BIAS-CORRECTED NONPARAMETRIC SPECTRAL ESTIMATION

Abstract. The theory of nonparametric spectral density estimation based on an observed stretch X₁,…, X_N from a stationary time series has been studied extensively in recent years. However, the most popular spectral estimators, such as the ones proposed by Bartlett, Daniell, Parzen, Priestley and Tukey, are plagued by the problem of bias, which effectively prohibits ?N-convergence of the estimator. This is true even in the case where the data are known to be m-dependent, in which case ?N-consistent estimation is possible by a simple plug-in method. In this report, an intuitive method for the reduction in the bias of a nonparametric spectral estimator is presented. In fact, applying the proposed methodology to Bartlett's estimator results in bias-corrected estimators that are related to kernel estimators with lag-windows of trapezoidal shape. The asymptotic performance (bias, variance, rate of convergence) of the proposed estimators is investigated; in particular, it is found that the trapezoidal lag-window spectral estimator is ?N-consistent in the case of moving-average processes, and ?(N/log/N)-consistent in the case of autoregressive moving-average processes. The finite-sample performance of the trapezoidal lag-window estimator is also assessed by means of a numerical simulation.     

Robust estimation for the covariance matrix of multi‐variate time series

In this article, we study the robust estimation for the covariance matrix of stationary multi‐variate time series. As a robust estimator, we propose to use a minimum density power divergence estimator (MDPDE) proposed by Basu et al. (1998) . Particularly, the MDPDE is designed to perform properly when the time series is Gaussian. As a special case, we consider the robust estimator for the autocovariance function of univariate stationary time series. It is shown that the MDPDE is strongly consistent and asymptotically normal under regularity conditions. Simulation results are provided for illustration.     

Common Breaks in Means for Cross‐Correlated Fixed‐T Panel Data

This note considers a panel data model in which the variable of interest has undergone a common structural break in the mean. The object of interest is the unknown breakpoint. The challenge is to device an estimator that is consistent when the data are cross‐correlated and the number of time periods T is fixed and cannot be increased without bound. The proposed solution involves taking an already existing estimator initially proposed for cross‐section uncorrelated panels and applying it to defactored data. Consistency is established as the number of cross‐section units N grows large, and is verified in small samples using Monte Carlo simulation.     

Oracle M‐Estimation for Time Series Models

We propose a thresholding M‐estimator for multivariate time series. Our proposed estimator has the oracle property that its large‐sample properties are the same as of the classical M‐estimator obtained under the a priori information that the zero parameters were known. We study the consistency of the standard block bootstrap, the centred block bootstrap and the empirical likelihood block bootstrap distributions of the proposed M‐estimator. We develop automatic selection procedures for the thresholding parameter and for the block length of the bootstrap methods. We present the results of a simulation study of the proposed methods for a sparse vector autoregressive VAR(2) time series model. The analysis of two real‐world data sets illustrate applications of the methods in practice.     

A Nonparametric Prewhitened Covariance Estimator

This paper proposes a new nonparametric spectral density estimator for time series models with general autocorrelation. The conventional nonparametric estimator that uses a positive kernel has mean squared error no better than n^?4/5. We show that the best implementation of our estimator has mean squared error of order n^?8/9, provided there is sufficient smoothness present in the spectral density. This is, of course, achieved by bias reduction; however, unlike most other bias reduction methods, like the kernel method with higher‐order kernels, our procedure ensures a positive definite estimate. Our method is a generalization of the well‐known prewhitening method of spectral estimation; we argue that this can best be interpreted as multiplicative bias reduction. Higher‐order expansions for the proposed estimator are derived, providing an improved bandwidth choice that minimizes the mean squared error to the second order. A simulation study shows that the recommended prewhitened kernel estimator reduces bias and mean squared error in spectral density estimation.     

Two-stage and sequential estimation of parameter N of binomial distribution when p is known

In this article, we propose two-stage and purely sequential procedures to construct bounded width and prescribed proportional closeness confidence intervals for the unknown parameter N of B(N,p) distribution where the parameter p is assumed to be known. The exact distributions of the stopping variables and the estimators of N at stopping are derived for all cases. The coverage probabilities of the proposed interval estimator are computed exactly and are shown to be nearly the same as the prescribed level.     

Monitoring of emulsion polymerisations: A study of reaction kinetics in the presence of secondary nucleation

A non‐linear state estimator that provides on‐line information on residual monomer concentration and radical concentration is experimentally validated under dynamic conditions, and in particular in the presence of secondary particle nucleation. The proposed estimator uses the heat of reaction as an input. Practical methods for determining the initial values of the parameters in the energy balance are also proposed and tested on laboratory and pilot scale installations. It is shown that this method provides a robust tool for the on‐line monitoring of batch (with and without shot additions of monomer, surfactant and initiator) and semi‐batch emulsion polymerisation reactors. The observer was shown to converge rapidly, even under conditions where the number of particles per litre of emulsion (N_p) changed rapidly.     

ESTIMATION OF THE MEMORY PARAMETER FOR NONSTATIONARY OR NONINVERTIBLE FRACTIONALLY INTEGRATED PROCESSES

Abstract. We consider the asymptotic characteristics of the periodogram ordinates of a fractionally integrated process having memory parameter d≥ 0.5, for which the process is nonstationary, or d≤ -.5, for which the process is noninvertible. Series having d outside the range (-.5,.5) may arise in practice when a raw series is modeled without preliminary consideration of the stationarity and invertibility of the series or when a wrong decision is made concerning the stationarity and invertibility of the series. We find that the periodogram of a nonstationary or noninvertible fractionally integrated process at the jth Fourier frequency ωj= 2πj/n, where n is the sample size, suffers from an asymptotic relative bias which depends on j. We also examine the impact of periodogram bias on the regression estimator of d proposed by Geweke and Porter-Hudak (1983) in finite samples. The results indicate that the bias in the periodogram ordinates can strongly affect the GPH estimator even when the number of Fourier frequencies used in the regression is allowed to depend on the length of the series. We find that data tapering and elimination of the first periodogram ordinate in the regression can reduce this bias, at the cost of an increase in variance for nonstationary series. Additionally, we find for nonstationary series that the GPH estimator is more nearly invariant to first-differencing when a data taper is used.     

Least Squares Bias in Time Series with Moderate Deviations from a Unit Root

This articles derives the approximate bias of the least squares estimator of the autoregressive coefficient in discrete autoregressive time series where the autoregressive coefficient is given by α_T=1+c/k_T, with k_T being a deterministic sequence increasing to infinity at a rate slower than T, such that k_T=o(T) as T→∞. The cases in which c<0, c=0 and c>0 are considered correspond to (moderately) stationary, non‐stationary and (moderately) explosive series respectively. The result is used to derive the limiting distribution of the indirect inference method for such processes with moderate deviations from a unit root and for explosive series with a fixed coefficient, which does not depend on the sample size. Second, the result demonstrates why the jackknife estimator cannot be constructed for explosive time series for values of the autoregressive parameter close to unity in view of the discontinuity of the bias function, which the article derives. Finally, the expression is used to construct a bias‐corrected estimator, and simulations are carried out to assess the three estimators' bias reduction capabilities.     

An Asymptotic F Test for Uncorrelatedness in the Presence of Time Series Dependence

We propose a simple asymptotically F-distributed Portmanteau test for zero autocorrelations in an otherwise dependent time series. By employing the orthonormal series variance estimator of the variance matrix of sample autocovariances, our test statistic follows an F distribution asymptotically under fixed-smoothing asymptotics. The asymptotic F theory accounts for the estimation error in the underlying variance estimator, which the asymptotic chi-squared theory ignores. Monte Carlo simulations reveal that the F approximation is much more accurate than the corresponding chi-squared approximation in finite samples. The asymptotic F test is as easy to use as the chi-squared test: there is no need to obtain critical values by simulations. Furthermore, it has more accurate empirical sizes and substantial power advantages, comparing to other competitors.     

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.     

Unit root testing with slowly varying trends

A unit root test is proposed for time series with a general nonlinear deterministic trend component. It is shown that asymptotically the pooled OLS estimator of overlapping blocks filters out any trend component that satisfies some Lipschitz condition. Under both fixed‐b and small‐b block asymptotics, the limiting distribution of the t‐statistic for the unit root hypothesis is derived. Nuisance parameter corrections provide heteroskedasticity‐robust tests, and serial correlation is accounted for by pre‐whitening. A Monte Carlo study that considers slowly varying trends yields both good size and improved power results for the proposed tests when compared to conventional unit root tests.     

Generalized Resampling Scheme With Application to Spectral Density Matrix in Almost Periodically Correlated Class of Time Series

The aim of this article is to introduce new resampling scheme for nonstationary time series, called generalized resampling scheme (GRS). The proposed procedure is a generalization of well known in the literature subsampling procedure and is simply related to existing block bootstrap techniques. To document the usefulness of GRS, we consider the example of model with almost periodic phenomena in mean and variance function, where the consistency of the proposed procedure was examined. Finally, we prove the consistency of GRS for the spectral density matrix for nonstationary, multivariate almost periodically correlated time series. We consider both zero mean and non‐zero mean case. The consistency holds under general assumptions concerning moment and α‐mixing conditions for multivariate almost periodically correlated time series. Proving the consistency in this case poses a difficulty since the estimator of the spectral density matrix can be interpreted as a sum of random matrixes whose dependence grow with the sample size.     

An Asymptotic Second-Order Lower Bound for the Bayes Risk of a Sequential Procedure

Abstract

Consider the problem of estimating the difference of the means of two populations, where each population distribution is a member of the one-parameter exponential family of probability distributions. A Bayesian approach is adopted in which the mean difference is estimated under the squared error loss and the prior distributions are of the form proposed by Diaconis and Ylivisaker [Diaconis, P.; Ylivisaker, D. Conjugate priors for exponential families. Ann. of Statist. 1979, 6, 269–281]. The main result determines an asymptotic second-order lower bound for the Bayes risk of a sequential procedure that takes N observations from one population and t ? N from the other population, and estimates the mean difference by the Bayes estimator, where N is determined according to a sequential design and t denotes the total number of observations sampled from both populations.     

Local Power of Fixed‐T Panel Unit Root Tests With Serially Correlated Errors and Incidental Trends

The asymptotic local power properties of various fixed T panel unit root tests with serially correlated errors and incidental trends are studied. Asymptotic (over N) local power functions are analytically derived, and through them, the effects of general forms of serial correlation are examined. We find that a test based on an instrumental variables (IV) estimator dominates the tests based on the within‐groups (WG) estimator. These functions also show that in the presence of incidental trends, an instrumental variables test based on the first differences of the model has non‐trivial local power in an N^?1/2 neighbourhood of unity. Furthermore, for a test based on the within‐groups estimator, although it is found that it has trivial power in the presence of incidental trends, this ceases to be the case if there is serial correlation as well.     

Reaction of Aliphatic Amines with 49% Formic Acid. 1-Hexylamine,di-1-hexylamine,N,N-dimethyl-1-hexylamine, 1-dodecylamine,N,N-dimethyl-1-dodecylamine and N,N-dimethyl-1-butylamine

Two primary amines, 1-hexylamine 2 , 1-dodecylamine 19 , one secondary amine, di-1-hexylamine 18 , and three tertiary amines, N,N-dimethyl-1-hexylamine 6 , N,N-dimethyl-1-butylamine 3 , and N,N-dimethyl-1-dodecylamine 22 were each heated at 150 °C, 250 °C or 350 °C with 49% aqueous formic acid for varying periods of time. The aliphatic primary amines underwent easy N-formylation and subsequent reduction to give N-methyl- and N,N-dimethylalkylamines. Especially at higher temperatures, other reactions intervened including elimination of NH₃ to the corresponding alkenes followed by partial double bond isomerization. Tertiary amines were more reactive at higher temperatures undergoing hydrolysis and reductive cleavages to secondary and primary amines, which subsequently followed the reaction sequences seen for primary amines. This series of saturated amines showed none of the cleavage into smaller fragments that was observed in the reductive alkylation of pyridine and 4-methylpyridine to a series of N-alkylpiperdines. This result reinforces the bis-aza-retro-Aldol-fragmentation mechanism postulated for the formation of the N-alkylpiperidines.     

Minimum α‐divergence estimation for arch models

Abstract. This paper considers a minimum α‐divergence estimation for a class of ARCH(p) models. For these models with unknown volatility parameters, the exact form of the innovation density is supposed to be unknown in detail but is thought to be close to members of some parametric family. To approximate such a density, we first construct an estimator for the unknown volatility parameters using the conditional least squares estimator given by Tjøstheim [Stochastic processes and their applications (1986) Vol. 21, pp. 251–273]. Then, a nonparametric kernel density estimator is constructed for the innovation density based on the estimated residuals. Using techniques of the minimum Hellinger distance estimation for stochastic models and residual empirical process from an ARCH(p) model given by Beran [Annals of Statistics (1977) Vol. 5, pp. 445–463] and Lee and Taniguchi [Statistica Sinica (2005) Vol. 15, pp. 215–234] respectively, it is shown that the proposed estimator is consistent and asymptotically normal. Moreover, a robustness measure for the score of the estimator is introduced. The asymptotic efficiency and robustness of the estimator are illustrated by simulations. The proposed estimator is also applied to daily stock returns of Dell Corporation.     

Local Whittle estimation of multi‐variate fractionally integrated processes

This article derives a semi‐parametric estimator of multi‐variate fractionally integrated processes covering both stationary and non‐stationary values of d. We utilize the notion of the extended discrete Fourier transform and periodogram to extend the multi‐variate local Whittle estimator of Shimotsu (2007) to cover non‐stationary values of d. Consistency and asymptotic normality is shown for d ∈ (?1/2,∞). A simulation study illustrates the performance of the proposed estimator for relevant sample sizes. Empirical justification of the proposed estimator is shown through an empirical analysis of log spot exchange rates. We find that the log spot exchange rates of Germany, United Kingdom, Japan, Canada, France, Italy and Switzerland against the US Dollar for the period January 1974 until December 2001 are well decribed as I(1) processes.     

Terpolymer films having semipolar structure: Preparation,wettability, and mechanical properties

A terpolymer, obtained by the free‐radical terpolymerization of 2‐(N,N‐dimethylamino)ethyl methacrylate (DMMA), methyl methacrylate(MMA), and isobutyl methacrylate (IBMA), was allowed to react with hydrogen peroxide, chloroacetic acid, and diethyl sulfate to form the corresponding modified terpolymers: (1) N,N‐dimethyl‐N‐(2‐methacryloyloxyethyl)amine N‐oxide, MMA and IBMA (DMANO series); (2) N‐(carboxymethyl)‐N,N‐dimethyl‐ N‐(2‐methacryloyloxyethyl)ethyl ammonium, MMA and IBMA (CDME series); and (3) N‐(ethyl)‐N,N‐dimethyl‐N‐(2‐methacryloyloxyethyl)ethyl ammonium ethylsulfonate, MMA and IBMA (EDMEES series), respectively. The terpolymer compositions were determined using ¹³C NMR spectrometry. Surface free energies of the terpolymers were estimated by measuring the contact angles of water and methylene iodide on the three series films (DMANO, CDME, and EDMEES), and the effect of the N‐oxide group on wettability was discussed. It was found that the upper surface of the films for the DMANO and CDME series are more hydrophobic than that for the EDMEES series. Notably, elongation to break for the DMANO series was relatively larger than that for the CDME series because of the water bound to the N‐oxide functional group. © 2005 Wiley Periodicals, Inc. J Appl Polym Sci 98: 1235–1243, 2005     

Robust estimation of the scale and of the autocovariance function of Gaussian short‐ and long‐range dependent processes

A desirable property of an autocovariance estimator is to be robust to the presence of additive outliers. It is well known that the sample autocovariance, being based on moments, does not have this property. Hence, the use of an autocovariance estimator which is robust to additive outliers can be very useful for time‐series modelling. In this article, the asymptotic properties of the robust scale and autocovariance estimators proposed by Rousseeuw and Croux (1993) and Ma and Genton (2000) are established for Gaussian processes, with either short‐ or long‐range dependence. It is shown in the short‐range dependence setting that this robust estimator is asymptotically normal at the rate , where n is the number of observations. An explicit expression of the asymptotic variance is also given and compared with the asymptotic variance of the classical autocovariance estimator. In the long‐range dependence setting, the limiting distribution displays the same behaviour as that of the classical autocovariance estimator, with a Gaussian limit and rate when the Hurst parameter H is less than 3/4 and with a non‐Gaussian limit (belonging to the second Wiener chaos) with rate depending on the Hurst parameter when H ∈ (3/4,1). Some Monte Carlo experiments are presented to illustrate our claims and the Nile River data are analysed as an application. The theoretical results and the empirical evidence strongly suggest using the robust estimators as an alternative to estimate the dependence structure of Gaussian processes.