UC San Diego

This thesis is concerned with nonparametric techniques for inferring properties of time series. First, we consider finite-order moving average and nonlinear autoregressive processes with no parametric assumption on the innovation distribution, and present a kernel density estimator of a bootstrap series that estimates their marginal densities root-$n$ consistently. This is equal to the rate of the best known convolution estimators, and faster than the standard kernel density estimator. We also conduct simulations to check the finite sample properties of our estimator, and the results are generally better than corresponding results for the standard kernel density estimator. Next, given stationary time series data, we study the problem of finding the best linear combination of a set of lag window spectral density estimators with respect to the mean squared risk. We present an aggregation procedure and prove a sharp oracle inequality for its risk. We also provide simulations demonstrating the performance of our aggregation procedure, given Bartlett and other estimators of varying bandwidths as input. This extends work by Rigollet and Tsybakov on aggregation of density estimators. The last part of this thesis introduces a class of robust autocorrelation estimators based on interpreting the sample autocorrelation function as a linear regression. We investigate the efficiency and robustness properties of the estimators that result from plugging on three common robust regression techniques. Construction of robust autocovariance and positive definite autocorrelation estimates is discussed, as well as application of the estimators to AR model fitting. We finish with simulations, which suggest that the estimators are especially well suited for AR model fitting