We develop a general theory for the goodness-of-fit test to non-linear models. In particular, we assume that observations are noisy samples of submanifold defined by sufficiently smooth map. The observation noise is additive Gaussian. Our main result shows “residual” model fit, solving least-square problem, follows (possibly noncentral) χ <sup xmlns:mml="http://www.w3.org/1998/Math/MathML" xmln...