On Algorithms for Restricted Maximum Likelihood Estimation
In: Mathematics Preprint Archive Vol. 2002, Issue 11, pp 124-156
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In: Mathematics Preprint Archive Vol. 2002, Issue 11, pp 124-156
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Working paper
In: Statistica Neerlandica, Band 45, Heft 3, S. 271-282
ISSN: 1467-9574
For a balanced two‐way mixed model, the maximum likelihood (ML) and restricted ML (REML) estimators of the variance components were obtained and compared under the non‐negativity requirements of the variance components by Lee and Kapadia (1984). In this note, for a mixed (random blocks) incomplete block model, explicit forms for the REML estimators of variance components are obtained. They are always non‐negative and have smaller mean squared error (MSE) than the analysis of variance (AOV) estimators. The asymptotic sampling variances of the maximum likelihood (ML) estimators and the REML estimators are compared and the balanced incomplete block design (BIBD) is considered as a special case. The ML estimators are shown to have smaller asymptotic variances than the REML estimators, but a numerical result in the randomized complete block design (RCBD) demonstrated that the performances of the REML and ML estimators are not much different in the MSE sense.
In: Statistical papers, Band 38, Heft 3, S. 343-349
ISSN: 1613-9798
In: Structural equation modeling: a multidisciplinary journal, Band 10, Heft 3, S. 352-379
ISSN: 1532-8007
In: Risk analysis: an international journal, Band 24, Heft 5, S. 1143-1151
ISSN: 1539-6924
The dose to human and nonhuman individuals inflicted by anthropogenic radiation is an important issue in international and domestic policy. The current paradigm for nonhuman populations asserts that if the dose to the maximally exposed individuals in a population is below a certain criterion (e.g., <10 mGy d−1) then the population is adequately protected. Currently, there is no consensus in the regulatory community as to the best statistical approach. Statistics, currently considered, include the maximum likelihood estimator for the 95th percentile of the sample mean and the sample maximum. Recently, the investigators have proposed the use of the maximum likelihood estimate of a very high quantile as an estimate of dose to the maximally exposed individual. In this study, we compare all of the above‐mentioned statistics to an estimate based on extreme value theory. To determine and compare the bias and variance of these statistics, we use Monte Carlo simulation techniques, in a procedure similar to a parametric bootstrap. Our results show that a statistic based on extreme value theory has the least bias of those considered here, but requires reliable estimates of the population size. We recommend establishing the criterion based on what would be considered acceptable if only a small percentage of the population exceeded the limit, and hence recommend using the maximum likelihood estimator of a high quantile in the case that reliable estimates of the population size are not available.
In: Journal of applied mathematics & decision sciences: JAMDS, Band 1, Heft 1, S. 53-65
ISSN: 1532-7612
The problem of assigning cell probabilities to maximize a multinomial likelihood with order restrictions on the probabilies and/or restrictions on the local odds ratios is modeled as a posynomial geometric program (GP), a class of nonlinear optimization problems with a well-developed duality theory and collection of algorithms. (Local odds ratios provide a measure of association between categorical random variables.) A constrained multinomial MLE example from the literature is solved, and the quality of the solution is compared with that obtained by the iterative method of El Barmi and Dykstra, which is based upon Fenchel duality. Exploiting the proximity of the GP model of MLE problems to linear programming (LP) problems, we also describe as an alternative, in the absence of special-purpose GP software, an easily implemented successive LP approximation method for solving this class of MLE problems using one of the readily available LP solvers.
In: Statistica Neerlandica: journal of the Netherlands Society for Statistics and Operations Research, Band 58, Heft 4, S. 440-465
ISSN: 1467-9574
The notion of cointegration has led to a renewed interest in the identification and estimation of structural relations among economic time series. This paper reviews the different approaches that have been put forward in the literature for identifying cointegrating relationships and imposing (possibly over‐identifying) restrictions on them. Next, various algorithms to obtain (approximate) maximum likelihood estimates and likelihood ratio statistics are reviewed, with an emphasis on so‐called switching algorithms. The implementation of these algorithms is discussed and illustrated using an empirical example.
In: Risk analysis: an international journal, Band 19, Heft 1, S. 109-130
ISSN: 1539-6924
Variability arises due to differences in the value of a quantity among different members of a population. Uncertainty arises due to lack of knowledge regarding the true value of a quantity for a given member of a population. We describe and evaluate two methods for quantifying both variability and uncertainty. These methods, bootstrapsimulation and a likelihood‐based method, are applied to three datasets. The datasetsinclude a synthetic sample of 19 values from a Lognormal distribution, a sample of nine values obtained from measurements of the PCB concentration in leafy produce, and asample of five values for the partitioning of chromium in the flue gas desulfurization system of coal‐fired power plants. For each of these datasets, we employ the two methods to characterize uncertainty in the arithmetic mean and standard deviation, cumulative distribution functions based upon fitted parametric distributions, the 95th percentile of variability, and the 63rd percentile of uncertainty for the 81st percentile of variability. The latter is intended to show that it is possible to describe anypoint within the uncertain frequency distribution by specifying an uncertainty percentile and a Variability percentile. Using the bootstrap method, we compare results based upon use of the method of matching moments and the method of maximum likelihood for fitting distributions to data. Our results indicate that with only 5‐19 data pointsas in the datasets we have evaluated, there is substantial uncertainty based upon random sampling error. Both the boostrap and likelihood‐based approaches yield comparable uncertainty estimates in most cases.
In: Statistica Neerlandica: journal of the Netherlands Society for Statistics and Operations Research, Band 56, Heft 1, S. 2-22
ISSN: 1467-9574
Recent developments in Markov chain Monte Carlo [MCMC] methods have increased the popularity of Bayesian inference in many fields of research in economics, such as marketing research and financial econometrics. Gibbs sampling in combination with data augmentation allows inference in statistical/econometric models with many unobserved variables. The likelihood functions of these models may contain many integrals, which often makes a standard classical analysis difficult or even unfeasible. The advantage of the Bayesian approach using MCMC is that one only has to consider the likelihood function conditional on the unobserved variables. In many cases this implies that Bayesian parameter estimation is faster than classical maximum likelihood estimation. In this paper we illustrate the computational advantages of Bayesian estimation using MCMC in several popular latent variable models.
In: Statistica Neerlandica, Band 42, Heft 4, S. 273-295
ISSN: 1467-9574
The similarities between the logistic regression model and the Rasch model (used in psychometric item response theory) are used to derive several methods based on logits that produce parameter estimates for the Rasch model. A result from LeCam and Dzhaparidze is used by which an initial consistent estimate is transformed by one scoring method iteration into an estimate that has the same asymptotic efficiency as the (in this case conditional) maximum likelihood estimate of the item parameters. Indirect evidence about the bias of this CML estimator is produced by studying the (more easily derived) bias of the estimator based on the unweighted logits. Finally, some simple weighted least squares logit‐based estimates are presented, and their performance is assessed. On the whole, the computationally simpler logit‐based estimates give a fairly good approximation to the CML estimates.
In: Statistica Neerlandica: journal of the Netherlands Society for Statistics and Operations Research, Band 58, Heft 2, S. 197-219
ISSN: 1467-9574
For a multilevel model with two levels and only a random intercept, the quality of different estimators of the random intercept is examined. Analytical results are given for the marginal model interpretation where negative estimates of the variance components are allowed for. Except for four or five level‐2 units, the Empirical Bayes Estimator (EBE) has a lower average Bayes risk than the Ordinary Least Squares Estimator (OLSE). The EBEs based on restricted maximum likelihood (REML) estimators of the variance components have a lower Bayes risk than the EBEs based on maximum likelihood (ML) estimators. For the hierarchical model interpretation, where estimates of the variance components are restricted being positive, Monte Carlo simulations were done. In this case the EBE has a lower average Bayes risk than the OLSE, also for four or five level‐2 units. For large numbers of level‐1 (30) or level‐2 units (100), the performances of REML‐based and ML‐based EBEs are comparable. For small numbers of level‐1 (10) and level‐2 units (25), the REML‐based EBEs have a lower Bayes risk than ML‐based EBEs only for high intraclass correlations (0.5).
In: Statistica Neerlandica, Band 44, Heft 4, S. 195-219
ISSN: 1467-9574
Statistical inference for fixed effects, random effects and components of variance in an unbalanced linear model with variance components will be discussed. Variance components will be estimated by Restricted Maximum Likelihood. Iterative procedures for computing the estimates, such as Fisher scoring and the EM‐algorithm, are described.
In: Wildlife research, Band 25, Heft 6, S. 643
ISSN: 1448-5494, 1035-3712
The Allen activity index, originally developed for monitoring dingo
populations, is statistically described as a mixed linear model, from which a
variance formula for the index is derived. The resulting formula requires
input of variance component estimates, the estimation of which is accomplished
using restricted maximum-likelihood estimation. An example is used to
demonstrate the calculation of the variance components and their use in the
variance formula. Application of the variance formula substantially enhances
the quantitative practicality of this useful index of wildlife populations.