A Comparison of Alternative Approaches to Supremum-Norm Goodness of Fit Tests with Estimated Parameters

Abstract

Goodness of fit tests based on empirical processes have nonstandard limiting distributions when the null hypothesis is composite -that is, when parameters of the null model are estimated. Several solutions to this problem have been suggested, including the calculation of adjusted critical values for these nonstandard distributions and the transformation of the empirical process such that statistics based on the transformed process are asymptotically distribution-free. The approximation methods proposed by Durbin (1985) can be applied to compute appropriate critical values for tests based on supremum-norm statistics. The resulting tests have quite accurate size, a fact which has gone unrecognized in the econometrics literature. Some justification for this accuracy lies in the similar features that Durbin's approximation methods share with the theory of extrema for Gaussian random fields and for Gauss-Markov processes. These adjustment techniques are also related to the transformation methodology proposed by Khmaladze (1981) through the score function of the parametric model. Simulation experiments suggest that these two testing strategies are roughly comparable to one another and more powerful than a simple bootstrap procedure.

Dual regression

“…However, simultaneous inference over regions of the joint support of Y and X is typically of interest in practice. Several approaches for uniform inference in the presence of non-pivotal limit processes have been considered in the literature (e.g., Koenker &Xiao, 2002, andParker, 2013), including simulation methods (Chernozhukov et al, 2013). Extension of existing results to dual regression is beyond the scope of this paper but they provide a natural direction for future study of uniform inference on the empirical dual regression process.…”

Approximations for weighted Kolmogorov–Smirnov distributions via boundary crossing probabilities

“…The latter appears in the vast literature devoted to goodness-of-fit tests when parameters are estimated: see del Barrio (2007), Parker (2013), and references therein.…”

“…On the contrary p 1,g (x), which is a linear combination of values of Φ is accurate even for very large values of x. Another calculation can be done for large x: Durbin's approximation (see Durbin (1985) and Parker (2013) for a useful review). Let v(t) denote the variance function of X : v(t) = R X (t, t), where R X is defined by (2).…”