Extended Neyman smooth goodness-of-fit tests, applied to competing heavy-tailed distributions

A simplified version of the Neyman (1937) “Smooth” goodness-of-fit test is extended to account for the presence of estimated model parameters, thereby removing overfitting bias. Using a Lagrange Multiplier approach rather than the Likelihood Ratio statistic proposed by Neyman greatly simplifies the calculations. Polynomials, splines, and the step function of Pearson’s test are compared as alternative perturbations to the theoretical uniform distribution. The extended tests have negligible size distortion and more power than standard tests. The tests are applied to competing symmetric leptokurtic distributions with US stock return data. These are generally rejected, primarily because of the presence of skewness.