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Estimation and Testing in Elliptical Functional Measurement Error Models
Communications in Statistics - Theory and Methods
Abstract
The purpose of this article is to investigate estimation and hypothesis testing by maximum likelihood and method of moments in functional models within the class of elliptical symmetric distributions. The main results encompass consistency and asymptotic normality of the method of moments estimators. Also, the asymptotic covariance matrix of the maximum likelihood estimator is derived, extending some existing results in elliptical distributions. A measure of asymptotic relative efficiency is reported. Wald-type statistics are considered and numerical results obtained by Monte Carlo simulation to investigate the performance of estimators and tests are provided for Student-t and contaminated normal distributions. An application to a real dataset is also included.
Many methods discussed in this book are motivated by research problems arising from various fields, including nutrition studies, cancer research and environmental studies. Methods and application of measurement error models are vast in the epidemiology literature. Although the book discusses some research in this field, the coverage is far from complete.
For many measurements made on processes and individuals, the true value being measured is unknown or unknowable. When data consist of two or more distinct methods of measurement, the problem of comparing measurements is often referred to as comparative calibration. A provisional analysis of such data can be made assuming one method or set of measurements is unbiased, by means of the classic functional relationship model. Assuming normality in the multivariate case (i.e., more than two methods of measurement), maximum likelihood estimates are usually calculated using generalized eigenvalues and eigenvectors. We propose calculating the same maximum likelihood estimates using a simple EM algorithm. The problems of estimating the variance of parameter estimated and testing the hypothesis of equal calibration lines are considered. An alternative error assumption for the basic model is also explored. The proposed methods are validated using simulation.
This paper presents necessary and sufficient conditions under which a random variable X may be generated as the ratio Z/V where Z and V are independent and Z has a standard normal distribution. This representation is useful in Monte Carlo calculations. It is established that when 1/2V2 is exponential, X is double exponential; and that when 1/2V has the asymptotic distribution of the Kolmogorov distance statistic, X is logistic.
Maximum likelihood estimation with nonnormal error distributions provides one method of robust regression. Certain families of normal/independent distributions are particularly attractive for adaptive, robust regression. This article reviews the properties of normal/independent distributions and presents several new results. A major virtue of these distributions is that they lend themselves to EM algorithms for maximum likelihood estimation. EM algorithms are discussed for least Lp regression and for adaptive, robust regression based on the t, slash, and contaminated normal families. Four concrete examples illustrate the performance of the different methods on real data.
This paper considers comparisons between several approaches producing consistent estimators in functional comparative calibration models, which can be seen as a special case of the linear multivariate measurement error models. Following Stefanski and Carroll (1987, Biometrika 74, 703–716) we also derive conditional and sufficiency scores by conditioning on some sufficient statistics for the incidental parameters. The asymptotic distributions associated with the different estimators are studied and it is shown that the conditional score is equivalent to the maximum likelihood estimator. Asymptotic relative efficiencies and results of a small-scale simulation study comparing the different approaches are reported.
The purpose of this article is to investigate hypothesis testing in functional comparative calibration models. Wald type statistics are considered which are asymptotically distributed according to the chi-square distribution. The statistics are based on maximum likelihood, corrected score approach, and method of moment estimators of the model parameters, which are shown to be consistent and asymptotically normally distributed. Results of analytical and simulation studies seem to indicate that the Wald statistics based on the method of moment estimators and the corrected score estimators are, as expected, less efficient than the Wald type statistic based on the maximum likelihood estimators for small n. Wald statistic based on moment estimators are simpler to compute than the other Wald statistics tests and their performance improves significantly as n increases. Comparisons with an alternative F statistics proposed in the literature are also reported.
Dolby's (1976) ultrastructural model with no replications is investigated within the class of the elliptical distributions. General asymptotic results are given for the sample covariance matrix S in the presence of incidental parameters. These results are used to study the asymptotic behaviour of some estimators of the slope parameter, unifying and extending existing results in the literature. In particular, under some regularity conditions they are shown to be consistent and asymptotically normal. For the special case of the structural model, some asymptotic relative efficiencies are also reported which show that generalized least squares and the method of moment estimators can be highly inefficient under nonnormality.
This paper considers the estimation of “structural” parameters when the number of unknown parameters increases with the sample size. Neyman and Scott (1948) had demonstrated that maximum likelihood estimators (MLE) of structural parameters may be inconsistent in this case. Patefield (1977) further observed that the asymptotic covariance matrix of the MLE is not equal to the inverse of the information matrix. In this paper we establish asymptotic properties of estimators (which include in particular the MLE) obtained via the usual likelihood approach when the incidental parameters are first replaced by their estimates (which are allowed to depend on the structural parameters). Conditions for consistency and asymptotic normality together with a proper formula for the asymptotic covariance matrix are given. The results are illustrated and applied to the problem of estimating linear functional relationships, and mild conditions on the incidental parameters for the MLE (or an adjusted MLE) to be consistent and asymptotically normal are obtained. These conditions are weaker than those imposed by previous authors.
The theory of elliptically contoured distributions is presented in an unrestricted setting, with no moment restrictions or assumptions of absolute continuity. These distributions are defined parametrically through their characteristic functions and then studied primarily through the use of stochastic representations which naturally follow from the work of Schoenberg [5] on spherically symmetric distributions. It is shown that the conditional distributions of elliptically contoured distributions are elliptically contoured, and the conditional distributions are precisely identified. In addition, a number of the properties of normal distributions (which constitute a type of elliptically contoured distributions) are shown, in fact, to characterize normality.
In this paper, functional models with not replications are investigated within the class of the elliptical distributions. Emphasis is placed on the special case of the Student-t distribution. Main results encompasses consistency and asymptotic normality of the maximum likelihood estimators. Due to the presence of incidental parameters, standard maximum likelihood methodology cannot be used to obtain the main results, which require extensions of some existing results related to elliptical distributions. Asymptotic relative efficiencies are reported which show that the generalized least squares estimator can be highly inefficient when compared with the maximum likelihood estimator under nonnormality.
Comments on " Functional comparative calibration using an EM algorithm
- H Bolfarine
- M Galea-Rojas
Bolfarine, H., Galea-Rojas, M. (1995). Comments on " Functional comparative calibration using an EM algorithm ". Biometrics 51(4):1579–1580.
Aspects of Multivariate Statistical Theory Elliptical functional models
- R J Muirhead
- F Vilca-Labra
- R B Arellano-Valle
- H Bolfarine
Muirhead, R. J. (1982). Aspects of Multivariate Statistical Theory. New York: John Wiley. Vilca-Labra, F., Arellano-Valle, R. B., Bolfarine, H. (1998). Elliptical functional models. J. Multivariate Anal. 65:36–57.