Nonparametric Statistics for Stochastic Processes

Abstract

This is a technical monograph in the European tradition aimed at a specialist rather than a general audience. Having said that, I have found it to be a useful resource in my dabbling with smoothing techniques.The exposition is aimed at deriving asymptotic properties of non-parametric alternatives to the common Box-Jenkins approach to time series analysis. The problems examined involve the estimation of the finite dimensional distributions of a stochastic process, regression estimators, and non-parametric prediction of future observations for both discrete and continuous time processes. The emphasis is on optimal asymptotic results and bandwidth choice.Kernel estimators previously applied to independent observations induce the estimators examined. This is set out clearly in the synopsis. The theory is based on mixing processes, which are reviewed in Chapter 1. Chapter 2 examines rates of convergence in density estimation for discrete time processes. Chapter 3 considers regression estimators and prediction for discrete time processes. The regression estimator is a Nadaraya-Watson kernel estimator applied to dependent bivariate observations. In prediction this estimator is applied to pairs of observations from the same stationary Markov process. The approach is then extended to continuous time processes in Chapters 4 and 5. Chapter 6 uses local time to estimate a density. The implementation of the method, including reviews of comparisons with ARMA and ARCH approaches, is given in Chapter 7.

Time-Varying Risk Premium in Large Cross-Sectional Equity Data Sets

“…Within blocks, covariances do not need to vanish asymptotically. The proof of Lemma 14 uses a result on almost sure convergence in Stout (1974), a large deviation theorem based on the Hoeffding inequality in Bosq (1998), and CLTs for martingale difference arrays in Davidson (1994) and White (2001).…”

CONVERGENCE RATES OF SUMS OF α-MIXING TRIANGULAR ARRAYS: WITH AN APPLICATION TO NONPARAMETRIC DRIFT FUNCTION ESTIMATION OF CONTINUOUS-TIME PROCESSES

“…Let n (l) := sup 1 k n l Cov Z nk ; Z n(k+l) for each l(= 0; 1; : : : ; n 1). Since jZ n;i j 2b n and n (l) (l) j2b n j 2 4Ab 2 n l , where the latter follows from Billingsley's inequality (Corollary 1.1 of Bosq, 1998 …”

“…In this example, the mixing rate and the moment existence rely directly on the parameter r. The condition (14) on Z ( ) represents a case in which the drift function's e¤ect of the mean-reversion is very weak. 7 If fX n;i g is an array of (normalized) discrete-time observations from fZ s g, say, X n;i = (Z i E[Z i ]), then its mixing coe¢ cients satisfy n (m) A (m ) and the uniform moment bound g n = g n (p) is well-de…ned for p 2 (0; 2r). Therefore, we can apply the rate results of (12) and (13).…”

A General Result on the Mean Integrated Squared Error of the Hard Thresholding Wavelet Estimator underα-Mixing Dependence

“…The problems where is the density of 1 or a regression function have received a lot of attention. A partial list of related works includes Robinson [4], Roussas [5,6], Truong and Stone [7], Tran [8], Masry [9,10], Masry and Fan [11], Bosq [12], and Liebscher [13].…”