Goodness-of-Fit Tests for Dependent Data

We establish some properties for a class of functional tests of goodness-of-fit for correlated observations generated by α-mixing discrete time stochastic processes. These tests are associated with projection density estimators. This class of functional tests contains in particular Pearson’s χ2 test and the ‘smooth test’ of Neyman, J. (1937). Smooth test for goodness of fit. Scand. Aktuar. 20, 119–128. Results in the case of independent observations, in a general framework, can be found in Bosq, D. (2002). Functional tests of fit. Goodness-of-Fit Tests and Model Validity, Statistics for Industry and Technology, Birkh¨auser, Boston, pp. 341–356. We analyze the asymptotic behavior of the test statistics, both under the null hypothesis and under the alternative, establishing the rate of convergence to their limit distributions. We determine the necessary and sufficient conditions for consistency of the test. Indications for implementing the test are provided.