomial regression models.     

Exact D-optimal designs for first-order trigonometric regression models on a partial circle

Recently, various approximate design problems for low-degree trigonometric regression models on a partial circle have been solved. In this paper we consider approximate and exact optimal design problems for first-order trigonometric regression models without intercept on a partial circle. We investigate the intricate geometry of the non-convex exact trigonometric moment set and provide characterizations of its boundary. Building on these results we obtain a solution of the exact $D$ -optimal design problem. It is shown that the structure of the optimal designs depends on both the length of the design interval and the number of observations.     

Bayesian robust designs for linear models with possible bias and correlated errors

Consider the design problem for the approximately linear model with serially correlated errors. The correlated structure is the qth degree moving average process, MA(q), especially for q = 1, 2. The optimal design is derived by using Bayesian approach. The Bayesian designs derived with various priors are compared with the classical designs with respect to some specific correlated structures. The results show that any prior knowledge about the sign of the MA(q) process parameters leads to designs that are considerately more efficient than the classical ones based on homoscedastic assumptions.