Exact optimal and adaptive inference in linear and nonlinear models under heteroskedasticity and non-normality of unknown forms

In this paper, we derive simple sign-based point-optimal test in linearand nonlinear regression models. The test is exact, distribution free, robustagainst heteroskedasticity of unknown form, and it may be inverted to obtaincon…dence regions for the vector of unknown parameters.Because our optimal test depends on the alternative hypothesis, we pro-pose an adaptive approach based on sample-split technique to choose analternative such that the power curve of point-optimal sign test is close tothat of power envelope curve. The simulation study shows that when usingapproximately 10% of sample to estimate the alternative and the rest tocalculate the test statistic, the power curve of the point-optimal sign test istypically close to the power envelope curve.We present a Monte Carlo study to assess the performance of the pro-posed “quasi”-point-optimal sign test by comparing its size and power tothose of some common tests which are supposed to be robust against het-eroskedasticity. The results show that our procedure is superior.