We address the issue of performing testing inference in generalized linear models when the sample size is small. This class of models provides a straightforward way of modeling normal and non-normal data and has been widely used in several practical
situations. The likelihood ratio, Wald and score statistics, and the recently proposed gradient statistic provide the basis for testing inference on the parameters in these models. We focus on the small-sample case, where the reference chi-squared distribution gives a poor approximation to the true null distribution of these test statistics. We derive a general Bartlett-type correction factor in matrix notation for the gradient test which reduces the size distortion of the test, and numerically compare the proposed test with the usual likelihood ratio, Wald, score and gradient tests, and with the Bartlett-corrected likelihood ratio and score tests. Our simulation results suggest that the corrected test we propose can be an interesting alternative to the other tests since it leads to very accurate inference even for very small samples. We also present an empirical application for illustrative purposes.
We obtain an asymptotic expansion for the null distribution function of thegradient statistic for testing composite null hypotheses in the presence of nuisance parameters. The expansion is derived using a Bayesian route based on the shrinkage argumen
t described in Ghosh and Mukerjee (1991). Using this expansion, we propose a Bartlett-type corrected gradient statistic with chi-square distribution up to an error of order o(n^{-1}) under the null hypothesis. Further, we also use the expansion to modify the percentage points of the large sample reference chi-square distribution. A small Monte Carlo experiment and various examples are presented and discussed.
The Birnbaum-Saunders regression model is commonly used in reliability studies. We address the issue of performing inference in this class of models when the number of observations is small. We show that the likelihood ratio test tends to be liberal
when the sample size is small, and we obtain a correction factor which reduces the size distortion of the test. The correction makes the error rate of he test vanish faster as the sample size increases. The numerical results show that the modified test is more reliable in finite samples than the usual likelihood ratio test. We also present an empirical application.
This paper develops a bias correction scheme for a multivariate heteroskedastic errors-in-variables model. The applicability of this model is justified in areas such as astrophysics, epidemiology and analytical chemistry, where the variables are subj
ect to measurement errors and the variances vary with the observations. We conduct Monte Carlo simulations to investigate the performance of the corrected estimators. The numerical results show that the bias correction scheme yields nearly unbiased estimates. We also give an application to a real data set.
This paper develops a bias correction scheme for a multivariate normal model under a general parameterization. In the model, the mean vector and the covariance matrix share the same parameters. It includes many important regression models available i
n the literature as special cases, such as (non)linear regression, errors-in-variables models, and so forth. Moreover, heteroscedastic situations may also be studied within our framework. We derive a general expression for the second-order biases of maximum likelihood estimates of the model parameters and show that it is always possible to obtain the second order bias by means of ordinary weighted lest-squares regressions. We enlighten such general expression with an errors-in-variables model and also conduct some simulations in order to verify the performance of the corrected estimates. The simulation results show that the bias correction scheme yields nearly unbiased estimators. We also present an empirical ilustration.
