  We consider the problem of designing experiments to detect the presence of a
specified heteroscedastity in a non-linear Gaussian regression model. In this
framework, we focus on the ${\rm D}_s$- and KL-criteria and study their
relationship with the noncentrality parameter of the asymptotic chi-squared
distribution of a likelihood-based test, for local alternatives. Specifically,
we found that when the variance function depends just on one parameter, the two
criteria coincide asymptotically and in particular, the ${\rm D}_1$-criterion
is proportional to the noncentrality parameter. Differently, if the variance
function depends on a vector of parameters, then the KL-optimum design
converges to the design that maximizes the noncentrality parameter.
Furthermore, we confirm our theoretical findings through a simulation study
concerning the computation of asymptotic and exact powers of the log-likelihood
ratio statistic.
