Testing a Regression Model when we Have Smooth Alternatives in Mind

Goodness-of-fit tests based on residual sum of squares are a standard procedure in fitiing regression models. Often we have a smooth alternative in mind, a qualitative feature that the X2-test does not take into account. We show that the power of detecting a smooth alternative increases when we smooth the current model as well. The proposed test is shown to be able to detect any continuous local alternative tending to zero slower than n-1/2. Theoretical results also address minimax nonparametric hypothesis testing in Sobolev spaces.