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.     

ROBUST REGRESSION AND INTERPOLATION FOR TIME SERIES

Abstract. In this paper we shall consider the interpolation problem under the condition that the spectral density of a stationary process concerned is vaguely known (i.e., Huber's ε -contaminated model). Then we can get a minimax robust interpolator for the class of spectral densities S ={ g:g(x)=(1-ε)f(x)+εh(x)ε Ar D_o, 0<ε<1}, where f(x) is a known spectral density and D ₀ is a certain class of spectral densities. Also we shall consider the time series regression problem under the condition that the residual spectral density is vaguely known. Then we can get a minimax robust regression coefficient estimate for the class of the residual spectral densities S .     

Simple Mathematical Model for the Electrical Conductivity of Highly Porous Ceramics

A novel approach has been proposed to describe the relationship between the conductivity and relative density of highly porous materials. As a first approximation, porous material was represented by a uniaxial string of spheres along the direction of the potential gradient. Then, a string of spheres was remodeled into a rotating body of sine-wave functions, f ( x ) = (1 + r ₀)/2 + [(1 – r ₀)/2] sin (π x / c ) for 0 ≤ x < c and f (π x ) = (1 + r ₀)/2 + [(1 – r ₀)/2] sin {π( x + 1 − 2 c )/(l − c )} for c ≤ x < 1, where the former represents the shape of a sphere, the latter that of the bottleneck between neighboring spheres, and r ₀ denotes the ratio of the minimum diameter at the bottleneck to the maximum diameter of the rotating body. It was shown that the calculated relationships reproduced the reported experimental results for the relationship between the porosity and conductivity of La_0.5Sr_0.5CoO₃, BaF₂, and (ZrO₂)_0.9(Y₂O₃)_0.1. The relative conductivity to the bulk material was close to zero at 45–60% relative density, which is the density of green wares. It steeply increased with an increase in the relative density and then gradually approached that of the bulk material.     

NON-NEGATIVE AUTOREGRESSIVE MODELS

Abstract. Consider a stationary non-negative autoregressive (AR) model given x _t = b ₁ x _t -1, +…+ b _p x _t-p + e _t , where the e _t are independent identically distributed non-negative variables and b ₁, …, b _p are non-negative parameters, and all the roots of the equation 1 – b ₁ u –…– b _p u ^p = 0 are outside the unit circle. The stationary solution of the above AR model is called a stationary non-negative AR process. Let x ₁, x ₂, … x _n be an example of a stationary non-negative AR process. Under very general conditions strongly consistent estimators of the AR parameters b ₁, b ₂, …, b _p have been studied. In this paper a new procedure is proposed to estimate not only b ₁, b ₂, …, b _p but also b _o which is the essential lower bound of the variable e _t . We shall show that the new estimators obtained using the new procedure are consistent estimators of b _o, b ₁, …, b _p under the weakest condition which guarantees that the stationary non-negative AR model has a stationary non-degenerative solution.     

THE DISTRIBUTION OF PERIODOGRAM ORDINATES

Abstract. The empirical distribution function of y _i = I (ω _j )/{2π f (ω _j )}, ω j = 2π j / T , where I (ω) is the periodogram for a set of observations from a stationary time series with spectral density f (ω), is shown to converge, almost surely, to the distribution with density exp(- x), under appropriate conditions. The same methods are used to prove the convergence, almost surely, of an estimate of the prediction error variance constructed from the I (ω_j) and of the complex empirical distribution function based on the Fourier coefficients.     

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.     

Structure and Microwave Dielectric Characteristics of Ca1+xNd1−xAl1−xTixO4 Ceramics

CaNdAlO₄ microwave dielectric ceramics were modified by Ca/Ti co-substitution, and their dielectric characteristics were evaluated along with their structure and microstructures. Ca_{1+ x} Nd_{1− x} Al_{1− x} Ti _x O₄ ( x =0, 0.025, 0.05, 0.10, 0.15, 0.20) ceramics with the relative density of over 95% theoretical density were obtained by sintering at 1400°–1450°C in air for 3 h, where the K₂NiF₄-type solid solution single phase was determined from the compositions of x <0.20, while a small amount of CaTiO₃ secondary phase was detected for x =0.20. With Ca/Ti co-substitution in CaNdAlO₄ ceramics, the dielectric constant (ɛ_r) increased with increasing x , and the temperature coefficient of resonant frequency (τ_f) was adjusted from negative to positive, while the Q × f ₀ value increased significantly at first and reached an extreme value at x =0.025 and the maximum at x =0.15. The best combination of microwave dielectric characteristics were achieved at x =0.15 (ɛ_r=19.5, Q × f ₀=93 400 GHz, τ_f=−2 ppm/°C). The improvement of the Q × f ₀ value primarily originated from the reduced interlayer polarization with Ca/Ti co-substitution, while the decreased tolerance factor, the subsequent increased interlayer stress, and the appearance of CaTiO₃ secondary phase brought negative effects upon the Q × f ₀ 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.     

Some Results Concerning the Asymptotic Distribution of Sample Fourier Transforms and Periodograms for a Discrete-time Stationary Process with a Continuous Spectrum

We consider a stationary process ( X_t , t = 0, ±1, ...) with a continuous spectrum. Denote by D_n (λ) a tapered Fourier transform of ( X ₀, X ₁, ..., X _{n −1}) at (angular) frequency λ. We obtain the asymptotic distribution of D_n (λ) and the joint asymptotic distribution of { D_n (λ _j ), 1 ≤ j ≤ k } with continuity of the spectral density f (.) at the relevant frequencies as the only assumption concerning the second-order structure of ( X_t ); all other assumptions required are easily stated. The results are extended to processes for which f (.) is continuous except at λ = 0, with lim_λ←0 f (λ)λ^{2 d} = K , a constant, where 0 < d < ½, as is typical of certain types of processes with long-range dependence. Results for the sample periodogram, proportional to | D_n (λ)|², follow immediately.     

DISCRIMINANT ANALYSIS FOR STATIONARY VECTOR TIME SERIES

Abstract. In this paper, we shall consider the case where a stationary vector process { X_t } belongs to one of two categories described by two hypotheses π ₁ and π ₂. These hypotheses specify that { X_t } has spectral density matrices f (Λ) and g (Λ) under π ₁ and π ₂, respectively. Although Gaussianity of { X_t } is not assumed, we can formally make the Gaussian likelihood ratio (GLR) based on X (1),… X ( T ). Then an approximation I ( f : g ) of the GLR is given in terms of f (Λ) and g (Λ). If f (Λ) and g (Λ) are known, we can use I ( f : g ) as a classification statistic. It is shown that I ( f : g ) is a consistent classification criterion in the sense that the misclassification probabilities converge to zero as T →∝. When g is contiguous to f , we discuss non-Gaussian robustness of I ( f : g ). A sufficient condition for the non-Gaussian robustness will be given. Also a numerical example will be given.     

ON LINEAR PROCESSES WITH GIVEN MOMENTS

Abstract. Let X _t = c ₀ Y _t + c ₁ Y _{t -1}+… be a linear process with known coefficients c _k , where Y _t is a strict white noise. Let m ₁, …, m _2r be given numbers. A method is presented to determine whether there exists a distribution of Y _t such that EX ^k _t = m _k for k = 1, …, 2 r . In the positive case, such a distribution of Y _t is described. Some explicit formulas for AR(1) and AR(2) models are derived. The results can be used for simulating a process with given moments of its stationary distribution. The procedure also enables proof that some stationary distributions cannot belong to the given linear process.     

NON-PARAMETRIC APPROACH IN TIME SERIES ANALYSIS

Abstract. Suppose that { X _t } is a Gaussian stationary process with spectral density f ( Λ ). In this paper we consider the testing problem , where K (Λ) is an appropriate function and c is a given constant. This test setting is unexpectedly wide and can be applied to many problems in time series. For this problem we propose a test based on K { f _n ( Λ )} dΛ where f _n ( Λ ) is a non-parametric spectral estimator of f ( Λ ), and we evaluate the asymptotic power under a sequence of non-parametric contiguous alternatives. We compare the asymptotic power of our test with the other and show some good properties of our test. It is also shown that our testing problem can be applied to testing for independence. Finally some numerical studies are given for a sequence of exponential spectral alternatives. They confirm the theoretical results and the goodness of our test.     

Pu, U, and Hf Incorporation in Gd Silicate Apatite

Trivalent Pu can be incorporated in the silicate apatite structure to form Ca₂Pu₈(SiO₄)₆O₂ by sintering under reducing conditions, while the incorporation of tetravalent Pu in the Ca/rare earth sites in oxidizing or neutral conditions is limited to only 0.6 formula units (f.u.). The d -spacings and intensities of the X-ray pattern of hexagonally structured Ca₂Pu₈(SiO₄)₆O₂ after firing at 1250°C are given, and the a and c lattice parameters are 0.9561₁ and 0.7028₁ nm, respectively. The respective solid solubility limits of U and Hf in Ca₂Gd_{8– x} (U/Hf) _x (SiO₄)₆O₂ apatite samples were 0.3 and 0.2 f.u.     

Influence of Non-Stoichiometry on the Structure and Properties of Ba(Zn1/3Nb2/3)O3 Microwave Dielectrics: I. Substitution of Ba3W2O9

A narrow region of Zn-vacancy-containing cubic perovskites was formed in the (1− x )Ba₃(ZnNb₂)O₉−( x )Ba₃W₂O₉ system up to 2 mol% substitution ( x =0.02). The introduction of cation vacancies enhanced the stability of the 1:2 B-site ordered form of the structure, Ba(Zn_{1− x} □ _x )_1/3(Nb_{1− x} W _x )_2/3O₃, which underwent an order–disorder transition at 1410°C, ∼35° higher than pure Ba(Zn_1/3Nb_2/3)O₃. The Zn vacancies also accelerated the kinetics of the ordering reaction, and samples with x =0.006 comprised large ordered domains with a high lattice distortion ( c/a =1.226) after a 12 h anneal at 1300°C. The tungstate-containing solid solutions can be sintered to a high density at 1390°C, and the resultant ordered ceramics exhibit some of the highest microwave dielectric Q factors ( Q × f =1 18 000 at 8 GHz) reported for a niobate-based perovskite.     

Synthesis of Phase-Pure Pb(ZnxMg1–x)1/3Nb2/3O3 up to x= 0.7 from a Single Mixture via a Soft-Mechanochemical Route

Phase-pure perovskite Pb(Zn _x Mg_{1– x} )_1/3Nb_2/3O₃ solid solution (PZ _x M_{1– x} N) is obtained for x ≦ 0.7 by heating a milled stoichiometric mixture of PbO, Mg(OH)₂, Nb₂O₅, and 2ZnCO₃·3Zn(OH)₂·H₂O at 1100°C for 1 h. Percent perovskite ( f _P) with respect to total crystalline phase decreases with increasing temperature of subsequent heating then increases to 900°C for the mixtures where x ≦ 0.8 and milled for 3 h. For mixtures with x = 0.9 and x = 1, f _P decreases monotonically. Curie temperature increases almost linearly with increasing x up to x = 0.7. The maximum dielectric constant at 1 kHz is 2×10⁴ and 1.7×10⁴ for the mixture with x = 0.4 and x = 0.7, respectively. The stabilization mechanism of strained perovskite is discussed.     

Reduced Dielectric Loss of Modified ZnNb2O6 Ceramics by Substituting Nb5+ with Ta5+

The effects of substituting Nb⁵⁺ with Ta⁵⁺ on the microwave dielectric properties of the ZnNb₂O₆ ceramics were investigated in this study. The forming of Zn(Nb_{1− x} Ta _x )₂O₆ ( x =0–0.09) solid solution was confirmed by the measured lattice parameters and the EDX analysis. By increasing x , not only could the Q × f of the Zn(Nb_{1− x} Ta _x )₂O₆ ( x =0–0.09) solid solution be tremendously boosted from 83 600 GHz at x =0 to a maximum 152 000 GHz at x =0.05, the highest ɛ_r∼24.6 could also be achieved simultaneously. It was mainly due to the uniform grain morphology and the highest relative density of the specimen. A fine combination of microwave dielectric properties (ɛ_r∼24.6, Q × f ∼152 000 GHz at 8.83 GHz, τ_f∼–71.1 ppm/°C) was achieved for Zn(Nb_0.95Ta_0.05)₂O₆ solid solution sintered at 1175°C for 2 h.     

On the Spectral Density of the Wavelet Transform of Fractional Brownian Motion

We consider the wavelet transform { W _a ( t ), −∞ < t } < ∞}, at scale a > 0, of a fractional Brownian motion. A simple and mathematically rigorous proof is given to establish the existence of the spectral density f _Wa (λ) of the wavelet transform and provide an expression for it.     

TIME SERIES RESIDUALS WITH APPLICATION TO PROBABILITY DENSITY ESTIMATION

Abstract. A linear stationary and invertible process y _t models the second-order properties of T observations on a discrete time series, up to finitely many unknown parameters θ. Two estimators of the residuals or innovations ɛ _t of y _t are presented, based on a θ estimator which is root- T consistent with respect to a wide class of ɛ _t distributions, such as a Gaussian estimator. One sets unobserved y _t equal to their mean, the other treats y _t as a circulant and may be best computed via two passes of the fast Fourier transform. The convergence of both estimators to ɛ _t is investigated. We apply the estimated ɛ _t to estimate the probability density function of ɛ _t . Kernel density estimators are shown to converge uniformly in probability to the true density. A new sub-class of linear time series models is motivated.     

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.     

ON THE INVERTIBILITY OF MULTIVARIATE LINEAR PROCESSES

Abstract. It is shown that a multivariate linear stationary process whose coefficients are absolutely summable is invertible if and only if its spectral density is regular everywhere. This general characterization of invertibility is applied later to the case of a linear process having an autoregressive moving-average (ARMA) representation. Under the usual assumptions, it is deduced that a process Y described by an ARMA(φ, TH) model is invertible if and only if the polynomial detTH( z ) has no roots on the unit circle. Given an invertible process Y which has an ARMA representation, it is finally shown that the process YT , where YT , =ε _i =0^l S _i Y _t-i , is invertible if and only if the matrix S ( z ) =ε _{i =0}^l S _i z ⁱ is of full rank for all z of modulus 1. It follows, in particular, that any subprocess of an invertible ARMA process is also invertible.