On bartlett and bartlett-type corrections francisco cribari-neto

This paper reviews the literature on Bartlett and Bartlett-type corrections. It focuses on the corrections to the likelihood ratio, score and Wald test statistics. Three different Bartlett-type corrections which are equivalent to order n-1, n being the sample size, are compared through simulation. One of the forms displayed superior behavior both in terms of size and power. We also use Monte Carlo simulation to examine the effect of independent variables and the impact of the number of nuisance parameters on the finite-sample behavior of some asymptotic econometric criteria in regression models.     

On the corrections to information matrix tests

This paper addresses the issue of designing finite-sample corrections to information matrix tests. We review a Cornish-Fisher correction that has been propowed elsewhere and propose an alternative, Bartlett-type correction. Simulation results for skewness, excess kurtosis, normality and heteroskedasticity tests are given.     

Higher-order Bartlett-type adjustment

This paper deals with Bartlett-type adjustment which makes all the terms up to order n^−k in the asymptotic expansion vanish, where k is an integer k ⩾ 1 and n depends on the sample size. Extending Cordeiro and Ferrari (1991, Biometrika, 78, 573–582) for the case of k = 1, we derive a general formula of the kth-order Bartlett-type adjustment for the test statistic whose kth-order asymptotic expansion of the distribution is given by a finite linear combination of chi-squared distribution with suitable degrees of freedom. Two examples of the second-order Bartlett-type adjustment are given. We also elucidate the connection between Bartlett-type adjustment and Cornish-Fisher expansion.     

Bartlett-type adjustments for empirical discrepancy test statistics

This paper derives two Bartlett-type adjustments that can be used to obtain higher-order improvements to the distribution of the class of empirical discrepancy test statistics recently introduced by Corcoran (Biometrika 85 (1998) 967) as a generalisation of Owen's (Biometrika 36 (1988) 237) empirical likelihood. The corrections are illustrated in the context of the so-called Cressie–Read goodness-of-fit statistic (Biometrika 85 (1998) 535), and their effectiveness in finite samples is evaluated using simulations.     

New results on the likelihood ratio and score tests for the von Mises distribution

In this paper, we derive Bartlett and Bartlett-type corrections [G.M. Cordeiro and S.L.P. Ferrari 1991, A modified score test statistic having chi-squared distribution to order n ^?1 , Biometrika 78 (1991), pp. 573–582] to improve the likelihood ratio and Rao's score statistics for testing the mean parameter and the concentration parameter in the von Mises distribution. Simple formulae are suggested for the corrections valid for small and large values of the concentration parameter that do not depend on the modified Bessel functions and can be useful in practical applications.     

Exploring the Impact of Multivariate GARCH Innovations on Hypothesis Testing for Cointegrating Vectors

This article investigates the impact of multivariate generalized autoregressive conditional heteroskedastic (GARCH) errors on hypothesis testing for cointegrating vectors. The study reviews a cointegrated vector autoregressive model incorporating multivariate GARCH innovations and a regularity condition required for valid asymptotic inferences. Monte Carlo experiments are then conducted on a test statistic for a hypothesis on the cointegrating vectors. The experiments demonstrate that the regularity condition plays a critical role in rendering the hypothesis testing operational. It is also shown that Bartlett-type correction and wild bootstrap are useful in improving the small-sample size and power performance of the test statistic of interest.     

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.     

Performance of a bartlett-type modification for the deviance

Cordeiro (1983) has derived the expected value of the deviance for generalized linear models correct to terms of order n ^-1 being the sample size. Then a Bartlett-type factor is available for correcting the first moment of the deviance and for fitting its distribution. If the model is correct, the deviance is not, in general, distributed as chi-squared even asymptotically and very little is known about the adequacy of the X ² approximation. This paper through simulation studies examines the behaviour of the deviance and a Bartlett adjusted deviance for testing the goodness-of-fit of a generalized linear model. The practical use of such adjustment is illustrated for some gamma and Poisson models. It is suggested that the null distribution of the adjusted deviance is better approximated by chi-square than the distribution of the deviance.     

Size distributions reconsidered

We consider tests of the hypothesis that the tail of size distributions decays faster than any power function. These are based on a single parameter that emerges from the Fisher–Tippett limit theorem, and discriminate between leading laws considered in the literature without requiring fully parametric models/specifications. We study the proposed tests taking into account the higher order regular variation of the size distribution that can lead to catastrophic distortions. The theoretical bias corrections realign successfully nominal and empirical test behavior, and inform a sensitivity analysis for practical work. The methods are used in an examination of the size distribution of cities and firms.     

Small-Sample Corrections for Score Tests in Birnbaum–Saunders Regressions

In this article, we deal with the issue of performing accurate small-sample inference in the Birnbaum–Saunders regression model, which can be useful for modeling lifetime or reliability data. We derive a Bartlett-type correction for the score test and numerically compare the corrected test with the usual score test and some other competitors.     

Goodness-of-fit tests for binomial AR(1) processes

The binomial AR(1) model describes a nonlinear process with a first-order autoregressive (AR(1)) structure and a binomial marginal distribution. To develop goodness-of-fit tests for the binomial AR(1) model, we investigate the observed marginal distribution of the binomial AR(1) process, and we tackle its autocorrelation structure. Motivated by the family of power-divergence statistics for handling discrete multivariate data, we derive the asymptotic distribution of certain categorized power-divergence statistics for the case of a binomial AR(1) process. Then we consider Bartlett's formula, which is widely used in time series analysis to provide estimates of the asymptotic covariance between sample autocorrelations, but which is not applicable when the underlying process is nonlinear. Hence, we derive a novel Bartlett-type formula for the asymptotic distribution of the sample autocorrelations of a binomial AR(1) process, which is then applied to develop tests concerning the autocorrelation structure. Simulation studies are carried out to evaluate the size and power of the proposed tests under diverse alternative process models. Several real examples are used to illustrate our methods and findings.     

Bias correction and improved residuals for non-exponential family nonlinear models

In this paper we derive general formulae for the biases to order n ^?1 of the parameter estimates in a general class of nonlinear regression models, where n is the sample size. The formulae are related to those of Cordeiro and McCullagh (1991) and Paula (1992) and may be viewed as extensions of their results, Correction factors are derived for the score and deviance component residuals in these models. The practical use of such corrections is illustrated for the log-gamma model.     

An Asymptotic Analysis of the Logrank Test

Asymptotic expansions for the null distribution of the logrank statistic and its distribution under local proportional hazards alternatives are developed in the case of iid observations. The results, which are derived from the work of Gu (1992) and Taniguchi (1992), are easy to interpret, and provide some theoretical justification for many behavioral characteristics of the logrank test that have been previously observed in simulation studies. We focus primarily upon (i) the inadequacy of the usual normal approximation under treatment group imbalance; and, (ii) the effects of treatment group imbalance on power and sample size calculations. A simple transformation of the logrank statistic is also derived based on results in Konishi (1991) and is found to substantially improve the standard normal approximation to its distribution under the null hypothesis of no survival difference when there is treatment group imbalance. This revised version was published online in July 2006 with corrections to the Cover Date.     

Corrected likelihood-ratio tests in logistic regression using small-sample data

Likelihood-ratio tests (LRTs) are often used for inferences on one or more logistic regression coefficients. Conventionally, for given parameters of interest, the nuisance parameters of the likelihood function are replaced by their maximum likelihood estimates. The new function created is called the profile likelihood function, and is used for inference from LRT. In small samples, LRT based on the profile likelihood does not follow χ² distribution. Several corrections have been proposed to improve LRT when used with small-sample data. Additionally, complete or quasi-complete separation is a common geometric feature for small-sample binary data. In this article, for small-sample binary data, we have derived explicitly the correction factors of LRT for models with and without separation, and proposed an algorithm to construct confidence intervals. We have investigated the performances of different LRT corrections, and the corresponding confidence intervals through simulations. Based on the simulation results, we propose an empirical rule of thumb on the use of these methods. Our simulation findings are also supported by real-world data.     

Rényi statistics for testing hypotheses in mixed linear regression models

Rényi divergences are used to propose some statistics for testing general hypotheses in mixed linear regression models. The asymptotic distribution of these tests statistics, of the Kullback–Leibler and of the likelihood ratio statistics are provided, assuming that the sample size and the number of levels of the random factors tend to infinity. A simulation study is carried out to analyze and compare the behavior of the proposed tests when the sample size and number of levels are small.     

New Lilliefors and Srinivasan tables with applications

Two approaches to the problem of goodness-of-fit with nuisance parameters are presented in this paper, both based on modifications of the Kolmogorov-Smirnov statistics. Improved tables of critical values originally computed by Lilliefors and Srinivasan are presented in the normal and exponential cases. Also given are tables for the uniform case, normal with known mean and normal with known variance. All tables were computed using Monte Carlo simulation with sample size n = 20000.     

Optimal Bootstrap Sample Size in Construction of Percentile Confidence Bounds

In traditional bootstrap applications the size of a bootstrap sample equals the parent sample size, n say. Recent studies have shown that using a bootstrap sample size different from n may sometimes provide a more satisfactory solution. In this paper we apply the latter approach to correct for coverage error in construction of bootstrap confidence bounds. We show that the coverage error of a bootstrap percentile method confidence bound, which is of order O ( n ^−2/2) typically, can be reduced to O ( n ⁻¹) by use of an optimal bootstrap sample size. A simulation study is conducted to illustrate our findings, which also suggest that the new method yields intervals of shorter length and greater stability compared to competitors of similar coverage accuracy.