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.     

An unexpected property of minimax estimation in the relative squared error approach to linear regression analysis

An unexpected property of the relative squared error approach to linear regression analysis is derived: It is shown that an estimator being minimax among all linear affine estimators is also minimax in the set of all estimators. Two illustrative special cases are mentioned, where a generalized least squares estimator and a general ridge or Kuks-Olman estimator turn out to be minimax.     

Optimale Planung eines Kovarianzanalyse- und eines Intraclass Regressions-Experiments

Summary The modeling and design of general experiments with qualitative and quantitative factors of influence is presented and discussed. A complete and ready to apply characterization of concrete optimal designs and their operational characteristics are given for the two basic models of analysis of covariance and of intra class regression under different experiment constraints.     

Asymptotic distributions of some scale estimators in nonlinear models

Often in the robust analysis of regression and time series models there is a need for having a robust scale estimator of a scale parameter of the errors. One often used scale estimator is the median of the absolute residuals s ₁. It is of interest to know its limiting distribution and the consistency rate. Its limiting distribution generally depends on the estimator of the regression and/or autoregressive parameter vector unless the errors are symmetrically distributed around zero. To overcome this difficulty it is then natural to use the median of the absolute differences of pairwise residuals, s ₂, as a scale estimator. This paper derives the asymptotic distributions of these two estimators for a large class of nonlinear regression and autoregressive models when the errors are independent and identically distributed. It is found that the asymptotic distribution of a suitably standardizes s ₂ is free of the initial estimator of the regression/autoregressive parameters. A similar conclusion also holds for s ₁ in linear regression models through the origin and with centered designs, and in linear autoregressive models with zero mean errors.  This paper also investigates the limiting distributions of these estimators in nonlinear regression models with long memory moving average errors. An interesting finding is that if the errors are symmetric around zero, then not only is the limiting distribution of a suitably standardized s ₁ free of the regression estimator, but it is degenerate at zero. On the other hand a similarly standardized s ₂ converges in distribution to a normal distribution, regardless of the errors being symmetric or not. One clear conclusion is that under the symmetry of the long memory moving average errors, the rate of consistency for s ₁ is faster than that of s ₂.     

The mixture index of fit and minimax regression

A measure of the fit of a statistical model can be obtained by estimating the relative size of the largest fraction of the population where a distribution belonging to the model may be valid. This is the mixture index of fit that was suggested for models for contingency tables by Rudas, Clogg, Lindsay (1994) and it is extended here for models involving continuous observations. In particular, the approach is applied to regression models with normal and uniform error structures. Best fit, as measured by the mixture index of fit, is obtained with minimax estimation of the regression parameters. Therefore, whenever minimax estimation is used for these problems, the mixture index of fit provides a natural approach for measuring model fit and for variable selection. Received: September 1997     

Model reference adaptive systems applied to regression analyses

The adaptive estimation procedure of model reference adaptive systems is modified and applied to linear models. In general the principle can be used for almost any time series model. Because of the recursive nature of the resulting estimator, it is computationally appealing, especially when a time series is considered as a flow of data. In addition, the estimator turns out to have certain statistical optimality properties.
In the linear regression setting, Ridge estimators turn out to constitute a subclass of the adaptive estimators considered, whereas for unknown measurement variance, the resulting estimators are related to J ames -S tkin type estimators, and have better properties than the latter. The estimator is shown to be strongly consistent and to converge in law to a normal variate under the standard assumptions of linear models. Further it is shown to be admissible and minimax in restricted parameter spaces. The connection between K alman filters and the classical least-squares estimator is also pointed out.     

Partial Minimax Estimation in Regression Analysis

The general minimax estimator of the linear regression model is applicable when the whole parameter vector is restricted to an ellipsoid. In many applications, however, it is more realistic to assume that only a part of the parameter set is constrained. For this case an alternative minimax approach is developed.     

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.     

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     

Blockwise bootstrap wavelet in nonparametric regression model with weakly dependent processes

In this paper, we suggest a blockwise bootstrap wavelet to estimate the regression function in the nonparametric regression models with weakly dependent processes for both designs of fixed and random. We obtain the asymptotic orders of the biases and variances of the estimators and establish the asymptotic normality for a modified version of the estimators. We also introduce a principle to select the length of data block. These results show that the blockwise bootstrap wavelet is valid for general weakly dependent processes such as α-mixing, φ-mixing and ρ-mixing random variables.     

Small‐sample testing inference in symmetric and log‐symmetric linear regression models

This paper deals with the issue of testing hypotheses in symmetric and log‐symmetric linear regression models in small and moderate‐sized samples. We focus on four tests, namely, the Wald, likelihood ratio, score, and gradient tests. These tests rely on asymptotic results and are unreliable when the sample size is not large enough to guarantee a good agreement between the exact distribution of the test statistic and the corresponding chi‐squared asymptotic distribution. Bartlett and Bartlett‐type corrections typically attenuate the size distortion of the tests. These corrections are available in the literature for the likelihood ratio and score tests in symmetric linear regression models. Here, we derive a Bartlett‐type correction for the gradient test. We show that the corrections are also valid for the log‐symmetric linear regression models. We numerically compare the various tests and bootstrapped tests, through simulations. Our results suggest that the corrected and bootstrapped tests exhibit type I probability error closer to the chosen nominal level with virtually no power loss. The analytically corrected tests as well as the bootstrapped tests, including the Bartlett‐corrected gradient test derived in this paper, perform with the advantage of not requiring computationally intensive calculations. We present a real data application to illustrate the usefulness of the modified tests.     

Two planar facility location problems with high-speed corridors and continuous demand

The planar minisam (‘median’) and minimax (‘center’) facility location problems are examined under the assumptions that: demand is continuously and uniformly distributed: the L₁ (right-angle) metric is in use; and the planar region is traversed by a high-speed corridor (highway) running parallel to one of the directions of travel. For the minisum problem in a rectangular region, it is shown that, for any combination of problem parameters, there are only two candidate points in the region for the optimum location of the facility. This is also shown to be true for any convex and symmetric planar region. For the minimax problem in a rectangular region, there are only three candidate points for the optimal location. Some extensions and conjectures for the minisum problem involving more than one highway are also discussed.     

R-optimal designs for multi-factor models with heteroscedastic errors

In this paper, we consider the R-optimal design problem for multi-factor regression models with heteroscedastic errors. It is shown that a R-optimal design for the heteroscedastic Kronecker product model is given by the product of the R-optimal designs for the marginal one-factor models. However, R-optimal designs for the additive models can be constructed from R-optimal designs for the one-factor models only if sufficient conditions are satisfied. Several examples are presented to illustrate and check optimal designs based on R-optimality criterion.     

Decision theory for continuous observations: Minimax solutions

This paper is concerned with the existence of minimax solutions in a decision theoretic model for continuous observations as described inIrle, Schmitz, [1914]. There is given a result on the existence of minimax solutions with constant time observation, followed by a discussion of the conditions which are used to prove the above statement.     

Intraurban variation in the price of housing

Theoretical models of spatial consumer behavior assume the existence of a housing price surface that increases with accessibility. It is often hypothesized that inadequate accessibility measures are responsible for the failure of most empirical studies to support this premise. A general accessibility measure consisting of a double power series in cartesian coordinates is developed and shown to represent a general formulation of traditional accessibility measures. This formulation is incorporated in a hedonic price regression model, and the model is estimated using census tract data for the city of Milwaukee. The hedonic prices are then used to construct intraurban housing price indexes. The results of this study provide strong support for the existence of the presumed price surface. A comparison of this new hedonic price model with traditional models suggests that the double power series formulation is a superior representation of accessibility.     

Model selection for hazard rate estimation in presence of censoring

This note presents an estimator of the hazard rate function based on right censored data. A collection of estimators is built from a regression-type contrast, in a general collection of linear models. Then, a penalised model selection procedure provides an estimator which satisfies an oracle inequality. In particular, we can prove that it is adaptive in the minimax sense on Hölder spaces.     

Existence of solutions to principal–agent problems with adverse selection under minimal assumptions

We prove an existence result for the principal–agent problem with adverse selection under general assumptions on preferences and allocation spaces. Instead of assuming that the allocation space is finite-dimensional or compact, we consider a more general coercivity condition which takes into account the principal’s cost and the agents’ preferences. Our existence proof is simple and flexible enough to adapt to partial participation models as well as to the case of type-dependent budget constraints.     

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.     

Efficiencies for optimum designs when transforming the response in nonlinear models with nonconstant variance

Often the responses from mechanistic models have to be transformed to achieve error distributions that are symmetric and have constant variance. Because of the nature of the relationship between the response and the mechanistic model, it is necessary to transform both sides of the model. Expressions are given for the parameter sensitivities in the transformed model and examples given of optimum designs for particular values of λ, together with the efficiency of these designs as λ varies. Approaches to finding designs robust to variations in λ are indicated and exemplified.     

Totally positive regression: E-optimal designs

E-optimality of approximate designs in linear regression models is paired with a dual problem of nonlinear Chebyshev approximation. When the regression functions form a totally positive system, then the information matrices of designs for subparameters turn out to be “almost” totally positive, a property which allows to solve the nonlinear Chebyshev problem. Thereby we obtain explicit formulae for E-optimal designs in terms of equi-oscillating generalized polynomials. The considerations unify and generalize known results on E-optimality for particular regression setups.