A Bias Correction for Cross-validation Bandwidth Selection when a Kernel Estimate is Based on Dependent Data

Least-squares cross-validation (LSCV) bandwidth selection for kernel density estimation has been shown to underestimate the optimal bandwidth if data are positively correlated. We calculate the asymptotic bias for the LSCV criterion under a continuous-time model and apply it as a correction term to discrete-time data that can be modeled as a smooth continuous-time process sampled at a high rate.     

A systematic comparison of PCA‐based Statistical Process Monitoring methods for high‐dimensional,time‐dependent Processes

High‐dimensional and time‐dependent data pose significant challenges to Statistical Process Monitoring. Most of the high‐dimensional methodologies to cope with these challenges rely on some form of Principal Component Analysis (PCA) model, usually classified as nonadaptive and adaptive. Nonadaptive methods include the static PCA approach and Dynamic Principal Component Analysis (DPCA) for data with autocorrelation. Methods, such as DPCA with Decorrelated Residuals, extend DPCA to further reduce the effects of autocorrelation and cross‐correlation on the monitoring statistics. Recursive Principal Component Analysis and Moving Window Principal Component Analysis, developed for nonstationary data, are adaptive. These fundamental methods will be systematically compared on high‐dimensional, time‐dependent processes (including the Tennessee Eastman benchmark process) to provide practitioners with guidelines for appropriate monitoring strategies and a sense of how they can be expected to perform. The selection of parameter values for the different methods is also discussed. Finally, the relevant challenges of modeling time‐dependent data are discussed, and areas of possible further research are highlighted. © 2016 American Institute of Chemical Engineers AIChE J, 62: 1478–1493, 2016     

Time‐Transformed Unit Root Tests for Models with Non‐Stationary Volatility

Abstract. Conventional unit root tests are known to be unreliable in the presence of permanent volatility shifts. In this paper, we propose a new approach to unit root testing which is valid in the presence of a quite general class of permanent variance changes which includes single and multiple (abrupt and smooth transition) volatility change processes as special cases. The new tests are based on a time transformation of the series of interest which automatically corrects their form for the presence of non‐stationary volatility without the need to specify any parametric model for the volatility process. Despite their generality, the new tests perform well even in small samples. We also propose a class of tests for the null hypothesis of stationary volatility in (near‐) integrated time‐series processes.     

Density estimation with locally identically distributed data and with locally stationary data

Abstract. We consider multivariate density estimation when the assumptions of identically distributed data or stationary data are relaxed to the assumptions of locally identically distributed data or locally stationary data. We assume that the distribution of the data is changing continuously as function of time. To estimate densities non‐parametrically with these local regularity conditions, we need time localization in addition to the usual space localization. We define a time‐localized kernel estimator that estimates the density non‐parametrically at any given point of time. The consistency of the time‐localized kernel estimator is proved and the rates of convergence of the estimator are derived under conditions on the β‐and α‐mixing coefficients. Both the time‐series setting and spatial setting are covered.     

Local Likelihood for non‐parametric ARCH(1) models

Abstract. We propose a non‐parametric local likelihood estimator for the log‐transformed autoregressive conditional heteroscedastic (ARCH) (1) model. Our non‐parametric estimator is constructed within the likelihood framework for non‐Gaussian observations: it is different from standard kernel regression smoothing, where the innovations are assumed to be normally distributed. We derive consistency and asymptotic normality for our estimators and show, by a simulation experiment and some real‐data examples, that the local likelihood estimator has better predictive potential than classical local regression. A possible extension of the estimation procedure to more general multiplicative ARCH(p) models with p > 1 predictor variables is also described.     

Functional Coefficient Autoregressive Models: Estimation and Tests of Hypotheses

In this paper, we study nonparametric estimation and hypothesis testing procedures for the functional coefficient AR (FAR) models of the form X_t = f ₁( X _{t − d} ) X _{t − 1}+ ... + f_p ( X _{t − d} ) X _{t − p} +ε _t , first proposed by Chen and Tsay (1993). As a direct generalization of the linear AR model, the FAR model is a rich class of models that includes many useful parametric nonlinear time series models such as the threshold AR models of Tong (1983) and exponential AR models of Haggan and Ozaki (1981). We propose a local linear estimation procedure for estimating the coefficient functions and study its asymptotic properties. In addition, we propose two testing procedures. The first one tests whether all the coefficient functions are constant, i.e. whether the process is linear. The second one tests if all the coefficient functions are continuous, i.e. if any threshold type of nonlinearity presents in the process. The results of some simulation studies as well as a real example are presented.     

ON M‐Estimation Under Long‐Range Dependence in Volatility

Abstract. We consider M‐estimation of a location parameter for processes with zero autocorrelations but long‐range dependence in volatility. The observed process is the product of i.i.d. Gaussian observations and a long‐memory Gaussian process. For nonlinear estimators, the rate of convergence depends on the type of the ψ‐function. For skew‐symmetric ψ‐functions, a central limit theorem with ‐rate of convergence holds, under suitable regularity assumptions. This is not true in general for M‐estimators where the ψ‐function is not skewsymmetric.     

Kernel deconvolution of stochastic volatility models

Abstract.  In this paper, we study the problem of the nonparametric estimation of the function m in a stochastic volatility model h _t  =  exp( X _t /2λ) ξ _t , X _t  =  m ( X _{t −1}) +  η _t , where ξ _t is a Gaussian white noise. We show that the model can be written as an autoregression with errors-in-variables. Then an adaptation of the deconvolution kernel estimator proposed by Fan and Truong [ Annals of Statistics , 21, (1993) 1900] estimates the function m with the optimal rate, which depends on the distribution of the measurement error. The rates vary from powers of n to powers of  ln( n ) depending on the rate of decay near infinity of the characteristic function of this noise. The performance of the method are studied by some simulation experiments and some real data are also examined.