Wavelets in Statistics

In this paper we give the main uses of wavelets in statistics, with emphasis in time series analysis. We include the fundamental work on non parametric regression, which motivated the development of techniques used in the estimation of the spectral density of stationary processes and of the evolutionary spectrum of locally stationary processes

Indirect inference for locally stationary ARMA processes with stable innovations

The class of locally stationary processes assumes that there is a time-varying spectral representation, that is, the existence of finite second moment. We propose the $\alpha$-stable locally stationary process by modifying the innovations into stable distributions and the indirect inference to estimate this type of model. Due to the infinite variance, some of interesting properties such as time-varying auto-correlation cannot be defined. However, since the $\alpha$-stable family of distributions is closed under linear combination which includes the possibility of handling asymmetry and thicker tails, the proposed model has the same tail behavior throughout the time. In this paper, we propose this new model, present theoretical properties of the process and carry out simulations related to the indirect inference in order to estimate the parametric form of the model. Finally, an empirical application is illustrated.Comment: 31 pages, 14 figures. Submitted to the Journal of Statistical Computation and Simulatio

State Space Markov Switching Models Using

We propose a state space model with Markov switching, whose regimes are associated with the model parameters and regime transition probabilities are time-dependent. The estimation is based on maximum likelihood method using the EM algorithm. The distribution of the estimators is assessed using bootstrap. To evaluate the state variables and regime probabilities, the Kalman filter and a probability filter procedure conditional to each possible regime at each instant are used. This procedure is illustrated with simulated data and the United States monthly industrial production index from January 1960 to January 1995

A Wavelet Analysis for Time Series

In this paper we develop a wavelet spectral analysis for a stationary discrete process. Some basic ideas on wavelets are given and the concept of wavelet spectrum is introduced. Asymptotic properties of the discrete wavelet transform of a sample of observed values from the process are derived and the wavelet periodogram is considered as an estimator of the wavelet spectrum. Applications to real and simulated series are given