Goodness of Fit Test for Dependent Observations

Introduction. The goodness of t tests of the empirical data to one theoretical distribution, and particularly the χ of Pearson, are very used in practice, also as preliminary step for the use of methods of parametric inference which are consistent with precise models of distribution. Gasser [10] explored the χ test for correlated data by simulations and, subsequently, Moore [14] and Gleser and Moore [11] studied the asymptotic properties for gaussian processes, while Chanda [6, 7] explored the e ects of dependence for observations generated by linear, bilinear and Volterra's processes. We establish some properties of a class of functional tests of goodness of t for correlated observations generated by α mixing discrete time stochastic processes. These tests are associated with projection density estimator. This class of functional tests contains in particular the Pearson's χ test (1900) [16] and the smooth test of Neyman (1937) [15]. Tests based on the deviation of a density estimator from the true density have been considered by several authors; particularly Bosq ([1, 2, 3, 4, 5]) has de ned such tests considering the special case of projection density estimators and he has established many results for independent observations in a general framework. Gadiaga [9] has obtained some results for correlated observations generated by α mixing discrete time stochastic processes. Lee and Na [13] and Tenreiro [18] analyze tests associated with kernel density estimator for, respectively, α mixing and β mixing processes. In this work, we analyze the asymptotic behaviour of the test statistics and we establish the rate of convergence to the limit distribution by using a theorem of Tikhomirov [19]. Since the limit distribution is a linear combination of independent χ1 r.v.'s with unknown coe cients, we de ne consistent estimators for these coefcients. We give the necessary and su cient conditions for the consistency of the test. Moreover we give some indications about implementation of the test and we present some results of a simulation to compare the χ test and the smooth test by evaluation of the empirical power.