Inference from PseudoLikelihoods with Plug‐In Estimates

Effective implementation of likelihood inference in models for high‐dimensional data often requires a simplified treatment of nuisance parameters, with these having to be replaced by handy estimates. In addition, the likelihood function may have been simplified by means of a partial specification of the model, as is the case when composite likelihood is used. In such circumstances tests and confidence regions for the parameter of interest may be constructed using Wald type and score type statistics, defined so as to account for nuisance parameter estimation or partial specification of the likelihood. In this paper a general analytical expression for the required asymptotic covariance matrices is derived, and suggestions for obtaining Monte Carlo approximations are presented. The same matrices are involved in a rescaling adjustment of the log likelihood ratio type statistic that we propose. This adjustment restores the usual chi‐squared asymptotic distribution, which is generally invalid after the simplifications considered. The practical implication is that, for a wide variety of likelihoods and nuisance parameter estimates, confidence regions for the parameters of interest are readily computable from the rescaled log likelihood ratio type statistic as well as from the Wald type and score type statistics. Two examples, a measurement error model with full likelihood and a spatial correlation model with pairwise likelihood, illustrate and compare the procedures. Wald type and score type statistics may give rise to confidence regions with unsatisfactory shape in small and moderate samples. In addition to having satisfactory shape, regions based on the rescaled log likelihood ratio type statistic show empirical coverage in reasonable agreement with nominal confidence levels.     

On testing inference in beta regressions

This article deals with testing inference in the class of beta regression models with varying dispersion. We focus on inference in small samples. We perform a numerical analysis in order to evaluate the sizes and powers of different tests. We consider the likelihood ratio test, two adjusted likelihood ratio tests proposed by Ferrari and Pinheiro [Improved likelihood inference in beta regression, J. Stat. Comput. Simul. 81 (2011), pp. 431–443], the score test, the Wald test and bootstrap versions of the likelihood ratio, score and Wald tests. We perform tests on the parameters that index the mean submodel and also on the parameters in the linear predictor of the precision submodel. Overall, the numerical evidence favours the bootstrap tests. It is also shown that the score test is considerably less size-distorted than the likelihood ratio and Wald tests. An application that uses real (not simulated) data is presented and discussed.     

Exact Inference in the Proportional Hazard Model: Possibilities and Limitations

It is suggested that inference under the proportional hazard model can be carried out by programs for exact inference under the logistic regression model. Advantages of such inference is that software is available and that multivariate models can be addressed. The method has been evaluated by means of coverage and power calculations in certain situations. In all situations coverage was above the nominal level, but on the other hand rather conservative. A different type of exact inference is developed under Type II censoring. Inference was then less conservative, however there are limitations with respect to censoring mechanism, multivariate generalizations and software is not available. This method also requires extensive computational power. Performance of large sample Wald, score and likelihood inference was also considered. Large sample methods works remarkably well with small data sets, but inference by score statistics seems to be the best choice. There seems to be some problems with likelihood ratio inference that may originate from how this method works with infinite estimates of the regression parameter. Inference by Wald statistics can be quite conservative with very small data sets.     

Bounded Estimation in the Presence of Nuisance Parameters

The aim of this paper is to extend in a natural fashion the results on the treatment of nuisance parameters from the profile likelihood theory to the field of robust statistics. Similarly to what happens when there are no nuisance parameters, the attempt is to derive a bounded estimating function for a parameter of interest in the presence of nuisance parameters. The proposed method is based on a classical truncation argument of the theory of robustness applied to a generalized profile score function. By means of comparative studies, we show that this robust procedure for inference in the presence of a nuisance parameter can be used successfully in a parametric setting.     

Hypothesis Testing in Functional Comparative Calibration Models

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.     

Likelihood-based tests in zero-inflated power series models

We address the issue of performing testing inference in the class of zero-inflated power series models. These models provide a straightforward way of modelling count data and have been widely used in practical situations. The likelihood ratio, Wald and score statistics provide the basis for testing the parameter of inflation of zeros in this class of models. In this paper, in addition to the well-known test statistics, we also consider the recently proposed gradient statistic. We conduct Monte Carlo simulation experiments to evaluate the finite-sample performance of these tests for testing the parameter of inflation of zeros. The numerical results show that the new gradient test we propose is more reliable in finite samples than the usual likelihood ratio, Wald and score tests. An empirical application to real data is considered for illustrative purposes.     

Empirical and simulated adjustments of composite likelihood ratio statistics

Composite likelihood inference has gained much popularity thanks to its computational manageability and its theoretical properties. Unfortunately, performing composite likelihood ratio tests is inconvenient because of their awkward asymptotic distribution. There are many proposals for adjusting composite likelihood ratio tests in order to recover an asymptotic chi-square distribution, but they all depend on the sensitivity and variability matrices. The same is true for Wald-type and score-type counterparts. In realistic applications, sensitivity and variability matrices usually need to be estimated, but there are no comparisons of the performance of composite likelihood-based statistics in such an instance. A comparison of the accuracy of inference based on the statistics considering two methods typically employed for estimation of sensitivity and variability matrices, namely an empirical method that exploits independent observations, and Monte Carlo simulation, is performed. The results in two examples involving the pairwise likelihood show that a very large number of independent observations should be available in order to obtain accurate coverages using empirical estimation, while limited simulation from the full model provides accurate results regardless of the availability of independent observations. This suggests the latter as a default choice, whenever simulation from the model is possible.     

Asymptotic likelihood inference for nonhomogeneous observations

This paper surveys asymptotic theory of maximum likelihood estimation for not identically distributed, possibly dependent observations. Main results on consistency, asymptotic normality and efficiency are stated within a unified framework. Limiting distributions of the likelihood ratio, Wald and score statistics for composite hypotheses are obtained under the same conditions by a generalization of existing theory. Modifications for maximum likelihood estimation under misspecification, containing the results for correctly specified models, are presented, and extensions to likelihood inference in the presence of nuisance parameters are indicated.     

Asymptotic relative efficiency of wald tests in measurement error models

In this paper, asymptotic relative efficiency (ARE) of Wald tests for the Tweedie class of models with log-linear mean, is considered when the aux¬iliary variable is measured with error. Wald test statistics based on the naive maximum likelihood estimator and on a consistent estimator which is obtained by using Nakarnura's (1990) corrected score function approach are defined. As shown analytically, the Wald statistics based on the naive and corrected score function estimators are asymptotically equivalents in terms of ARE. On the other hand, the asymptotic relative efficiency of the naive and corrected Wald statistic with respect to the Wald statistic based on the true covariate equals to the square of the correlation between the unobserved and the observed co-variate. A small scale numerical Monte Carlo study and an example illustrate the small sample size situation.     

Robust estimation in the normal mixture model

This paper focuses on the inference of the normal mixture model with unequal variances. A feature of the model is flexibility of density shape, but its flexibility causes the unboundedness of the likelihood function and excessive sensitivity of the maximum likelihood estimator to outliers. A modified likelihood approach suggested in Basu et al. [1998, Biometrika 85, 549–559] can overcome these drawbacks. It is shown that the modified likelihood function is bounded above under a mild condition on mixing proportions and the resultant estimator is robust to outliers. A relationship between robustness and efficiency is investigated and an adaptive method for selecting the tuning parameter of the modified likelihood is suggested, based on the robust model selection criterion and the cross-validation. An EM-like algorithm is also constructed. Numerical studies are presented to evaluate the performance. The robust method is applied to single nuleotide polymorphism typing for the purpose of outlier detection and clustering.     

A modified likelihood ratio test for the mean direction in the von mises distribution

The likelihood ratio test (LRT) for the mean direction in the von Mises distribution is modified for possessing a common asymptotic distribution both for large sample size and for large concentration parameter. The test statistic of the modified LRT is compared with the F distribution but not with the chi-square distribution usually employed, Good performances of the modified LRT are shown by analytical studies and Monte Carlo simulation studies, A notable advantage of the test is that it takes part in the unified likelihood inference procedures including both the marginal MLE and the marginal LRT for the concentration parameter.     

Likelihood‐based Inference with Missing Data Under Missing‐at‐Random

Likelihood‐based inference with missing data is challenging because the observed log likelihood is often an (intractable) integration over the missing data distribution, which also depends on the unknown parameter. Approximating the integral by Monte Carlo sampling does not necessarily lead to a valid likelihood over the entire parameter space because the Monte Carlo samples are generated from a distribution with a fixed parameter value. We consider approximating the observed log likelihood based on importance sampling. In the proposed method, the dependency of the integral on the parameter is properly reflected through fractional weights. We discuss constructing a confidence interval using the profile likelihood ratio test. A Newton–Raphson algorithm is employed to find the interval end points. Two limited simulation studies show the advantage of the Wilks inference over the Wald inference in terms of power, parameter space conformity and computational efficiency. A real data example on salamander mating shows that our method also works well with high‐dimensional missing data.     

Likelihood Ratio Tests for Fixed Model Terms using Residual Maximum Likelihood

Likelihood ratio tests for fixed model terms are proposed for the analysis of linear mixed models when using residual maximum likelihood estimation. Bartlett-type adjustments, using an approximate decomposition of the data, are developed for the test statistics. A simulation study is used to compare properties of the test statistics proposed, with or without adjustment, with a Wald test. A proposed test statistic constructed by dropping fixed terms from the full fixed model is shown to give a better approximation to the asymptotic χ²-distribution than the Wald test for small data sets. Bartlett adjustment is shown to improve the χ²-approximation for the proposed tests substantially.     

On the wald,lagrangian multiplier and likelihood ratio tests when the information matrix is singular

Summary Modified formulas for the Wald and Lagrangian multiplier statistics are introduced and considered together with the likelihood ratio statistics for testing a typical null hypothesisH ₀ stated in terms of equality constraints. It is demonstrated, subject to known standard regularity conditions, that each of these statistics and the known Wald statistic has the asymptotic chi-square distribution with degrees of freedom equal to the number of equality constraints specified byH ₀ whether the information matrix is singular or nonsingular. The results of this paper include a generalization of the results of Sively (1959) concerning the equivalence of the Wald, Lagrange multiplier and likelihood ratio tests to the case of singular information matrices.     

Testing overdispersion in the zero-inflated Poisson model

The zero-inflated negative binomial (ZINB) model is used to account for commonly occurring overdispersion detected in data that are initially analyzed under the zero-inflated Poisson (ZIP) model. Tests for overdispersion (Wald test, likelihood ratio test [LRT], and score test) based on ZINB model for use in ZIP regression models have been developed. Due to similarity to the ZINB model, we consider the zero-inflated generalized Poisson (ZIGP) model as an alternate model for overdispersed zero-inflated count data. The score test has an advantage over the LRT and the Wald test in that the score test only requires that the parameter of interest be estimated under the null hypothesis. This paper proposes score tests for overdispersion based on the ZIGP model and illustrates that the derived score statistics are exactly the same as the score statistics under the ZINB model. A simulation study indicates the proposed score statistics are preferred to other tests for higher empirical power. In practice, based on the approximate mean–variance relationship in the data, the ZINB or ZIGP model can be considered, and a formal score test based on asymptotic standard normal distribution can be employed for assessing overdispersion in the ZIP model. We provide an example to illustrate the procedures for data analysis.     

Robustness and maximum likelihood estimation

The usual concept of robustness is called "criterion" or "non-adaptive" robustness to distinguish it from "inference" or "adaptive" robustness. The former term is appled to describe relative insensitivity to changes in the parent distribution, while the latter specifically implies dependence on and hence adaptation to changes in the parent distribution. It is argued that knowledge of, and sensitivity to the parent distribution is an important aspect of inference, and thus the latter concept of robustness is more relevant than the former. This focuses attention on adaptive procedures that use most of the sample information, that is, are efficient. Maximum likelihood has been criticized as depending critically on knowledge of the exact parent distribution, and hence of lacking criterion or non-adaptive robustness. This might have been justified when computational parameter to allow for uncertainly of shape. then the method of maximim likelihood is hsown to possess the more important requirement of being adaptive and efficent, capable of assessing the more relevant creiterion of inference or adaptive robustness.     

Bias reduction of maximum likelihood estimates for a modified skew-normal distribution

ABSTRACT

This paper presents a modified skew-normal (SN) model that contains the normal model as a special case. Unlike the usual SN model, the Fisher information matrix of the proposed model is always non-singular. Despite of this desirable property for the regular asymptotic inference, as with the SN model, in the considered model the divergence of the maximum likelihood estimator (MLE) of the skewness parameter may occur with positive probability in samples with moderate sizes. As a solution to this problem, a modified score function is used for the estimation of the skewness parameter. It is proved that the modified MLE is always finite. The quasi-likelihood approach is considered to build confidence intervals. When the model includes location and scale parameters, the proposed method is combined with the unmodified maximum likelihood estimates of these parameters.     

Which Wald statistic? Choosing a parameterization of the Wald statistic to maximize power in k-sample generalized estimating equations

The Wald statistic is known to vary under reparameterization. This raises the question: which parameterization should be chosen, in order to optimize power of the Wald statistic? We specifically consider k-sample tests of generalized linear models (GLMs) and generalized estimating equations (GEEs) in which the alternative hypothesis contains only two parameters. An example is presented in which such an alternative hypothesis is of interest. Amongst a general class of parameterizations, we find the parameterization that maximizes power via analysis of the non-centrality parameter, and show how the effect on power of reparameterization depends on sampling design and the differences in variance across samples. There is no single parameterization with optimal power across all alternatives. The Wald statistic commonly used under the canonical parameterization is optimal in some instances but it performs very poorly in others. We demonstrate results by example and by simulation, and describe their implications for likelihood ratio statistics and score statistics. We conclude that due to poor power properties, the routine use of score statistics and Wald statistics under the canonical parameterization for GEEs is a questionable practice.     

Odds ratio inference from stratified samples

One of the common problems encountered in applied statistics is that of comparing two proportions from stratified samples. One approach to this problem is via inference on the corresponding odds ratio. In this paper, the various point and interval estimators of and hypothesis testing procedures for a common odds ratio from multiple 2 ×2 tables are reviewed. Based On research to date, the conditional maximum likelihood and Mantel-Haenszel estimators are recommended as the point estimators of choice. Neither confidence intervals nor hypothesis testing metthods have been studied as well as the point estimators, but there is a confidence interval method associated with the Mantel-Haenszel estimator that is a good choice.     

Profile likelihood in directed graphical models from BUGS output

We present a method for using posterior samples produced by the computer program BUGS (Bayesian inference Using Gibbs Sampling) to obtain approximate profile likelihood functions of parameters or functions of parameters in directed graphical models with incomplete data. The method can also be used to approximate integrated likelihood functions. It is easily implemented and it performs a good approximation. The profile likelihood represents an aspect of the parameter uncertainty which does not depend on the specification of prior distributions, and it can be used as a worthwhile supplement to BUGS that enable us to do both Bayesian and likelihood based analyses in directed graphical models.