Wavelet goodness-of-fit test for dependent data

We introduce a new goodness-of-fit test which can be applied to hypothesis testing about the marginal distribution of dependent data. We derive a new test for the equivalent hypothesis in the space of wavelet coefficients. Such properties of the wavelet transform as orthogonality, localisation and sparsity make the hypothesis testing in wavelet domain easier than in the domain of distribution functions. We propose to test the null hypothesis separately at each wavelet decomposition level to overcome the problem of bi-dimensionality of wavelet indices and to be able to find the frequency where the empirical distribution function differs from the null in case the null hypothesis is rejected. We suggest a test statistic and state its asymptotic distribution under the null and under some of the alternative hypotheses.