Likelihood Ratio Type Statistic Research Articles

Change-point analysis with bathtub shape for the exponential distribution

ABSTRACTLikelihood ratio type test statistic and Schwarz information criterion statistics are proposed for detecting possible bathtub-shaped changes in the parameter in a sequence of exponential distributions. The asymptotic distribution of likelihood ratio type statistic under the null hypothesis and the testing procedure based on Schwarz information criterion are derived. Numerical critical values and powers of two methods are tabulated for certain selected values of the parameters. The tests are applied to detect the change points for the predator data and Stanford heart transplant data.

Inference from PseudoLikelihoods with Plug‐In Estimates

SummaryEffective 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.

Likelihood Ratio Type Statistics for Repeated Measures Designs with Heterogeneous Covariance Matrices

Because of their simplicity, Wald statistics are typically used in complex experimental designs. Likelihood ratio statistics in factorial designs are more flexible than Wald statistics in the sense of adapting to non-saturated designs by fitting only as many parameters as the model calls for. This leads to a significant gain in power. Here we propose likelihood ratio type statistics for testing hypotheses in repeated measures designs with heterogeneous covariance matrices, and derive their asymptotic distribution in one general theorem that does not require normality or even continuity of the responses. Simulation studies demonstrate their advantages over the Wald statistics.

A powerful test for comparing multiple regression functions

In this article, we address the important problem of comparison of two or more population regression functions. Recently, Pardo-Fernández, Van Keilegom and González-Manteiga [2007, ‘Testing for Equality of k Regression Curves’, Statistica Sinica, 17, 1115–1137] developed test statistics for simple nonparametric regression models: Y ij =θ j (Z ij )+σ j (Z ij )ε ij , based on empirical distributions of the errors in each population j=1, …, J. In this article, we propose a test for equality of the θ j (·) based on the concept of generalised likelihood ratio type statistics. We also generalise our test for other nonparametric regression set-ups, for example, nonparametric logistic regression, where the log-likelihood for population j is any general smooth function ℒ{Y j , θ j (Z j )}. We describe a resampling procedure to obtain the critical values of the test. In addition, we present a simulation study to evaluate the performance of the proposed test and compare our results to those in Pardo-Fernández et al. [2007, ‘Testing for Equality of k Regression Curves’, Statistica Sinica, 17, 1115–1137].

Inference for a skew extension of the Grubbs model

In this paper, we discuss inferential aspects for the Grubbs model when the unknown quantity x (latent response) follows a skew-normal distribution, extending early results given in Arellano-Valle et al. (J Multivar Anal 96:265–281, 2005b). Maximum likelihood parameter estimates are computed via the EM-algorithm. Wald and likelihood ratio type statistics are used for hypothesis testing and we explain the apparent failure of the Wald statistics in detecting skewness via the profile likelihood function. The results and methods developed in this paper are illustrated with a numerical example.

Estimating and Testing Structural Changes in Multivariate Regressions

This paper considers issues related to estimation, inference, and computation with multiple structural changes that occur at unknown dates in a system of equations. Changes can occur in the regression coefficients and/or the covariance matrix of the errors. We also allow arbitrary restrictions on these parameters, which permits the analysis of partial structural change models, common breaks that occur in all equations, breaks that occur in a subset of equations, and so forth. The method of estimation is quasi-maximum likelihood based on Normal errors. The limiting distributions are obtained under more general assumptions than previous studies. For testing, we propose likelihood ratio type statistics to test the null hypothesis of no structural change and to select the number of changes. Structural change tests with restrictions on the parameters can be constructed to achieve higher power when prior information is present. For computation, an algorithm for an efficient procedure is proposed to construct the estimates and test statistics. We also introduce a novel locally ordered breaks model, which allows the breaks in different equations to be related yet not occurring at the same dates.

Tests for an Epidemic Change in a Sequence of Exponentially Distributed Random Variables

AbstractConsider a sequence of independent exponential random variables that is susceptible to a change in the means. We would like to test whether the means have been subjected to an epidemic change after an unknown point, for an unknown duration in the sequence. The likelihood ratio statistic and a likelihood ratio type statistic are derived. The distribution theories and related properties of the test statistics are discussed. Percentage points and powers of the tests are tabulated for selected values of the parameters. The powers of these two tests are then compared to the two statistics proposed by Aly and Bouzar. The tests are applied to find epidemic changes in the set of Stanford heart transplant data and air traffic arrival data.

Criterion-based inference for GMM in autoregressive panel data models

In this paper we examine the properties of a simple criterion-based, likelihood ratio type test of parameter restrictions for standard GMM estimators in autoregressive panel data models. A comparison is made with recent test proposals based on the continuously-updated GMM criterion [Journal of Business & Economic Statistics 14 (1996) 262] or exponential tilting parameters [Econometrica 66 (1998) 333]. The likelihood ratio type statistic is computed simply as the difference between the standard GMM tests of overidentifying restrictions in the restricted and unrestricted models. In Monte Carlo simulations we find this test has similar properties to the two criterion-based alternatives, whilst being much simpler to compute. All three criterion-based tests outperform conventional Wald tests in this context.

Change points with linear trend for the exponential distribution

A sequence of independent exponential random variables is hypothesized to have equal means, and we would like to test whether the means have been subjected to a linear trend change at an unknown point, for an unknown duration in the sequence. Two likelihood ratio-type statistics and Rao's efficient score statistics are derived. The distribution theory and related properties of the test statistics are discussed. Numerical critical points and powers of the test statistics are tabulated for certain selected values of the parameters. The tests are then applied to find linear trend changes in the set of Stanford heart transplant data and air traffic arrival data.

Criterion-Based Inference for GMM in Autoregressive Panel Data Models

In this paper we examine the properties of a simple criterion-based, likelihood ratio type test of parameter restrictions for standard GMM estimators in autoregressive panel data models. A comparison is made with recent test proposals based on the continuously-updated GMM criterion (Hansen, Heaton and Yaron, 1996) or exponential tilting parameters (Imbens, Spady and Johnson, 1998). The likelihood ratio type statistic is computed simply as the difference between the standard GMM tests of overidentifying restrictions in the restricted and unrestricted models. In Monte Carlo simulations we find this test has similar properties to the two criterion-based alternatives, whilst being much simpler to compute. All three criterion-based tests outperform conventional Wald tests in this context.