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.
翻译:我们考虑了设计实验以检测非线性高斯回归模型中存在特定超混凝土的问题。 在此框架内, 我们侧重于 $@rm D ⁇ s$ 和 KL 标准, 并研究它们与基于可能性的测试的不集中参数分布的非集中参数之间的关系, 用于本地替代品。 具体地说, 我们发现, 当差异函数仅取决于一个参数时, 两种标准不同时, 特别是 $_rm D ⁇ 1$- criticion 与非集中参数是成比例的 。 不同的是, 如果差异函数依赖于参数矢量, 那么 KL- optimm 设计就会与最大限度地增加非集中参数的设计相匹配。 此外, 我们通过模拟研究来确认我们关于对日志比比值进行计算时的理论结论 。