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.     

RATE OF CONVERGENCE OF CENTRED ESTIMATES OF AUTOREGRESSIVE PARAMETERS FOR INFINITE VARIANCE AUTOREGRESSIONS

Abstract. Let Y n =μ+Σβ_j ( Y _n–j–μ) +ɛ_n be a p th order autoregressive process with innovations {ɛ_n} in the domain of attraction of a stable law with index α < 2. Hannan and Kanter (1977) showed that when the location parameter μ is known that least squares estimates of the autoregressive parameters have a very fast rate of convergence; specifically, N ^1/δ(β_j–β_j) → O almost surely for δ > α. It is shown here that if α is estimated by the sample mean, N _1/δ(β_j–β_j) → O almost surely for δ > max(1, α). In addition, some statements are made regarding estimators of α which will give the full (Hannan and Kanter) rate of convergence, in particular when α < 1.     

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.     

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 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.     

OPTIMALITY OF THE MAXIMUM LIKELIHOOD ESTIMATOR IN FIRST-ORDER AUTOREGRESSIVE PROCESSES

Abstract. Let ρρ* be the maximum likelihood estimator (MLE) of the parameter ρ in the first-order autoregressive process with normal errors. The problem of optimality in the sense of weighted squared error is considered rather than moments of asymptotic distributions. Many unbiased estimators can be constructed, but the two-dimensional sufficient statistic is incomplete. It is shown that dEρ* - ρ→ 0 uniformly in ρ, and that dE (ρ*)/ d ρ→ 1 for all |ρ| > 1. For |ρ| > 1, it is known that the asymptotic distribution of {I(ρ)}¹²⁽ρ* - ρ), where I (ρ) is the Fisher information, is Cauchy. It follows that the Cramèr-Rao inequality will not yield useful results for investigating the limit of exact efficiency of any asymptotically unbiased estimator, including the MLE. For all |ρ| < 1, ρ* is asymptotically optimal in the sense of minimizing expected weighted squared error. In addition, for |ρ| < 1, ρ* minimizes the variance asymptotically.     

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.     

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.     

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.     

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.     

Estimation in nonstationary random coefficient autoregressive models

Abstract.  We investigate the estimation of parameters in the random coefficient autoregressive (RCA) model X _k  = ( φ  +  b _k ) X _{k −1} +  e _k , where ( φ ,  ω ²,  σ ²) is the parameter of the process,     ,     . We consider a nonstationary RCA process satisfying E  log | φ  +  b ₀| ≥ 0 and show that σ ² cannot be estimated by the quasi-maximum likelihood method. The asymptotic normality of the quasi-maximum likelihood estimator for ( φ ,  ω ²) is proven so that the unit root problem does not exist in the RCA model.     

Electrical Resistivities of Monocrystalline and Polycrystalline TiB2

The electrical resistivity of monocrystalline and polycrystalline TiB₂, was measured under an inert atmosphere by a four-point ac impedance technique over the range 298 to 1373 K. The results are expressed in the form ρ-ρ₂₉₈= m(T -298). The following values of ρ₂₉₈ (μω.cm) and m (nω.cm-K^-1) were determined: for polycrystalline TiB₂ (69% dense) 18.2 and 95; for polycrystalline TiB₂ (99% dense) 7.4 and 42; and for monocrystalline TiB₂, 6.6 and 34.9.     

Sinterability of β-Calcium Orthophosphate Powder Prepared by Spray-Pyrolysis

Mixed solutions of Ca(NO₃)₂ and (NH₄)₂HPO₄ with Ca/P = 1.50 were spray-pyrolyzed at 600°C to produce β-calcium orthophosphate (β-Ca₃(PO₄)₂) powder; the spray-pyrolyzed powder was ground and then calcined at 600°C for 1 h. The best crystalline β-Ca₃(PO₄)₂ powder was obtained from the solution with 1.80 mol.L^–1 Ca(NO₃)₂, 1.20 mol.L^–1 (NH₄)₂HPO₄. The resulting powder was composed of primary particles with sizes of <0.5 μm. Dense β-Ca₃(PO₄)₂ ceramics with a relative density of 96.1% could be fabricated by firing this compressed powder at 1070°C for 5 h.     

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.     

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.     

Preparation of Porous Ca10(PO4)6(OH)2 and β-Ca3(PO4)2 Bioceramics

Submicrometer-sized, pure calcium hydroxyapatite (HA, (Ca₁₀(PO₄)₆(OH)₂)) and β-tricalcium phosphate (β-TCP, Ca₃(PO₄)₂) bioceramic powders, that have been synthesized via chemical precipitation techniques, were used in the preparation of aqueous slurries that contained methyl cellulose to manufacture porous (70%–95% porosity) HA or β-TCP ceramics. The pore sizes in HA bioceramics of this study were 200–400 μm, whereas those of β-TCP bioceramics were 100–300 μm. The pore morphology and total porosity of the HA and β-TCP samples were investigated via scanning electron microscopy, water absorption, and computerized tomography.     

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.     

Optical Properties of Dysprosium-Doped Low-Phonon-Energy Glasses for a Potential 1.3 μm Optical Amplifier

Dysprosium-doped glasses were prepared in the system of gallium-based sulfide, tellurite, zirconium-baed and indium-based fluorides and their optical properties were studied. From the absorption cross sections of five f-f bands, three Judd-Ofelt parameters, ω _t ( t = 2, 4, 6), of Dy³⁺ ion were determined. The compositional variaton of the ω₂value showed the order sulfide > tellurite > fluorozirconate > fluoroindate, whereas the ω₆ value showed the opposite tendency. Compositional variaton of the fluorescence intensity ratio of the (⁴F_9/2⁶H_13/2)/(⁴F_9/2)→⁶H_15/2) is explained by the ratio of ω₂/ω₆ of doped Dy³⁺. The emission probabilities A and the branching ratio β from ⁶H_9/2 and ⁶F_11/2 levels, which are the doublet initial level of the 1.3 μm luminescence, were calculated for the glasses, and it was found that both values showed a tendency similar to that of ω₂ against the glass composition. In the sulfide glass, A was 2.6 × 10³S^-1 and β was 93%, both the highest in all of the glasses investigated. By 1.06 μm pumping of a Nd: YAG laser, the sulfide glass showed strong 1.3 μm emission and the lifetime was 25 μs, resulting in a quantum efficiency of 7%. This value is higher than that of the Pr³⁺:¹G₄ level in ZBLAN glass with β= 60%.     

Effect of Al(OH)3 Addition on β-LiAlO2 Crystal Growth

Crystal growth of rod-shaped β-LiAlO₂ was previously reported by us, and the rod-shaped β-LiAlO₂ crystals were 1.5 μ in diameter and 10 to 15 μm long. In the present study needle-shaped β-LiAlO₂ crystals which were thinner and had larger aspect ratios (length/diameter) than the rodshaped β-LiAlO₂ crystals were grown by using LiOH–Al₂O₃–Al(OH)₃–NaOH as the raw material. These crystals were 0.7 to 1 μm in diameter, 9 to 13 μm long, and had aspect ratios of about 10 to 13.     

Zinc Vanadates in Vanadium Oxide-Doped Zinc Oxide Varistors

Convergent-beam electron diffraction has been used to determine the space groups of β- and γ-Zn₃(VO₄)₂ particles in vanadium oxide-doped zinc oxide varistors. The crystal structure of β-Zn₃(VO₄)₂ has been determined to be monoclinic with space group P 2₁ and lattice parameters of a = 9.80 Å, b = 8.34 Å, c = 10.27 Å, and β= 115.8°, whereas that of γ-Zn₃(VO₄)₂ is monoclinic with space group Cm and a = 10.40 Å, b = 8.59 Å, c = 9.44 Å, and β= 98.8°. Energy-dispersive X-ray microanalysis of these two phases shows significant deviations from their expected stoichiometry. It is apparent that the β-phase is, in fact, the metastable Zn₄V₂O₉ phase, whereas the γ-phase either is a new oxide that consists of zinc, vanadium, and manganese or, more likely, is a zinc vanadate phase with a Zn:V atomic ratio of 1:1 that has the ability to go into solid solution with manganese.