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In this paper we address the issue of testing inference of the dispersion parameter in heteroscedastic symmetric nonlinear regression models considering small samples. We derive Bartlett corrections to improve the likelihood ratio as well modified profile likelihood ratio tests. Our results extend some of those obtained in Cordeiro (J Stat Comput Simul 74:609-620, 2004) and Ferrari et al. (J Stat Plan Inference 124:423-437, 2004), who consider a symmetric nonlinear regression model and normal linear regression model, respectively. We also present the bootstrap and bootstrap Bartlett corrected likelihood ratio tests. Monte Carlo simulations are carried out to compare the finite sample performances of the three corrected tests and their uncorrected versions. The numerical evidence shows that the corrected modified profile likelihood ratio test, the bootstrap and bootstrap Bartlett corrected likelihood ratio test perform better than the other ones. We also present an empirical application.

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This paper provides general expressions for Bartlett and Bartlett-type correction factors for the likelihood ratio and gradient statistics to test the dispersion parameter vector in heteroscedastic symmetric nonlinear models. This class of regression models is potentially useful to model data containing outlying observations. Furthermore, we develop Monte Carlo simulations to compare size and power of the proposed corrected tests to the original likelihood ratio, score, gradient tests, corrected score test, and bootstrap tests. Our simulation results favor the score and gradient corrected tests as well as the bootstrap tests. We also present an empirical application.

In this paper, we consider the beta prime regression model recently proposed by Bourguignon et al. (2018), which is tailored to situations where the response is continuous and restricted to the positive real line with skewed and long tails and the regression structure involves regressors and unknown parameters. We consider two different strategies of bias correction of the maximum-likelihood estimators for the parameters that index the model. In particular, we discuss bias-corrected estimators for the mean and the dispersion parameters of the model. Furthermore, as an alternative to the two analytically bias-corrected estimators discussed, we consider a bias correction mechanism based on the parametric bootstrap. The numerical results show that the bias correction scheme yields nearly unbiased estimates. An example with real data is presented and discussed.