Showing papers on "Likelihood principle published in 1988"

Empirical likelihood ratio confidence intervals for a single functional

Abstract: SUMMARY The empirical distribution function based on a sample is well known to be the maximum likelihood estimate of the distribution from which the sample was taken. In this paper the likelihood function for distributions is used to define a likelihood ratio function for distributions. It is shown that this empirical likelihood ratio function can be used to construct confidence intervals for the sample mean, for a class of M-estimates that includes quantiles, and for differentiable statistical functionals. The results are nonparametric extensions of Wilks's (1938) theorem for parametric likelihood ratios. The intervals are illustrated on some real data and compared in a simulation to some bootstrap confidence intervals and to intervals based on Student's t statistic. A hybrid method that uses the bootstrap to determine critical values of the likelihood ratio is introduced.

Selection Models and the File Drawer Problem

Abstract: Meta-analysis consists of quantitative methods for combining evidence from different studies about a particular issue A frequent criticism of meta-analysis is that it may be based on a biased sample of all studies that were done In this paper, we use selection models, or weighted distributions, to deal with one source of bias, namely, the failure to report studies that do not yield statistically significant results We apply selection models to two approaches that have been suggested for correcting the bias The fail-safe sample size approach calculates the minimum number of unpublished studies showing nonsignificant results that must have been carried out in order to overturn the conclusion reached from the published studies The maximum likelihood approach uses a weighted distribution to model the selection bias in the generation of the data and estimates various parameters of interest We suggest the use of families of weight functions to model plausible biasing mechanisms to study the sensitivity of inferences about effect sizes By using an example, we show that the maximum likelihood approach has several advantages over the fail-safe sample size approach

On the level-error after Bartlett adjustment of the likelihood ratio statistic

Abstract: SUMMARY It is well known that Bartlett adjustment reduces level-error of the likelihood ratio statistic from order n-l to order n3/2. In the present note we show that level-error of the adjusted statistic is actually order n2.

Consistency of Maximum Likelihood and Pseudo-Likelihood Estimators for Gibbs Distributions

Abstract: We prove that the Maximum Likelihood and Pseudo-likelihood estimators for the parameters of Gibbs distributions (equivalently Markov Random Fields) over ℤd, d≥l, are consistent even at points of “first” or “higher-order” phase transitions. The distributions are parametrized by points in a finite-dimensional Euclidean space ℝm, m≥l, and the single spin state space is either a finite set or a compact metric space. Also, the underlying interactions need not be of finite range.

Item Parameter Estimation Via Marginal Maximum Likelihood and an EM Algorithm: A Didactic

Abstract: The Bock and Aitkin (1981) Marginal Maximum Likelihood/EM approach to item parameter estimation is an alternative to the classical joint maximum likelihood procedure of item response theory. Unfortunately, the complexity of the underlying mathematics and the terse nature of the existing literature has made understanding of the approach difficult. To make the approach accessible to a wider audience, the present didactic paper provides the essential mathematical details of a marginal maximum likelihood/EM solution and shows how it can be used to obtain consistent item parameter estimates. For pedagogical purposes, a short BASIC computer program is used to illustrate the underlying simplicity of the method. Since 1968 the primary approach to item parameter estimation has been the joint maximum likelihood estimation (JMLE) paradigm attributed to Dr. Alan Birnbaum. A distinguishing characteristic of this paradigm is that examinee abilities are unknown, and hence must be estimated along with the item parameters. This approach has been implemented in the widely used LOGIST computer program (Wingersky, Barton, & Lord, 1982), as well as other programs. Under the Birnbaum paradigm, the item parameters are the "structural" parameters, which are fixed in number by the size of the test. The ability parameters of the examinees are the "incidental" parameters, the number of which depends upon the sample size. From a theoretical point of view, this paradigm has an inherent problem first recognized in another context by Neyman and Scott (1948) (see also Little & Rubin, 1983). They showed that when structural parameters are estimated simultaneously with the incidental parameters, the maximum likelihood estimates of the former need not be consistent as sample size increases. If sufficient statistics are available for the incidental parameters, a conditional maximum likelihood estimation procedure (Anderson, 1972) can be established for the consistent estimation of the structural parameters. In item The authors are indebted to an associate editor and several anonymous reviewers for suggestions that were of great value in structuring the paper.

The Rasch Model and Multistage Testing

Abstract: This paper concerns the problem of estimating the item parameters of latent trait models in a multistage testing design. It is shown that using the Rasch model and conditional maximum likelihood estimates does not lead to solvable estimation equations. It is also shown that marginal maximum likelihood estimation, which assumes a sample of subjects from a population with a specified distribution of ability, will lead to solvable estimation equations, both in the Rasch model and in the Birnbaum model. As has been pointed out by Lord (1980) two-stage and multistage testing procedures bear the possibility of improving the measurement of a subject's ability, if the number of items that can be administered is fixed, and we have a large pool of items at our disposal. The same holds for calibrating an item pool: if the difficulty of the test items is matched with the ability level of the different groups of subjects, estimation errors of the item parameters can be reduced. In order to get an impression of the ability level of the subjects, a so-called routing test is administered first. In two-stage testing the performance on the routing test governs the choice of the next test to be administered: the better the performance, the more difficult the follow-up test. In multistage testing this procedure is generalized in the sense that more tests are subsequently administered and that the next test to be administered depends on the performance on the previous tests. Tailored testing can be viewed as a limiting case of this principle, where each subtest consists of one item only. In a so-called incomplete design with common items, that is, a design where different samples of subjects take different, though overlapping, subtests, two possibilities are open to calibrate the complete set of item parameters on one scale. The first possibility starts with carrying out estimation in each sample separately, to arrive at disjunct scales. These scales are then combined to a common scale by transforming the estimates of the item parameters. The transformation, which must belong to the family of admissible transformations of the item response model under consideration, is such that the difference in the estimates of the parameters of

An operative extension of the likelihood ratio test from fuzzy data

Abstract: In the present paper we are going to extend the likelihood ratio test to the case in which the available experimental information involves fuzzy imprecision (more precisely, the observable events associated with the random experiment concerning the test may be characterized as fuzzy subsets of the sample space, as intended by Zadeh, 1965). In addition, we will approximate the immediate intractable extension, which is based on Zadeh’s probabilistic definition, by using the minimum inaccuracy principle of estimation from fuzzy data, that has been introduced in previous papers as an operative extension of the maximum likelihood method.

Asymptotic expansion of the nonnull distribution of likelihood ratio statistic for testing multisample sphericity

Abstract: Asymptotic expansion of the nonnull distribution of the likelihood ratio statistic for testing muitisample sphericity in q multinormal populations is derived for the alternatives close to the null hypothesis

Recursive maximum likelihood identification of a non-linear output-affine model

Abstract: On-line identification of the parameters in a non-linear output-affine model is considered. The recursive maximum likelihood estimator, which was originally derived for linear systems, is extended to work for non-linear systems which can be described by a stochastic form of Sontag's output-affine model. Some numerical implementation aspects are discussed, a brief convergence analysis is given and a computer simulation study is included.

The Local Power of the Tests of Overidentifying Restrictions

Abstract: The local power function of the size-corrected likelihood ratio, linearized likelihood ratio, and Lagrange multiplier tests of overidentifying restrictions on a structural equation is the same to the order 1/T. Moreover, this local power function doe s not depend on the k-class estimator used in the calculation of the test statistic. When the author does not use size-corrected tests, a degrees of freedom corrected likelihood ratio test seems to have the best size and power properties. Finally, the implicit null hypothesis of these tests indicates that they can be interpreted as testing the validity of the structural specification of the equation against any other identified structural equation that encompasses the original e quation. Copyright 1988 by Economics Department of the University of Pennsylvania and the Osaka University Institute of Social and Economic Research Association.

Maximum likelihood estimates and likelihood ratio criteria for location and scale parameters of the multivariate l 1 -norm symmetric distributions

Abstract: In this paper we introduce three families of multivariate and matrixl 1-norm symmetric distributions with location and scale parameters and discuss their maximum likelihood estimates and likelihood ratio criteria. It is shown that under certain condition sthey have the same form as those for independent exponential variates.

Maximum likelihood estimation using differences in an autoregressive-1 process

Abstract: In a first-order autoregressive (AR-1) process with unknown mean, conventional maximum likelihood analysis requires joint estimation of the mean and AR coefficient. Differencing the series removes the mean, and for short series it should be more efficient to estimate the AR coefficient from the likelihood function of the differences. The exact likelihood function of the differences is given. A computer simulation study compares the behavior of the estimator obtained by maximizing the likelihood of the differences with that of the conventional maximum likelihood estimator. A root-mean-squared-error criterion shows superiority of the estimator based on differences for series of 50 time points or less.

Applications of the Generalized Likelihood to Time Series Analysis

Abstract: : Model-critical procedures provide a means to scrutinize as assumed parametric statistical model by varying the way the data are processed for repeated fits to the model. The criticism of the data is accomplished using the generalized likelihood function for the assumed probability density of the data. The degree of criticism is controlled by a user specified constant, c. The model-critical parameter estimates are obtained by maximization of the generalized likelihood function. When c=O, no criticism is performed and maximum likelihood estimates are obtained. Model-critical estimation procedures are presented for univariate and multivariate autoregressive-moving average processes. The procedures use a Kalman filter in evaluating the generalized likelihood function. A model selection criterion, based on the generalized likelihood, is also presented. A statistical test of fit for multivariate Gaussianity is presented; the test compares the model-critical and maximum likelihood estimates of the covariance matrix. Keywords: Statistical data, Tables data.

Comparison of Maximum Likelihood and Pearson Chi-Square Statistics for Assessing Latent Class Models.

Abstract: When latent class parameters are estimated, maximum likelihood and Pearson chi-square statistics can be derived for assessing the fit of the model to the data. This study uses simulated data to compare these two statistics. Data were generated for a 2-class unconstrained model and a 3-class constrained model. Data were also generated under three independent variables. Those variables are sample she and two parameters that define latent class models--conditional response probability and latent class proportion. Parameters were estimated for the generation models and a series of subsumed and subsuming models. Maximum likelihood and Pearson chi-square statistics were derived for each estimation. Distributions of fit statistics were prcduced by 1000 replications. For each distribution of statistics, the overall fit to tht appropriate chi-square distribution is assessed. In addition, the mean, variance, and tail weights are examined. The distributions of the statistics vary according to the parameter values of the models and the type of models estimated. When the estimated model is an accurate reflection of the data, the Pearson statistic is generally distributed as chi-square for both large and small samples, while the maximum likelihood statistic is distributed as chi-square only at the large sample