Robustness of the autoregressive spectral estimate for linear processes with infinite variance

Consider a discrete-time linear process { x _t }, a one-sided moving average of independent identically distributed random variables {ε _t }, with the common distribution in the domain of attraction of a symmetric stable law of index δ∈ (0, 2) and the moving-average coefficients b ( j ) such that ε _t is invertible in terms of the present and possibly infinite past values of { x _t }. By treating { x _t } as if it is second-order stationary, a normalized spectral density function f (μ) is defined in terms of the b ( j ) and, having observed x ₁, ..., x _T , an autoregression of order k is fitted by the well-known Yule–Walker and least squares methods and the normalized autoregressive spectral estimators are constructed. On letting k ←∞ as T ←∞, but sufficiently slowly, these estimators are shown to be uniformly consistent for f (μ), the convergence rate being T ^−1/φ, φ > δ. The finite sample behaviour is investigated by a simulation study which also examines possible effects of considering 'non-invertible' models.     

HIGHER-ORDER ASYMPTOTIC PROPERTIES OF A WEIGHTED ESTIMATOR FOR GAUSSIAN ARMA PROCESSES

Abstract. Let { X _t } be a Gaussian ARMA process with spectral density f _θ(Λ), where θ is an unknown parameter. To estimate θ, we propose an estimator θC_w of the Bayes type. Since our standpoint in this paper is different from Bayes's original approach, we call it a weighted estimator. We then investigate various higher-order asymptotic properties of θC_w. It is shown that θC_w is second-order asymptotically efficient in the class of second-order median unbiased estimators. Furthermore, if we confine our discussions to an appropriate class D of estimators, we can show that θC_w is third-order asymptotically efficient in D . We also investigate the Edgeworth expansion of a transformation of θC_w. We can then give the transformation of θC_w which makes the second-order part of the Edgeworth expansion vanish. Finally we consider the problem of testing a simple hypothesis H:θ=θ_o against the alternative A:θ#θ_o. For this problem we propose a class of tests δ_A which are based on the weighted estimator. We derive the X ² type asymptotic expansion of the distribution of S (ζδ_A) under the sequence of alternatives A n :θ=θ_o+ε n ^1/2, ε > 0. We can then compare the local powers of various tests on the basis of their asymptotic expansions.     

A Test of Linearity for Functional Autoregressive Models

We propose a new test for linearity in time series. We consider an asymptotically stationary functional AR( p ) model on ℜ ^d of the form
X _n = f ( X _{n −1}, ..., X _{n − p} ) + ξ _n ( n ∈ N).
The testing procedure is based on a suitably normalized sum of quadratic deviations between two different estimates of the function f evaluated at q distinct points of ℜ ^dp . The estimators are f^ _n , a recursive version of the non-parametric kernel estimator of f , and  _n , a least squares estimator well suited to the linear case. The main result states that the test statistic has a χ² limit distribution under the null hypothesis. A similar result is derived under the alternative hypothesis for the test statistic corrupted by a non-linear term. Our simulations indicate that our asymptotic results hold for moderate sample sizes when the testing procedure is used carefully     

Limiting distributions of unconditional maximum likelihood unit root test statistics in seasonal time–series models

Abstract.  The likelihood function of a seasonal model, Y _t  =  ρ Y _{t − d}  +  e _t as implemented in computer algorithms under the assumption of stationary initial conditions is a function of ρ which is zero at the point ρ  = 1. It is a smooth function for ρ in the above seasonal model with a well-defined maximum regardless of the data-generating mechanism. Gonzalez-Farias (PhD Thesis, North Carolina State University, 1992) proposed tests for unit roots based on maximizing the stationary likelihood function in nonseasonal time series. We extend it to seasonal time series. The limiting distribution of seasonal unit root test statistics based on the unconditional maximum likelihood estimators are shown. Models having a single mean, seasonal means, and a single-trend variable across the seasons are considered.     

ON THE CONSISTENCY OF LEAST SQUARES ESTIMATORS FOR A THRESHOLD AR(1) MODEL

Abstract. For the SETAR (2; 1,1) model

where {a_t(i)} are i.i.d. random variables with mean 0 and variance σ²(i), i = 1,2, and {a_t(l)} is independent of {a_t(2)}, we consider estimators of φ₁, φ ₂ and r which minimize weighted sums of the sum of squares functions for σ²(1) and σ²(2). These include as a special case the usual least squares estimators. It is shown that the usual least squares estimators of φ₁, φ₂ and r are consistent. If σ²(1) ≠σ²(2) conditions on the weights are found under which the estimators of r and φ₁ or φ₂ are not consistent.     

ESTIMATION OF THE COEFFICIENTS OF A MULTIVARIATE LINEAR FILTER USING THE INNOVATIONS ALGORITHM

It is shown under mild conditions that the estimators of the coefficient matrices obtained by applying the innovations algorithm to the sample covariances of observations of the multivariate linear time series X _t = ∑^∞ _{j =0}ψ _i Z _t , t = 0, ±1, ±2, . . ., are consistent. The asymptotic distribution of the estimators is found to have a very simple form which generalizes the corresponding univariate result of Brockwell and Davis (Simple consistent estimation of the coefficients of a linear filter. In Stochastic Processes and Their Applications . Amsterdam: North- Holland, pp. 47--59). The asymptotic distribution of the corresponding estimator of the spectral density matrix is also derived. Some simulation results are presented to illustrate the small-sample behaviour of the estimators.     

FREQUENCY DOMAIN TESTS OF MULTIVARIATE GAUSSIANITY AND LINEARITY

A stationary multivariate time series { X _t } is defined as linear if it can be written in the form X _t = ∑^∞ _{j =−∞} A _j e _{t − j} where A _j are square matrices and e _t are independent and identically distributed random vectors. If the e _t } are normally distributed, then { X _t is a multivariate Gaussian linear process. This paper is concerned with the testing of departures of a vector stationary process from multivariate Gaussianity and linearity using the bispectral approach. First the definition and properties of cumulants of random matrices are used to obtain the expressions for the higher-order cumulant and spectral vectors of a linear vector process as defined above. Then it is shown that linearity of a vector process implies constancy of the modulus square of its normalized higher-order spectra whereas the component of such a vector process does not necessarily have a linear representation. Finally, statistics for the testing of multivariate Gaussianity and linearity are proposed.     

NONPARAMETRIC ESTIMATORS FOR TIME SERIES

Abstract. Kernel multivariate probability density and regression estimators are applied to a univariate strictly stationary time series X _r We consider estimators of the joint probability density of X _t at different t -values, of conditional probability densities, and of the conditional expectation of functionals of X _v given past behaviour. The methods seem of particular relevance in light of recent interest in non-Gaussian time series models. Under a strong mixing condition multivariate central limit theorems for estimators at distinct points are established, the asymptotic distributions being of the same nature as those which would derive from independent multivariate observations.     

A note on estimation by least squares for harmonic component models

Abstract. Let observations ( X ₁,…, X _n ) be generated by a harmonic model such that X _t = A ₀ cos  ω ₀ t + B ₀ sin  ω ₀ t + ε _t , where A ₀, B ₀, ω ₀ are constants and ( ε _t ) is a stationary process with zero mean and finite variance. The estimation of A ₀, B ₀, ω ₀ by the method of least squares is considered. It is shown that, without any restriction on ω in the minimization procedure, the estimate     is an n -consistent estimate of ω ₀, and hence (     ) has the usual asymptotic distribution.
The extension to a harmonic model with k >1 components is discussed. The case k =2 is considered in detail, but it was only found possible to establish the result under the restriction that both angular frequencies lie in the interval         

THE COMPARISON OF LEAST SQUARES AND THIRD-ORDER PERIODOGRAM PROCEDURES IN THE ESTIMATION OF BIFREQUENCY

Abstract. Given a stretch of time series values, the third-order periodogram is investigated as a criterion for use in the estimation of bifrequencies, that is of frequency triples (ω₁ω₂, ω₃) with ω₃= 2ω -ω₁ω₂. The least squares estimates of such frequencies are compared with estimates derived by maximizing the modulus of the third-order periodogram. It is found that neither estimation procedure is uniformly better than the other.     

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 THE AUTOCORRELATION STRUCTURE AND IDENTIFICATION OF SOME BILINEAR TIME SERIES

Abstract. For the bilinear time series X _t =β X _t-k e _t-l + e _v , k ≥ l , formulas for the first k -1 autocorrelations of X ² _t are obtained. These results fill in a gap in Granger and Andersen (1978). Simulation experiments are used to study the applicability of theoretical results and to investigate some more general situations. It is found that if ß is not too small, k and l may be identified using the autocorrelations of X ² _t . Application to more general situations is also briefly discussed.     

Estimation of the Dominating Frequency for Stationary and Nonstationary Fractional Autoregressive Models

This paper was motivated by the investigation of certain physiological series for premature infants. The question was whether the series exhibit periodic fluctuations with a certain dominating period. The observed series are nonstationary and/or have long-range dependence. The assumed model is a Gaussian process X _t whose m th difference Y_t = (1 − B ) ^m X_t is stationary with a spectral density f that may have a pole (or a zero) at the origin. the problem addressed in this paper is the estimation of the frequency ω_max where f achieves the largest local maximum in the open interval (0, π). The process X_t is assumed to belong to a class of parametric models, characterized by a parameter vector θ, defined in Beran (1995). An estimator of ω_max is proposed and its asymptotic distribution is derived, with θ being estimated by maximum likelihood. In particular, m and a fractional differencing parameter that models long memory are estimated from the data. Model choice is also incorporated. Thus, within the proposed framework, a data driven procedure is obtained that can be applied in situations where the primary interest is in estimating a dominating frequency. A simulation study illustrates the finite sample properties of the method. In particular, for short series, estimation of ω_max is difficult, if the local maximum occurs close to the origin. The results are illustrated by two of the data examples that motivated this research.     

Sign Invariance in Goodness-of-Fit Tests for Time Series

A goodness-of-fit test for a stationary stochastic process may be based on a functional of the difference between the sample standardized spectral distribution and a hypothesized standardized spectral distribution. Theorems are given to show that under certain conditions the distribution of such a functional based on observations from a process { y_t } indexed by a parameter θ is the same for θ=θ₀ and for θ=−θ₀. The results are illustrated by three examples of time series processes.     

STATIONARY PROCESSES WITH A FINITE NUMBER OF NON-ZERO CANONICAL CORRELATIONS BETWEEN FUTURE AND PAST

Abstract. The spectral densities f (γ) are determined for stationary random processes X (t) with continuous time which have the property that the number of non-zero canonical correlations between the past X(t) ( t ≤ 0) (more accurately the present and the past) and the future X(t) ( t ≥θ) is finite (equal to N ) at any θ≥τ for some r > 0. A method for finding all the corresponding canonical correlations P₁, …, P_N and the canonical variables X ₁^-, …, X _N^- and X ₁⁺, …, X _N⁺ is given. Similar results related to processes X(t) with discrete (integral) time are briefly considered.     

PARTIAL AUTOCORRELATION PROPERTIES FOR NON-STATIONARY AUTOREGRESSIVE MOVING-AVERAGE MODELS

Abstract. We are primarily interested in relating the partial autocorrelation behaviour of an autoregressive integrated moving-average process of order ( p, d, q ), { Z _i } say, with those of its D -differenced processes {(1 - B ) ^D Z _i } ( D = 1, …, d ). To this end, we evaluate the early partial correlations corresponding to serial correlations which initially follow a slow linear decline from unity. These partials, to a first approximation, take a small constant negative value from lag 2 onwards. We also demonstrate a relationship between the theoretical partials π _k and π _k (Δ) for a once-integrated process { Z _i } and its first-differenced process {(1 - B ) Z _i } respectively. These results carry over to cases where the non-stationary zeros are at -1, and differencing is replaced by the corresponding 'simplifying' transformation implicit in the operator 1 + B.     

A note on L1 density estimation for linear processes

In this paper, we consider the L ₁ performance of a kernel estimator, f^_n of the density of a linear process X_t ∑^∞ _{k =0} a _k Z _t−k , a ₀ = 1, where { Z _t } is a sequence of independent and identically distributed (i.i.d.) random variables with E | Z ₁|^ε< ∞, for some ε > 1, and { a_k } is a sequence of reals converging to zero at a certain rate. Asymptotic minimizations of the integrated L ₁ risk of f_n and its upper bounds are considered. This paper extends the earlier results for the i.i.d. case by Devroye and Gyorfi ( Nonparametric Density Estimation: The L₁ View. New York: Wiley, 1985) and by Hall and Wand (Minimizing the L ₁ distance in nonparametric density estimation, J. Multivariate Anal. 26 (1988), 59–88) to the linear process case. Numerical examples to illustrate the performance of f_n are also presented.     

Relation Between Power and Exponential Laws of Slow Crack Growth

The coefficients ₀, and N of the power law of slow crack growth, =₀ exp [–/(RT)](K/K_C)^N, are evaluated in terms of the fundamental parameters V₀. Q₀. and B of the exponential law, V=V₀ exp [(-Q₀+BK)/(RT)]. It is shown that N =θ(BK_C)/(RT),=₀−θBK_C, and ₀=V₀θ^−N, where θ is a dimensionless coefficient with a value ranging from 0.2858 to 1.0, depending on the ratio of stress intensity at the fatigue limit to the critical stress intensity factor.     

A- and B Site Cosubstituted Ba6−3xSm8+2xTi18O54Microwave Dielectric Ceramics

Tin (Sn) substitution into the B-site and Nd/Sn cosubstitution into the A- and B-sites were investigated in a Ba _{6−3 x} Sm_{8+2 x} Ti₁₈O₅₄solid solution ( x = 2/3). A small amount of tin substitution for titanium improved the temperature coefficient of resonant frequency (τ_f) but led to a decrease of the relative dielectric constant (ɛ) and the quality factor ( Qf ). The Ba_{6−3 x} Sm_{8+2 x} (Ti_{1− z} Sn_z)₁₈O₅₄-based tungsten-bronze phase became unstable for compositions with a tin content of ≥10 mol%, where BaSm₂O₄and Sm₂(Sn,Ti)₂O₇appeared, and finally, these phases became the major phases. On the other hand, Nd/Sn cosubstitution led to a good combination of high ɛ, high Qf , and near-zero τ_f. Excellent microwave dielectric properties were achieved in Ba_{6−3 x} (Sm_{1− y} Nd _y )_{8+2 x} (Ti_{1− z} Sn _z )₁₈O₅₄ceramics with y = 0.8 and z = 0.05 sintered at 1360°C for 3 h: ɛ= 82, Qf = 10 000 GHz, and calculated τ_f=+17 ppm/°C. The tolerance factor and electronegativity difference exhibited important effects on the microwave dielectric properties, especially the Qf value. A large tolerance factor and high electronegativity difference generally led to a higher Qf value.     

NON-NEGATIVE AUTOREGRESSIVE PROCESSES

Abstract. Consider a stationary autoregressive process given by X _t = b ₁ X _{t -1}+…+ b _p X _t-p + Y _t , where the Y _t are independent identically distributed positive variables and b ₁,…, b _p are non-negative parameters. Let the variables X ₁,…, X _n be given. If p = 1 then it is known that b ₁*= min( X _t / X _{t -1}) is a strongly consistent estimator for b ₁ under very general conditions. In this paper the case p = 2 is analysed in detail. It is proved that min( X _t / X _{t -1})→ b ₁ almost surely (a.s.) and min( X _t / X _{t -2})→ b ₂+ b ₁² a.s. as n → 8. The convergence is very slow. Denote by b ₁* and b ₂* values of b ₁ and b ₂ respectively which maximize b ₂+ b ₂ under the conditions X _t - b ₁ X _{t -1}- b ₂ X _{t -2}≥ 0 for t = 3,…, n . We prove that b ₁* b ₁ and b ₂* b ₂ a.s. Simulations show that b ₁* and b ₂* are better than the least-squares estimators of the autoregressive coefficients when the distribution of Y _t is exponential.