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Title: Optimal design to test for heteroscedasticity in a regression model Authors:  Samantha Leorato - University of Milan (Italy) [presenting]
Chiara Tommasi - University of Milan (Italy)
Alessandro Lanteri - University of Turin (Italy)
Jesus Lopez-Fidalgo - University of Navarra (Spain)
Abstract: The goal is to design an experiment to detect a specific kind of heteroscedasticity in a non-linear regression model, i.e. $y_i = \eta(x_i; \beta) + \varepsilon_i$, $\varepsilon_i \sim N(0; \sigma^2 h(x_i; \gamma))$, $i = 1, \ldots , n$, where $\eta(x_i; \beta)$ is a possibly non-linear mean function, depending on a vector of regression coefficients $\beta \in \mathbb{R}^p$, and $\sigma^2 h(x_i; \gamma)$ is the error variance depending on an unknown constant $\sigma^2$ and on a continuous positive function $h$, completely known except for the parameter vector $\gamma \in \mathbb{R}^s$ and satisfies $h(\cdot; 0)=1$. We consider the testing problem $H_0 : \gamma=0$ against a local alternative $H_1 : \gamma=\lambda/\sqrt{n}$, for some $\lambda \not=0$ and $n$ the sample size. The application of a likelihood-based test is a common approach to this problem, since its asymptotic distribution is known. The aim consists in designing an experiment with the goal of maximizing (in some sense) the asymptotic power of a likelihood-based test. Few papers in optimal design of experiments are related to hypothesis testing most of which concern designing to check an adequate fit to the true mean function. We justify the use of the $D_s-$criterion and the KL-optimality to design an experiment with the inferential goal of checking for heteroscedasticity.