On flat-top kernel spectral density estimators for homogeneous random fields

The problem of nonparametric estimation of the spectral density function of a partially observed homogeneous random field is addressed. In particular, a class of estimators with favorable asymptotic performance (bias, variance, rate of convergence) is proposed. The proposed estimators are actually shown to be √N-consistent if the autocovariance function of the random field is supported on a compact set, and close to √N-consistent if the autocovariance function decays to zero sufficiently fast for increasing lags.     

Variance function estimation of a one-dimensional nonstationary process

We propose a flexible nonparametric estimation of a variance function from a one-dimensional process where the process errors are nonstationary and correlated. Due to nonstationarity a local variogram is defined, and its asymptotic properties are derived. We include a bandwidth selection method for smoothing taking into account the correlations in the errors. We compare the proposed difference-based nonparametric approach with Anderes and Stein(2011)’s local-likelihood approach. Our method has a smaller integrated MSE, easily fixes the boundary bias, and requires far less computing time than the likelihood-based method.     

Autocovariance Estimation in Regression with a Discontinuous Signal and m‐Dependent Errors: A Difference‐Based Approach

We discuss a class of difference‐based estimators for the autocovariance in nonparametric regression when the signal is discontinuous and the errors form a stationary m‐dependent process. These estimators circumvent the particularly challenging task of pre‐estimating such an unknown regression function. We provide finite‐sample expressions of their mean squared errors for piecewise constant signals and Gaussian errors. Based on this, we derive biased‐optimized estimates that do not depend on the unknown autocovariance structure. Notably, for positively correlated errors, that part of the variance of our estimators that depend on the signal is minimal as well. Further, we provide sufficient conditions for ‐consistency; this result is extended to piecewise Hölder regression with non‐Gaussian errors. We combine our biased‐optimized autocovariance estimates with a projection‐based approach and derive covariance matrix estimates, a method that is of independent interest. An R package, several simulations and an application to biophysical measurements complement this paper.     

EXPERIMENTAL DESIGN FOR CONTINUOUS PROCESSES

Measurements taken on a continuous process (e.g. production of chemicals, ore processing, steel production) often exhibit autocorrelation. Such correlation between successive measurements can be avoided only by taking measurements at widely separated times, but this delays the results. We discuss the impact of this correlation on experimental designs which for a continuous process must include the sequence in which treatment combinations are applied and the spacing between samples. The variogram of the process is a convenient Summary of the correlation. Formulae for the variance of the mean of measurements taken on a continuous process and of contrasts calculated from such measurements are simple expressions in terms of the variogram. These formulae allow a simple approach to experimental designs that avoid unnecessary delays.     

A comparison of multivariate signal discrimination techniques

This paper investigates several techniques to discriminate two multivariate stationary signals. The methods considered include Gaussian likelihood ratio tests for variance equality, a chi-squared time-domain test, and a spectral-based test. The latter two tests assess equality of the multivariate autocovariance function of the two signals over many different lags. The Gaussian likelihood ratio test is perhaps best viewed as principal component analyses (PCA) without dimension reduction aspects; it can be modified to consider covariance features other than variances via dimension augmentation tactics. A simulation study is constructed that shows how one can make inappropriate conclusions with PCA tests, even when dimension augmentation techniques are used to incorporate non-zero lag autocovariances into the analysis. The various discrimination methods are first discussed. A simulation study then illuminates the various properties of the methods. In this pursuit, calculations are needed to identify several multivariate time series models with specific autocovariance properties. To demonstrate the applicability of the methods, nine US and Canadian weather stations from three distinct regions are clustered. Here, the spectral clustering perfectly identified distinct regions, the chi-squared test performed marginally, and the PCA/likelihood ratio method did not perform well.     

A Short Note on the Dependence Structure of a Stochastic Process Generated by the Asymmetric Cusp Map

Intermittency maps are well-known to be capable of generating stochastic processes with slowly decaying cross covariances. The present paper considers the asymmetric cusp map and derives an asymptotic lower bound of the autocovariance function of a stochastic process generated by this map. As a consequence, such a stochastic process is shown to belong to the class of long memory processes.     

Dependence Estimation for High‐frequency Sampled Multivariate CARMA Models

The paper considers high‐frequency sampled multivariate continuous‐time autoregressive moving average (MCARMA) models and derives the asymptotic behaviour of the sample autocovariance function to a normal random matrix. Moreover, we obtain the asymptotic behaviour of the cross‐covariances between different components of the model. We will see that the limit distribution of the sample autocovariance function has a similar structure in the continuous‐time and in the discrete‐time model. As a special case, we consider a CARMA (one‐dimensional MCARMA) process. For a CARMA process, we prove Bartlett's formula for the sample autocorrelation function. Bartlett's formula has the same form in both models; only the sums in the discrete‐time model are exchanged by integrals in the continuous‐time model. Finally, we present limit results for multivariate MA processes as well, which are not known in this generality in the multivariate setting yet.     

Some efficient computational procedures for high order ARMA models

Recursive methods are commonly used to solve Yule—Walker equations for autoregrsssive parameters given an autocovariance function. The reverse procedure can be extended to the efficient solution of various sets of equations which arise in time series analysis. Those presented in this paper include computation of the autocovariance function of an ARMA model, and the Cramer—Wold factorization.     

LOCATION OF THE OPTIMAL SAMPLING POINT FOR THE QUALITY ASSESSMENT OF CONTINUOUS STREAMS

The paper makes an appraisal of the most appropriate sampling point for situations where a single sample must be used to estimate the mean flow of a continuous stream during a set time interval. Taking ‘optimal’ to mean the point at which the estimation error variance is minimised, optimal sampling locations are obtained for constant, linear and exponential flow rates when the process variogram is assumed linear or exponential. Numerical results illustrate the significance of failing to sample at the optimal point.     

Time Series Properties of the Class of Generalized First-Order Autoregressive Processes with Moving Average Errors

A new class of time series models known as Generalized Autoregressive of order one with first-order moving average errors has been introduced in order to reveal some hidden features of certain time series data. The variance and autocovariance of the process is derived in order to study the behaviour of the process. It is shown that in special cases these new results reduce to the standard ARMA results. Estimation of parameters based on the Whittle procedure is discussed. We illustrate the use of this class of model by using two examples.     

Symbolic cumulant calculations for frequency domain time series

With time series data, there is often the issue of finding accurate approximations for the variance of such quantities as the sample autocovariance function or spectral estimate. Smith and Field (J. Time. Ser. Anal 14: 381–395, 1993) proposed a variance estimate motivated by resampling in the frequency domain. In this paper we present some results on the cumulants of this and other frequency domain estimates obtained via symbolic computation. The statistics of interest are linear combinations of products of discrete Fourier transforms. We describe an operator which calculates the joint cumulants of such statistics, and use the operator to deepen our understanding of the behaviour of the resampling based variance estimate. The operator acts as a filter for a general purpose operator described in Andrews and Stafford (J.R. Statist. Soc. B55, 613–627).     

Block bootstrap for periodic characteristics of periodically correlated time series

This research is dedicated to the study of periodic characteristics of periodically correlated time series such as seasonal means, seasonal variances and autocovariance functions. Two bootstrap methods are used: the extension of the usual Moving Block Bootstrap (EMBB) and the Generalised Seasonal Block Bootstrap (GSBB). The first approach is proposed, because the usual Moving Block Bootstrap does not preserve the periodic structure contained in the data and cannot be applied for the considered problems. For the aforementioned periodic characteristics the bootstrap estimators are introduced and consistency of the EMBB in all cases is obtained. Moreover, the GSBB consistency results for seasonal variances and autocovariance function are presented. Additionally, the bootstrap consistency of both considered techniques for smooth functions of the parameters of interest is obtained. Finally, the simultaneous bootstrap confidence intervals are constructed. A simulation study to compare their actual coverage probabilities is provided. A real data example is presented.     

Spectral representation and autocovariance structure of Markov switching DSGE models

Abstract

We investigate the L²-structure of Markov switching Dynamic Stochastic General Equilibrium (MS DSGE) models and derive conditions for strict and second-order stationarity. Then we determine the autocovariance function of the process driven by a stationary MS DSGE model and give a stable VARMA representation of it. It turns out that the autocovariance structure of the process coincides with that of a standard VARMA. Finally, we propose a method to derive the spectral density in a matrix closed-form of MS DSGE models. Our results relate with the works of Francq and Zakoian, Krolzig, Zhang and Stine. Numerical and empirical illustrations complete the article.     

Testing for changes in autocovariances of nonparametric time series models

In the literature on change-point analysis, much attention has been paid to detecting changes in certain marginal characteristics, such as mean, variance, and marginal distribution. For time series data with nonparametric time trend, we study the change-point problem for the autocovariance structure of the unobservable error process. To derive the asymptotic distribution of the cumulative sum test statistic, we develop substantial theory for uniform convergence of weighted partial sums and weighted quadratic forms. Our asymptotic results improve upon existing works in several important aspects. The performance of the test statistic is examined through simulations and an application to interest rates data.     

Periodic integer-valued bilinear time series model

This paper deals with the study of some probabilistic and statistical properties of a periodic integer-valued diagonal bilinear model. The existence of a periodically strict stationary integer-valued process is shown. Sufficient conditions for the periodically stationary, both in the first and second orders, are established. The closed-forms of the mean and the second moment are obtained. The closed-form of the periodic autocovariance function is established. The Yule–Walker estimations of the underlying parameters are obtained. A simulation study is provided.     

Continuous processes derived from the solution of generalized Langevin equation: theoretical properties and estimation

ABSTRACT

In this paper we present a class of continuous-time processes arising from the solution of the generalized Langevin equation and show some of its properties. We define the theoretical and empirical codifference as a measure of dependence for stochastic processes. As an alternative dependence measure we also consider the spectral covariance. These dependence measures replace the autocovariance function when it is not well defined. Results for the theoretical codifference and theoretical spectral covariance functions for the mentioned process are presented. The maximum likelihood estimation procedure is proposed to estimate the parameters of the process arising from the classical Langevin equation, i.e. the Ornstein–Uhlenbeck process, and of the so-called Cosine process. We also present a simulation study for particular processes arising from this class showing the generation, and the theoretical and empirical counterpart for both codifference and spectral covariance measures.     

Nonparametric Estimation of Average Growth Curve with General Nonstationary Error Process

The nonparametric estimation of the growth curve has been extensively studied in both stationary and some nonstationary particular situations. In this work, we consider the statistical problem of estimating the average growth curve for a fixed design model with nonstationary error process. The nonstationarity considered here is of a general form, and this article may be considered as an extension of previous results. The optimal bandwidth is shown to depend on the singularity of the autocovariance function of the error process along the diagonal. A Monte Carlo study is conducted in order to assess the influence of the number of subjects and the number of observations per subject on the estimation.     

Improved estimation for the autocovariances of a Gaussian stationary process

For a Gaussian stationary process with mean μ and autocovariance function γ(·), we consider to improve the usual sample autocovariances with respect to the mean squares error (MSE) loss. For the cases μ=0 and μ≠0, we propose sort of empirical Bayes type estimators Γ? and Γ?, respectively. Then their MSE improvements upon the usual sample autocovariances are evaluated in terms of the spectral density of the process. Concrete examples for them are provided. We observe that if the process is near to a unit root process the improvement becomes quite large. Thus, consideration for estimators of this type seems important in many fields, e.g., econometrics.     

Spectral analysis of Markov switching GARCH models with statistical inference

We derive matrix expressions in closed form for the autocovariance function and the spectral density of Markov switching GARCH models and their powers. For this, we apply the Riesz–Fischer theorem which defines the spectral representation as the Fourier transform of the autocovariance function. Under suitable assumptions, we prove that the sample estimator of the spectral density is consistent and asymptotically normally distributed. Further statistical implications in terms of order identification and parameter estimation are discussed. A simulation study confirms the validity of the asymptotic properties. These methods are also well suited for financial market applications, and in particular for the analysis of time series in the frequency domain, as shown in some proposed real-world examples.     

Nonnegative Definiteness of the Sample Autocovariance Function

Various textbooks on time series analysis assert that the usual version of the sample autocovariance function (1) is nonnegative definite. Two simple proofs of this result are presented.