Nonparametric estimation of a multivariate probability density for mixing sequences by the method of wavelets

Abstract

The mathematical theory of wavelet and their applications in statistics have become a well-known technique for non-parametric curve estimation: see e.g. Meyer (1990), Daubachies (1992), Chui (1992), Donoho and Johnstone (1995) and Vidakovic (1999). We consider the problem of estimation of the partial derivatives of a multivariate probability density f of mixing sequences, using wavelet-based method.  Many stochastic processes and time series are known to be mixing. Under certain weak assumptions autoregressive and more generally bilinear time series models are strongly mixing with exponential mixing coeffcients.  The problem of density estimation from dependent samples is often considered.  For instance quadratic losses were considered by Ango Nze and Doukhan (1993). Bosq (1995) and Doukhan and Loen (1990). We investigate the variance and the rate of the almost convergence of wavelet-based estimators. Rate of convergence of estimators when f belongs to the Besov space is also established.