Skew-normal distribution for growth curve models in presence of a heteroscedasticity structure

In general, growth models are adjusted under the assumptions that the error terms are homoscedastic and normally distributed. However, these assumptions are often not verified in practice. In this work we propose four growth models (Morgan–Mercer–Flodin, von Bertalanffy, Gompertz, and Richards) considering different distributions (normal, skew-normal) for the error terms and three different covariance structures. Maximum likelihood estimation procedure is addressed. A simulation study is performed in order to verify the appropriateness of the proposed growth curve models. The methodology is also illustrated on a real dataset.     

ON TESTS FOR INTERACTION IN A NONREPLICATED TWO-WAY LAYOUT

The paper proposes tests for interaction in a two-way table with one observation per cell. The power of these tests is independent of the additive main effects in the linear model. This is an advantage compared to a test suggested earlier, which has low power if main effects are very variable.     

Testing heteroscedasticity in nonlinear and nonparametric regressions

This article presents new nonparametric tests for heteroscedasticity in nonlinear and nonparametric regression models. The tests have an asymptotic standard normal distribution under the null hypothesis of homoscedasticity and are robust against any form of heteroscedasticity. A Monte Carlo simulation with critical values obtained from the wild bootstrap procedure is provided to asses the finite sample performances of the tests. A real application of testing interest rate volatility functions illustrates the usefulness of the tests proposed. The Canadian Journal of Statistics © 2009 Statistical Society of Canada     

Robust multivariate analysis of variability

A general testing procedure is proposed to multivariately test for equality of p variances among k groups. The procedure applies a multivariate analysis of variance on an appropriate measure of spread for the uncensored original observations. Three such measures of spread are compared in a simulation experiment which considered two and three variables with equal and unequal sample sizes for the null and alternative hypotheses for Gaussian, Student's t (8, 12, and 20 degrees of freedom) and gamma (α=2,4,6 and 10) distributions . The likelihood ratio test (Box, 1949) was included in the above simulations. The results suggest that if one chooses a measure of spread appropriate for the distribution of the original observations, the proposed MANOVA-based testing procedure is robust and reasonably powerful. Using this procedure for the normal distribution, similar power was observed to that of the likelihood ratio test when the variables were uncorrelated or had little positive correlation.     

Robust extrapolation designs and weights for biased regression models with heteroscedastic errors

We consider the construction of designs for the extrapolation of regression responses, allowing both for possible heteroscedasticity in the errors and for imprecision in the specification of the response function. We find minimax designs and correspondingly optimal estimation weights in the context of the following problems: (1) for ordinary least squares estimation, determine a design to minimize the maximum value of the integrated mean squared prediction error (IMSPE), with the maximum being evaluated over both types of departure; (2) for weighted least squares estimation, determine both weights and a design to minimize the maximum IMSPE; (3) choose weights and design points to minimize the maximum IMSPE, subject to a side condition of unbiasedness. Solutions to (1) and (2) are given for multiple linear regression with no interactions, a spherical design space and an annular extrapolation space. For (3) the solution is given in complete generality; as one example we consider polynomial regression. Applications to a dose-response problem for bioassays are discussed. Numerical comparisons, including a simulation study, indicate that, as well as being easily implemented, the designs and weights for (3) perform as well as those for (1) and (2) and outperform some common competitors for moderate but undetectable amounts of model bias.