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.     

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 .     

Use of Crack Branching Data for Measuring Near-Surface Residual Stresses in Tempered Glass

An analytical procedure based on fracture mechanics is used to obtain the amount of residual stress in glass from measurements on the fracture surface. The technique utilizes the measurement of microcrack branching distances, known as the mirror — mist boundary, which occur at a critical crack branching stress intensity (K _m ) value. This procedurre shows that σ _A r _m ^1/2 Y _F (θ) =σ _R r ^1/2 _m +Ψ₀, where σ _A is the applied stress, r _m is the microcrack branching radius, σ _R is the residual stress, Y _F ( θ ) is the crack-border correction factor, and Ψ₀ is a material constant based on K _m . Thus, the equation is that of a straight line with the slope equal to the magnitude of the residual stress. Data for tempered glass from the literature are used to demonstrate the applicability of the technique.     

Kinetic Model for α–Tricalcium Phosphate Hydrolysis

A mechanistic model for the kinetics of hydrolysis of α-tricalcium phosphate (α–Ca₃(PO₄)₂ or α-TCP) to hydroxyapatite (Ca_{10− x} (HPO₄) _x (PO₄)_{6− x} (OH)_{2− x} or HAp) has been developed. The model is based on experimental hydrolysis rate data obtained using isothermal calorimetry. Analysis of the kinetic data according to the general kinetics models in terms of the fractional degree of reaction and time suggests the hydrolysis to be controlled by different rate-limiting mechanisms as reaction proceeds. Initially, the hydrolysis kinetics depend on the surface area of the anhydrous α-TCP. Subsequently, they change to a dependence on the rate of HAp product formation controlled by a nucleation and growth mechanism. The model predicts that HAp nuclei form at essentially one time and growth occurs in two dimensions, leading to a platelike morphology. The change in the reaction mechanism occurs at a fractional degree of hydrolysis, which does not change significantly with temperature in the range of 37°–56°C.     

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.     

Choice of thresholds for wavelet shrinkage estimate of the spectrum

We study the problem of estimating the log-spectrum of a stationary Gaussian time series by thresholding the empirical wavelet coefficients. We propose the use of thresholds t _{j , n} depending on sample size n , wavelet basis ψ and resolution level j . At fine resolution levels ( j = 1, 2, ...) we propose
t _{j , n} = α _j log n
where {α _j } are level-dependent constants and at coarse levels ( j ≫ 1)
t _{j , n} = (π/√3)(log n )^1/2.
The purpose of this thresholding level is to make the reconstructed log-spectrum as nearly noise-free as possible. In addition to being pleasant from a visual point of view, the noise-free character leads to attractive theoretical properties over a wide range of smoothness assumptions. Previous proposals set much smaller thresholds and did not enjoy these properties.     

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.     

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         

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.     

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.     

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.     

New MAX-Phase Compounds in the V–Cr–Al–C System

New layered compounds, (V_0.5Cr_0.5)₃AlC₂, (V_0.5Cr_0.5)₄AlC₃, and (V_0.5Cr_0.5)₅Al₂C₃ were synthesized by reactive hot pressing V, Cr, Al, and graphite powders. The crystal structures of these new phases were determined using a combination of X-ray diffraction and scanning transmission electron microscopy. (V_0.5Cr_0.5)₃AlC₂ is isotypical to Ti₃AlC₂; while (V_0.5Cr_0.5)₄AlC₃ has the Ti₄AlN₃ or α-Ta₄AlC₃-type crystal structure. (V_0.5Cr_0.5)₅Al₂C₃ is formed by periodically stacking of half-unit cells of (V_0.5Cr_0.5)₂AlC and (V_0.5Cr_0.5)₃AlC₂.     

State-space Models with Finite Dimensional Dependence

We consider nonlinear state-space models, where the state variable (ζ _t ) is Markov, stationary and features finite dimensional dependence (FDD), i.e. admits a transition function of the type: π(ζ _t |ζ _{t −1}) =π(ζ _t ) a '(ζ _t ) b (ζ _{t −1}), where π(ζ _t ) denotes the marginal distribution of ζ _t , with a finite number of cross-effects between the present and past values. We discuss various characterizations of the FDD condition in terms of the predictor space and nonlinear canonical decomposition. The FDD models are shown to admit explicit recursive formulas for filtering and smoothing of the observable process, that arise as an extension of the Kitagawa approach. The filtering and smoothing algorithms are given in the paper.
JEL. C4.     

Thermochemistry of the Aluminas and Aluminum Trihalides

Consistent standard free energies of formation of gibbsite, bayerite, boehmite, and diaspore; their respective transition aluminas, α-Al₂O₃; and AlF₃, AlCl₃, H₂O₃ HF, and HCl are compiled from 298.16 to 2100 K from literature review, computations, and estimates. Significant adjustments and additions to earlier compilations are included. Revised analysis is made of the gibbsite-to-α-A1₂O₃ transition series and of reactions of appropriate aluminas with HF and HCl, comparing them with experimental data. These updated Δ G ° tables should also yield accurate ΔG° data for many other alumina reactions, e.g. with SiO₂, M₂O, MO, etc.     

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.     

Thermal Expansion, Compressibility, and Polymorphism in Hafnium and Zirconium Titanates

Using X-ray diffraction techniques, thermal expansion and compressibility were measured on the orthorhombic compounds HfTiO₄, Hf_1.26Ti_0.74O₄, and ZrTiO₄ (both quenched and cooled slowly from 1300°C). The thermal expansion of HfTiO₄ is highly anisotropic; the thermal expansion coefficients along the crystallographic axes are α _a =+(8.7±0.5)×10⁻⁶°C⁻¹, α _b =−(5.2±0.5)×10⁻⁶°C⁻¹, and α _c =+ (5.3±0.5)×10⁻⁶°C⁻¹. The thermal expansion of Hf_1.26Ti_0.74O₄ was similar to that of HfTiO₄ but that of ZrTiO₄ was markedly less anisotropic. The compressibilities of HfTiO₄ and ZrTiO₄ also differed markedly. All compounds investigated, however, behaved similarly in exhibiting a polymorphic transition to a high-pressure phase having the monoclinic baddeleyite (ZrO₂) structure. The polymorphism can be explained qualitatively on the basis of crystal structure.     

Ca Segregation to Basal Surfaces in α-Alumina

High-resolution transmission electron microscopy (HRTEM) and analytical electron microscopy (AEM) were used to investigate the structure and chemistry of (0001) α-A1₂O₃/ A1 interfaces in melt-infiltrated polycrystalline alumina composites. HRTEM revealed an interfacial region different from both Al and α-A1₂O₃, with a structural width of 0.8 ± 0.2 nm. AEM of the same interfaces revealed a Ca excess of Γ = 2.5 ± 0.5 Ca atoms per nm² (Ca/nm²). AEM of a basal twin boundary in the α-A1₂O₃ also revealed a Ca excess (Γ= 1.0 ± 0.5 Ca/nm²). Since the metal-ceramic interfaces were the free surfaces of pores before melt infiltration, it can be concluded that Ca segregates to the basal surface of alumina, as well as to basal twin boundaries. Furthermore, the Ca at the free surfaces does not reside on only one cation plane, but is spread over 4 ± 1 basal cation layers and forms an interfacial phase with a nominal composition of CaO-6A1₂O₃.     

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.     

Thermal Expansion of the Low-Temperature Form of BaB2O4

Thermal expansion of the low-temperature form of BaB₂O₄ (β-BaB₂O₄) crystal has been measured along the principal crystallographic directions over a temperature range of 9° to 874°C by means of high-temperature X-ray powder diffraction. This crystal belongs to the trigonal system and exhibits strongly anisotropic thermal expansions. The expansion along the c axis is from 12.720 to 13.214 Å (1.2720 to 1.3214 nm), whereas it is from 12.531 to 12.578 Å (1.2531 to 1.2578 nm) along the a axis. The expansions are nonlinear. The coefficients A, B , and C in the expansion formula L _t = L ₀(1 + At + Bt ²+ Ct ³) are given as follows: a axis, A = 1.535 × 10⁻⁷, B = 6.047 × 10⁻⁹, C = -1.261 × 10⁻¹²; c axis, A = 3.256 × 10⁻⁵, B = 1.341 × 10⁻⁸, C = -1.954 × 10⁻¹²; and cell volume V, A = 3.107 × 10⁻⁵, B = 3.406 × 10⁻⁸, C = -1.197 × 10⁻¹¹. Based on α _t = (d L _t /d t )/ L ₀, the thermal expansion coefficients are also given as a function of temperature for the crystallographic axes a , c , and cell volume V.