Suchergebnisse

This paper deals with the issue of testing hypotheses in symmetric and log‐symmetric linear regression models in small and moderate‐sized samples. We focus on four tests, namely, the Wald, likelihood ratio, score, and gradient tests. These tests rely on asymptotic results and are unreliable when the sample size is not large enough to guarantee a good agreement between the exact distribution of the test statistic and the corresponding chi‐squared asymptotic distribution. Bartlett and Bartlett‐type corrections typically attenuate the size distortion of the tests. These corrections are available in the literature for the likelihood ratio and score tests in symmetric linear regression models. Here, we derive a Bartlett‐type correction for the gradient test. We show that the corrections are also valid for the log‐symmetric linear regression models. We numerically compare the various tests and bootstrapped tests, through simulations. Our results suggest that the corrected and bootstrapped tests exhibit type I probability error closer to the chosen nominal level with virtually no power loss. The analytically corrected tests as well as the bootstrapped tests, including the Bartlett‐corrected gradient test derived in this paper, perform with the advantage of not requiring computationally intensive calculations. We present a real data application to illustrate the usefulness of the modified tests.

We consider the issue of performing residual and local influence analyses in beta regression models with varying dispersion, which are useful for modelling random variables that assume values in the standard unit interval. In such models, both the mean and the dispersion depend upon independent variables. We derive the appropriate matrices for assessing local influence on the parameter estimates under different perturbation schemes. An application using real data is presented and discussed.

We develop three corrected score tests for generalized linear models with dispersion covariates, thus generalizing the results ofCordeiro,FerrariandPaula(1993)andCribari‐NetoandFerrari(1995). We present, in matrix notation, general formulae for the coefficients which define the corrected statistics. The formulae only require simple operations on matrices and can be used to obtain analytically closed‐form corrections for score test statistics in a variety of special generalized linear models with dispersion covariates. They also have advantages for numerical purposes since our formulae are readily computable using a language supporting numerical linear algebra. Two examples, namely, iid sampling without covariates on the mean or dispersion parameter oand one‐way classification models, are given. We also present some simulations where the three corrected tests perform better than the usual score test, the likelihood ratio test and its Bartlett corrected version. Finally, we present a numerical example for a data set discussed bySimonoffandTsai(1994).