AUTOREGRESSIVE PROCESSES WITH NORMAL STATIONARY DISTRIBUTIONS

Abstract. For the strictly stationary AR( k ) process Z _t = Λ ( Z _{t -1}) + α _t , with Λ : R ^k → R , Z _{t -1}= [ Z _{t -1}, Z _{t -2},…, Z _t-k ] and { α _t } an independent identically distributed white noise process, we partially characterize the Λ for which the stationary distribution of Z _t is normal.     

Anisotropic Thermal Expansion of β-Ca2SiO4 Monoclinic Crystal

Crystals of β-Ca₂SiO₄ (space group P 12₁/ n 1) were examined by high-temperature powder X-ray diffractometry to determine the change in unit-cell dimensions with temperature up to 645°C. The temperature dependence of the principal expansion coefficients (α_i) found from the matrix algebra analysis was as follows: α₁= 20.492 × 10⁻⁶+ 16.490 × 10⁻⁹ ( T - 25)°C⁻¹, α₂= 7.494 × 10⁻⁶+ 5.168 × 10⁻⁹( T - 25)°C⁻¹, α₃=−0.842 × 10⁻⁶− 1.497 × 10⁻⁹( T - 25)°C⁻¹. The expansion coefficient α₁, nearly along [302] was approximately 3 times α₂ along the b -axis. Very small contraction (α₃) occurred nearly along [     01]. The volume changes upon martensitic transformations of β↔α_L' were very small, and the strain accommodation would be almost complete. This is consistent with the thermoelasticity.     

ASYMPTOTIC EXPANSIONS FOR THE DISTRIBUTIONS OF SERIAL CORRELATIONS

Abstract. Let X ₁, …, X _n be a random sample from a population with a distribution function F and let E ( X ₁) = 0, E ( X ₁²) < ∞. Let r ₁=Σ _{t =1} ^{n -1} X _t X _{t +1}/Σ _{t =1} ^{n -1}( X _t ²+ X _{t +1}²). We derive a proper Edgeworth type expansion for the sampling distribution of r ₁ under the assumption that F is a mixture of Gaussian distributions of one of two given types. The result can easily be extended to the sampling distributions of serial correlations of arbitrary lag s .     

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.     

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.     

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         

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.     

Preparation and Luminescence Spectra of Calcium- and Rare-Earth (R = Eu, Tb, and Pr)-Codoped α-SiAlON Ceramics

Rare-earth-doped oxynitride or nitride compounds have been reported to be luminescent and may then serve as new phosphors with good thermal and chemical stabilities. In this work, we report the photoluminescence (PL) spectra of europium-, terbium-, and praseodymium-doped Ca-α-SiAlON ceramics. The highly dense ceramics were prepared by hot pressing at 1750°C for 1 h under 20 MPa in a nitrogen atmosphere. Europium-doped Ca-α-SiAlON displayed a single broad emission band peaking at λ= 550–590 nm depending on the europium concentration. The emission bands in the spectra of europium-doped Ca-α-SiAlONs were assigned to the allowed transition of Eu²⁺ from the lowest crystal field component of 4 f ⁶5 d to ⁸S_7/2 (4 f ⁷) ground-state level. The emission spectra of terbium- and praseodymium-doped Ca-α-SiAlON ceramics both consisted of several sharp lines, which were attributed to the ⁵D₄→⁷F _j ( j = 3, 4, 5, 6) transitions of Tb³⁺ and ³P₀→³H _j ( j = 3, 4, 5) transitions of Pr³⁺, respectively. In particular, the terbium-doped α-SiAlON ceramics showed a strong green emission among these phosphors.     

Anisotropic Thermal Expansion of Tetragonal Zirconia Polycrystals

Thermal expansion coefficients (α _a and α _c ) in two crystallographic axes ( a and c ) of the tetragonal phase are measured at 25°–1200°C in ZrO₂–M₂O₃ (M = Sc, In, Yb) and in ZrO₂–YTaO₄. The difference between these two thermal expansion coefficients, α _c –α _a , decreases with M₂O₃ or YTaO₄ composition even though the tetragonality ( c/a ) behaves differently in these two systems. The locus of α _c =α _a represents a maximum tetragonality for the tetragonal phase, but not the phase boundary for the cubic phase. The relationships among thermal expansion, temperature, and composition are discussed.     

A Median-Unbiased Estimator of the AR(1) Coefficient

A proof is given that the median of the ratios of consecutive observations of a stationary first-order autoregressive process X_t = α X _{t −1} + Y_t with P ( Y_t ≥ 0) = P ( Y_t ≤ 0) = 1/2 and P ( X_t = 0) = 0 is a median-unbiased estimator of α.     

Calorimetric Study of Nickel Molybdate: Heat Capacity, Enthalpy, and Gibbs Energy of Formation

The thermodynamic properties of the α and β polymorphs of NiMoO₄ were directly investigated by calorimetry. The standard entropies of formation, Δ_f S ° _T , of α and β were determined from measuring the molar heat capacity, C _p,m, from near absolute zero (2 K) to high temperature (1380 K) by a relaxation method and differential scanning calorimetry. The standard enthalpies of formation, Δ_f H ° _T , of α and β were determined by combining C _p,m with the standard enthalpy of formation, Δ_f H °₂₉₈, at 298 K obtained from drop solution calorimetry in molten sodium molybdate at 973 K. The standard Gibbs energies of formation, Δ_f G ° _T , of α and β were determined from their Δ _f S ° _T and Δ _f H ° _T values. The Δ _f G ° _T values indicate that the polymorphic transformation from α to β occurs at 1000 K, consistent with the observed phase transformation at 1000 K.     

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.     

TESTING FOR CYCLICAL NON-STATIONARITY IN AUTOREGRESSIVE PROCESSES

This paper deals with the distributions evolving from the likelihood-ratio test for the factor 1 − B ⁿ in the lag polynomial Φ( B ) under the basic assumption that the data series is generated by the autoregressive model Φ( B ) X _t = ε _t where {ε _t } denotes Gaussian white noise. A characterization of the statistic and its asymptotic properties is given. Asymptotic and finite-sample significance points are tabulated. The test procedure is illustrated by an economics example.     

Numerical Data for Some Commonly Used Solid State Reaction Equations

Many solid state reactions can be represented by equations of the type F (α) = kt , where α is the fraction of material reacted in time, t. These equations can be expressed in the form F (α) = A ( t/t _0.5) where t _0.5 is the time for 50% reaction and A is a calculable constant depending on the form of F (α). Numerical tables are given of F (α) in relation to α, and to ( t/t _0.5), for nine equations corresponding to reactions which are diffusion controlled, or are reaction-rate controlled, or obey first order kinetics, or follow the equations of Avrami and Erofe'ev. The application of the tables to the analysis of experimental data is described.     

New Approach to the β→α Polymorphic Transformation in Magnesium-Substituted Tricalcium Phosphate and its Practical Implications

The transformation β→α in Mg-substituted Ca₃(PO₄)₂ was studied. The results obtained showed that, contrary to common belief, there is, in the system Mg₃(PO₄)₂–Ca₃(PO₄)₂, a binary phase field where β+α-Ca₃(PO₄)₂ solid solutions coexist. This binary field lies between the single-phase fields of β- and α-Ca₃(PO₄)₂ solid solution in the Ca₃(PO₄)₂-rich zone of the mentioned system. In the light of the results and the Palatnik–Landau's Contact Rule of Phase Regions, a corrected phase equilibrium diagram has been proposed. The practical implications of these findings with regard to the synthesis of pure α- and β- Mg-substituted Ca₃(PO₄)₂ powders and to the sintering of related bioceramics with improved mechanical properties are pointed out.     

THIRD-ORDER ASYMPTOTIC PROPERTIES OF ESTIMATORS IN GAUSSIAN ARMA PROCESSES WITH UNKNOWN MEAN

Abstract. This paper deals with the third-order asymptotic theory for Gaussian autoregressive moving-average (ARMA) processes with unknown mean μ. We are interested in the estimation of ρ = ( α₁…, α_p, β₁…, β_q ), where α ₁…, α_ρ and β ₁…, β_q are the coefficients of the autoregressive part and the moving-average part, respectively. First, we investigate the third-order asymptotic optimality of the bias adjusted maximum likelihood estimator (MLE) of ρ in the presence of the nuisance parameters μ and ² (innovation variance). Next, for a Gaussian AR(1μ μ, ²), we propose a mean corrected estimator α_c1c2 of the autoregressive coefficient. We make a comparison between the bias adjusted estimator α_c1c2* and the bias adjusted MLE, in terms of their probabilities of concentration around the true value, or equivalently, in terms of their mean squared errors. Finally some numerical studies are provided in order to verify the third-order asymptotic theory.     

Estimation and testing for the parameters of ARCH(q) under ordered restriction

Abstract.  In this paper, we study a stationary ARCH( q ) model with parameters α ₀, α ₁, α ₂,…, α _q . It is known that the model requires all parameters α _i to be non-negative, but sometimes the usual algorithm based on Newton–Raphson's method leads us to obtain some negative solutions. So this study proposes a method of computing the maximum likelihood estimator (MLE) of parameters under the non-negative restriction. A similar method is also proposed for the case where the parameters are restricted by a simple order: α ₁≥ α ₂≥⋯≥ α _p . The strong consistency of the above two estimators is discussed. Furthermore, we consider the problem of testing homogeneity of parameters against the simple order restriction. We give the likelihood ratio (LR) test statistic for the testing problem and derive its asymptotic null distribution.     

The System Zirconia-Scandia

The system zirconia-scandia was investigated using X-ray diffraction analysis, differential thermal analysis, metallographic analysis, and melting point studies. Results reveal the monoclinic α₁ phase (0 to 2 mol% Sc₂O₃), the tetragonal α₂'phase (5 to 8% Sc₂O₃), the rhombohedral β phase (9 to 13% Sc₂O₃), the rhombohedral γ phase (15 to 23% Sc₂O₃), the rhombohedral δ phase (24 to 40% Sc₂O₃), and the cubic % phase (77.5 to 100% Sc₂O₃). The monoclinic α₁ phase and the tetragonal α₂'phase were found to transform to the tetragonal α₂ phase over a wide temperature range depending on composition. The β, γ, and α phases transformed to a cubic phase at temperatures of %600^%, 1100^%, and 1300^%C, respectively. A maximum melting point of %2870^%C was found at %10% Sc₂O₃ and a eutectic at %2400^%C at 55% Sc₂O₃.     

Interpretation of the Parabolic and Nonparabolic Oxidation Behavior of Silicon Oxynitride

The oxidation process of Si₂N₂O, prepared by a hot isostatic pressing technique, has been studied by the thermogravimetric method. The oxidation has been performed in oxygen for 20 h in the temperature range 1300° to 1600°C, producing oxide scales of amorphous SiO₂ and α-cristobalite. The weight gain for T 1350°C does not begin to follow a parabolic rate law, until a certain time, t ₀. The A ₀ parameter in the parabolic rate law, (Δ w / A ₀)²= K _p t + B , represents the cross section area, A , through which the oxygen diffuses; in the derivation of this law A is assumed to be constant during the experiment. If crystallization occurs during the oxidation process, A will decrease with time. A function, A ( t ), describing the time dependence, has been developed and incorporated into the parabolic rate law, yielding a new rate law, which reads Δ W/A ₀= a arctan √ bt + c √ t . This new rate law is valid in the time interval t < t ₀, whereas, for t > t ₀, the oxidation process follows the equation (Δ w/A ₀)²= K °_p t + B ₀. The relation of the latter equation to the common parabolic rate law is described. All of the oxidation curves are described by these equations. The activation energy of the oxygen diffusion (and of the oxidation ( K _p)) is found to be 245 ± 25 kJ/mol, which is consistent with literature values reported for oxygen diffusion.     

ASYMPTOTICS FOR THE LOW-FREQUENCY ORDINATES OF THE PERIODOGRAM OF A LONG-MEMORY TIME SERIES

Abstract. We consider the asymptotic distribution of the normalized periodogram ordinates I(ω_j)/f(ω_j) ( j = 1,2,…) of a general long-memory time series. Here, I (ω;) is the periodogram based on a sample size n , f (ω) is the spectral density and ω_j= 2π j/n. We assume that n →∝ with j held fixed, and so our focus is on low frequencies; these are the most important frequencies for the periodogram-based estimation of the memory parameter d. Contrary to popular belief, the normalized periodogram ordinates obtained from a Gaussian process are asymptotically neither independent identically distributed nor exponentially distributed. In fact, lim n E{I(ω_j)/f(ω_j)} depends on both j and d and is typically greater than unity, implying a positive asymptotic relative bias in I(ω_j) as an estimator of f(ω_j). Tapering is found to reduce this bias dramatically, except at frequency ω₁. The asymptotic distribution of I(ω_j)/f(ω_j) for a Gaussian process is, in general, that of an unequally weighted linear combination of two independent X²₁ random variables. The asymptotic mean of the log normalized periodogram depends on j and d and is not in general equal to the negative of Euler's constant, as is commonly assumed. Consequently, the regression estimator of d proposed by Geweke and Porter-Hudak will be asymptotically biased if the number of frequencies used in the regression is held fixed as n →∝.