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Log-likelihood Ratio (log-likelihood + ratio)

Distribution by Scientific Domains
Mathematics and Statistics75%

Selected Abstracts

Bayesian Hypothesis Testing: a Reference Approach

INTERNATIONAL STATISTICAL REVIEW, Issue 3 2002
José M. Bernardo
Summary For any probability model M={p(x|,, ,), ,,,, ,,,} assumed to describe the probabilistic behaviour of data x,X, it is argued that testing whether or not the available data are compatible with the hypothesis H0={,=,0} is best considered as a formal decision problem on whether to use (a0), or not to use (a0), the simpler probability model (or null model) M0={p(x|,0, ,), ,,,}, where the loss difference L(a0, ,, ,) ,L(a0, ,, ,) is proportional to the amount of information ,(,0, ,), which would be lost if the simplified model M0 were used as a proxy for the assumed model M. For any prior distribution ,(,, ,), the appropriate normative solution is obtained by rejecting the null model M0 whenever the corresponding posterior expectation ,,,(,0, ,, ,),(,, ,|x)d,d, is sufficiently large. Specification of a subjective prior is always difficult, and often polemical, in scientific communication. Information theory may be used to specify a prior, the reference prior, which only depends on the assumed model M, and mathematically describes a situation where no prior information is available about the quantity of interest. The reference posterior expectation, d(,0, x) =,,,(,|x)d,, of the amount of information ,(,0, ,, ,) which could be lost if the null model were used, provides an attractive nonnegative test function, the intrinsic statistic, which is invariant under reparametrization. The intrinsic statistic d(,0, x) is measured in units of information, and it is easily calibrated (for any sample size and any dimensionality) in terms of some average log-likelihood ratios. The corresponding Bayes decision rule, the Bayesian reference criterion (BRC), indicates that the null model M0 should only be rejected if the posterior expected loss of information from using the simplified model M0 is too large or, equivalently, if the associated expected average log-likelihood ratio is large enough. The BRC criterion provides a general reference Bayesian solution to hypothesis testing which does not assume a probability mass concentrated on M0 and, hence, it is immune to Lindley's paradox. The theory is illustrated within the context of multivariate normal data, where it is shown to avoid Rao's paradox on the inconsistency between univariate and multivariate frequentist hypothesis testing. Résumé Pour un modèle probabiliste M={p(x|,, ,) ,,,, ,,,} censé décrire le comportement probabiliste de données x,X, nous soutenons que tester si les données sont compatibles avec une hypothèse H0={,=,0 doit être considéré comme un problème décisionnel concernant l'usage du modèle M0={p(x|,0, ,) ,,,}, avec une fonction de coût qui mesure la quantité d'information qui peut être perdue si le modèle simplifiéM0 est utilisé comme approximation du véritable modèle M. Le coût moyen, calculé par rapport à une loi a priori de référence idoine fournit une statistique de test pertinente, la statistique intrinsèque d(,0, x), invariante par reparamétrisation. La statistique intrinsèque d(,0, x) est mesurée en unités d'information, et sa calibrage, qui est independante de la taille de léchantillon et de la dimension du paramètre, ne dépend pas de sa distribution à l'échantillonage. La règle de Bayes correspondante, le critère de Bayes de référence (BRC), indique que H0 doit seulement êetre rejeté si le coût a posteriori moyen de la perte d'information à utiliser le modèle simplifiéM0 est trop grande. Le critère BRC fournit une solution bayésienne générale et objective pour les tests d'hypothèses précises qui ne réclame pas une masse de Dirac concentrée sur M0. Par conséquent, elle échappe au paradoxe de Lindley. Cette théorie est illustrée dans le contexte de variables normales multivariées, et on montre qu'elle évite le paradoxe de Rao sur l'inconsistence existant entre tests univariés et multivariés. [source]

Comparing density forecast models,

JOURNAL OF FORECASTING, Issue 3 2007
Yong Bao
Abstract In this paper we discuss how to compare various (possibly misspecified) density forecast models using the Kullback,Leibler information criterion (KLIC) of a candidate density forecast model with respect to the true density. The KLIC differential between a pair of competing models is the (predictive) log-likelihood ratio (LR) between the two models. Even though the true density is unknown, using the LR statistic amounts to comparing models with the KLIC as a loss function and thus enables us to assess which density forecast model can approximate the true density more closely. We also discuss how this KLIC is related to the KLIC based on the probability integral transform (PIT) in the framework of Diebold et al. (1998). While they are asymptotically equivalent, the PIT-based KLIC is best suited for evaluating the adequacy of each density forecast model and the original KLIC is best suited for comparing competing models. In an empirical study with the S&P500 and NASDAQ daily return series, we find strong evidence for rejecting the normal-GARCH benchmark model, in favor of the models that can capture skewness in the conditional distribution and asymmetry and long memory in the conditional variance.,,Copyright © 2007 John Wiley & Sons, Ltd. [source]

Empirical likelihood for linear regression models under imputation for missing responses

THE CANADIAN JOURNAL OF STATISTICS, Issue 4 2001
Qihua Wang
Abstract The authors study the empirical likelihood method for linear regression models. They show that when missing responses are imputed using least squares predictors, the empirical log-likelihood ratio is asymptotically a weighted sum of chi-square variables with unknown weights. They obtain an adjusted empirical log-likelihood ratio which is asymptotically standard chi-square and hence can be used to construct confidence regions. They also obtain a bootstrap empirical log-likelihood ratio and use its distribution to approximate that of the empirical log-likelihood ratio. A simulation study indicates that the proposed methods are comparable in terms of coverage probabilities and average lengths of confidence intervals, and perform better than a normal approximation based method. [source]

Bayesian Hypothesis Testing: a Reference Approach

INTERNATIONAL STATISTICAL REVIEW, Issue 3 2002
José M. Bernardo
Summary For any probability model M={p(x|,, ,), ,,,, ,,,} assumed to describe the probabilistic behaviour of data x,X, it is argued that testing whether or not the available data are compatible with the hypothesis H0={,=,0} is best considered as a formal decision problem on whether to use (a0), or not to use (a0), the simpler probability model (or null model) M0={p(x|,0, ,), ,,,}, where the loss difference L(a0, ,, ,) ,L(a0, ,, ,) is proportional to the amount of information ,(,0, ,), which would be lost if the simplified model M0 were used as a proxy for the assumed model M. For any prior distribution ,(,, ,), the appropriate normative solution is obtained by rejecting the null model M0 whenever the corresponding posterior expectation ,,,(,0, ,, ,),(,, ,|x)d,d, is sufficiently large. Specification of a subjective prior is always difficult, and often polemical, in scientific communication. Information theory may be used to specify a prior, the reference prior, which only depends on the assumed model M, and mathematically describes a situation where no prior information is available about the quantity of interest. The reference posterior expectation, d(,0, x) =,,,(,|x)d,, of the amount of information ,(,0, ,, ,) which could be lost if the null model were used, provides an attractive nonnegative test function, the intrinsic statistic, which is invariant under reparametrization. The intrinsic statistic d(,0, x) is measured in units of information, and it is easily calibrated (for any sample size and any dimensionality) in terms of some average log-likelihood ratios. The corresponding Bayes decision rule, the Bayesian reference criterion (BRC), indicates that the null model M0 should only be rejected if the posterior expected loss of information from using the simplified model M0 is too large or, equivalently, if the associated expected average log-likelihood ratio is large enough. The BRC criterion provides a general reference Bayesian solution to hypothesis testing which does not assume a probability mass concentrated on M0 and, hence, it is immune to Lindley's paradox. The theory is illustrated within the context of multivariate normal data, where it is shown to avoid Rao's paradox on the inconsistency between univariate and multivariate frequentist hypothesis testing. Résumé Pour un modèle probabiliste M={p(x|,, ,) ,,,, ,,,} censé décrire le comportement probabiliste de données x,X, nous soutenons que tester si les données sont compatibles avec une hypothèse H0={,=,0 doit être considéré comme un problème décisionnel concernant l'usage du modèle M0={p(x|,0, ,) ,,,}, avec une fonction de coût qui mesure la quantité d'information qui peut être perdue si le modèle simplifiéM0 est utilisé comme approximation du véritable modèle M. Le coût moyen, calculé par rapport à une loi a priori de référence idoine fournit une statistique de test pertinente, la statistique intrinsèque d(,0, x), invariante par reparamétrisation. La statistique intrinsèque d(,0, x) est mesurée en unités d'information, et sa calibrage, qui est independante de la taille de léchantillon et de la dimension du paramètre, ne dépend pas de sa distribution à l'échantillonage. La règle de Bayes correspondante, le critère de Bayes de référence (BRC), indique que H0 doit seulement êetre rejeté si le coût a posteriori moyen de la perte d'information à utiliser le modèle simplifiéM0 est trop grande. Le critère BRC fournit une solution bayésienne générale et objective pour les tests d'hypothèses précises qui ne réclame pas une masse de Dirac concentrée sur M0. Par conséquent, elle échappe au paradoxe de Lindley. Cette théorie est illustrée dans le contexte de variables normales multivariées, et on montre qu'elle évite le paradoxe de Rao sur l'inconsistence existant entre tests univariés et multivariés. [source]