The adjustment of prediction intervals to account for errors in parameter estimation

Abstract.  Standard approximate 1 −  α prediction intervals (PIs) need to be adjusted to take account of the error in estimating the parameters. This adjustment may be aimed at setting the (unconditional) probability that the PI includes the value being predicted equal to 1 −  α . Alternatively, this adjustment may be aimed at setting the probability that the PI includes the value being predicted equal to 1 −  α , conditional on an appropriate statistic T . For an autoregressive process of order p , it has been suggested that T consist of the last p observations. We provide a new criterion by which both forms of adjustment can be compared on an equal footing. This new criterion of performance is the closeness of the coverage probability, conditional on all of the data, of the adjusted PI and 1 −  α . In this paper, we measure this closeness by the mean square of the difference between this conditional coverage probability and 1 −  α . We illustrate the application of this new criterion to a Gaussian zero-mean autoregressive process of order 1 and one-step-ahead prediction. For this example, this comparison shows that the adjustment which is aimed at setting the coverage probability equal to 1 −  α conditional on the last observation is the better of the two adjustments.     

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.     

The High-Temperature Proton Conductor Strontium Zirconate: Thermogravimetry of Water Uptake

The water uptake of nominally 5%-Yb-doped strontium zirconate (SrZr_0.95Yb_0.05O_3−α) was studied in situ by thermogravimetry at different temperatures and water vapor pressures. The process could be described in terms of an ideal solution behavior, showing a lower effective vacancy concentration than that expected from the nominal dopant content. The solution reaction was found to have ΔS°=−127 J/(mol·K) and ΔH°=−106 kJ/mol.     

Relaxation Dynamics in Superionic Glass Nanocomposites

We have studied the structural and electrical properties of several glass nanocomposite systems. The formation of nanowires, nanoparticles, and nanorods embedded in heat-treated x Ag₂O–(1− x )MoO₃ ( x =0.2, 0.3, and 0.4) glass matrix and the growth of α-AgI nanocrystals and ZnO nanoparticles in the 0.7AgI–0.15Ag₂O–0.15( x ZnO–(1− x )MoO₃) ( x =0.05, 0.20, and 0.30) glass matrix and α-AgI nanocrystals in the x AgI–(1− x )(0.5Ag₂O–0.5MoO₃) ( x =0.70, 0.75, and 0.80) glass matrices have been confirmed from X-ray diffraction and high-resolution transmission electron microscopic studies. We have investigated the electrical conductivity of these glass nanocomposites in a wide frequency and temperature range, and correlated them with their structures. We have analyzed the ac conductivity using a power law model. It has been observed that mobile ion concentration is independent of composition as well as temperature. It was also observed that the conductivity depends on the size of the nanoparticles grown in the host glass matrix. The scaling of the conductivity spectra reveals that the relaxation dynamics of Ag⁺ ions is independent of temperature and is also independent of composition at higher frequencies.     

β→α Transformation in Polycrystalline Sic: II, Interfacial Energetics

The β→α transformation in polycrystalline Sic occurs by the rapid growth of composite grains consisting of α-SiC plates "sandwiched" between grains of recrystallizedp-SiC. Growth of these composite grains into the fine-grained β matrix occurs much more rapidly than thickening of the α plates into their β"envelopes." A phenomenological analysis of the energetics of these several growth processes is presented and it is shown that the observations can be explained by the extreme anisotropy of the interfacial energy between β− and α−SiC; {lll}_β(0001)^α interfaces have much lower energies than random, β/α interfaces. In reaction-sintered Sic, this anisotropy is manifested by rapid growth of a seeds along their basal planes into the epitaxial β reaction product; slow growth occurs perpendicular to the basal planes.     

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.     

Phase Assemblages of Pr α-sialon Derived from SHS-ed Powders and TEM Study on the Nucleation of Pr α-sialon

The Pr α-sialon powders prepared by self-propagating high-temperature synthesis (SHS), consisting of 55 wt% Pr α-sialon and 45 wt% of β-sialon (abbreviated as α' and β'), were hot-pressed at 1800°C for 1 h. The results showed that Pr α' phase would transfer to β' with the appearance of JEM phase (Pr(Si_{6− z} Al _z )(N_{10− z} O _z )) after sintering, thus resulting in the increase of β' phase to 86 wt%. The addition of Y₂O₃ into SHS-ed Pr α' powders as the starting materials restrains the transformation of α' to β' and prevents the formation of JEM phase as well. The nucleation mechanism of Pr α' grain during hot-pressing was investigated in terms of transmission electron microscope and energy-dispersive spectrometer analysis. Two nucleation modes of Pr α' grains were found, i.e., nucleating on the undissolved Pr α' grains and on the nuclei of (Pr, Y) α' grains precipitated from liquid phase.     

Duplex α,β-Sialon Ceramics Stabilized by Dysprosium and Samarium

Duplex αβ,-sialon ceramics with a minimum volume fraction of residual intergranular glass have been prepared using Dy or Sm as the α-sialon stabilizing element. These microstructures contained high aspect ratio β-sialon grains homogeneously distributed in an α-sialon matrix. A number of the larger α-sialon grains contained dislocations and showed a core/shell structure. Dy gave an α-sialon which was stable over a wide temperature range (1350–1800°C) for long holding times, while the use of Sm resulted in less stable α-sialon structures at medium temperatures (1450°C) and the formation of melilite, R₂Si_3−xAl_xO_3+xN_4−x, β-sialon, and the 21R sialon polytype during prolonged heating. High α-phase contents gave a very high hardness ( H _V10 is approximately 22 GPa) but a comparatively low indentation fracture toughness (around 4.4 MPam^1/2). Duplex sialons fabricated from powder mixtures corresponding to an α-to-β sialon ratio of around 50:50 resulted in a sialon material with a favorable combination of high hardness (around 22 GPa) and increased toughness (to around 5.5 MPam^1/2).     

Stability and Oxidation Properties of RE-α-Sialon Ceramics (RE = Y, Nd, Sm, Yb)

Oxidation studies of hot-pressed RE-α-sialons, RE _x -Si_{12-4.5 x} Al_{4.5 x} O_{1.5 x} N_{16-1.5 x} (with x = 0.40 for RE = Nd, Sm, Yb; and x = 0.48 for RE = Y) were carried out in oxygen in a TG apparatus for ca. 20 h. Very good oxidation resistance was found for the Yb-doped samples, with parabolic rate constants K _p similar/congruent 0.09 10^-6-3 10^-6 mg²cm^-4s^-1 in the temperature range 1250-1350°C. The promising performance of this material was corroborated by long-term oxidation experiments (5 days) in air at 1350°C. Although the oxidation kinetics can be described by simple equations related to the parabolic rate law (e.g., the arctan equation, Δ W / A ₀=α arctan bt + c t ), the oxidation process in these materials is likely to be complex. The significantly lower oxidation resistance of the RE = Nd, Sm doped α-sialons, especially at higher temperatures, is related to the formation of melilite, RE₂Si_{3− y} Al _y O_{3+ y} N_{4− y} ( y ∼ 1), in these systems. The melilite phase is also responsible for the thermal instability of the Nd- and Sm-α-sialons.     

Zirconia-Related Phases in the Zirconia/Titanium Diffusion Couple After Annealing at 1100°–1550°C

A diffusion couple of 3 mol% Y₂O₃–ZrO₂ and titanium was isothermally annealed in argon at temperatures between 1100° and 1550°C. The phases and microstructure in the ceramic side were investigated using scanning electron microscopy and transmission electron microscopy, both attached to an energy-dispersive spectrometer. After annealing at 1100°C/6 h, zirconia grains did not grow conspicuously and evolved only traces of oxygen, resulting in t -ZrO_{2− x} but not α-Zr. At temperatures above 1300°C, a significant amount of oxygen evolved from zirconia, reducing the O/Zr ratio, such that α-Zr was excluded from t -ZrO_{2− x} during cooling, yielding a higher O/Zr ratio (≈2). When held at 1550°C/6 h, zirconia grains grew rapidly. The α-Zr was segregated on grain boundaries during cooling by the exsolution of zirconium from ZrO_{2− x} , while twinned t '-ZrO_{2− x} or lenticular t -ZrO_{2− x} , which was embedded in ordered c- ZrO_{2− x} , was found. The ordered c -ZrO_{2− x} was identified by the     {113} superlattice reflections of its electron diffraction patterns.     

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.     

Dimensionless Load—Displacement Relation and Its Application to Crack Propagation Problems

The dimensionless load (     ) and load-point displacement (û) are introduced to discuss crack propagation problems. The     −ũ relation for the case of equilibrium crack propagation in any linear elastic material is demonstrated to lie on a single universal fracture curve independent of the material. The concept of the universal     −ũ relation is extended to yield a direct and simple form of the dimensionless total energy (     _eq (α)) which is applicable to many types of instability problems for crack extension, when the shape factor ( Y ) of the specimen is known as a function of crack length ( a ). In addition, by applying the deviation of the experimentally observed     —ũ curve and the theoretical universal fracture curve, the evaluation of the nonlinear fracture resistance parameter of a polycrystalline graphite material, as an example, was demonstrated.     

Effects of Annealing Temperature on Microstructural Development at the Interface Between Zirconia and Titanium

The interfacial reaction layers in the Ti/ZrO₂ diffusion couples, isothermally annealed in argon at temperatures ranging from 1100° to 1550°C for 6 h, were characterized using scanning electron microscopy and transmission electron microscopy, both attached with an energy-dispersive spectrometer. Very limited reaction occurred between Ti and ZrO₂ at 1100°C. A β'-Ti(Zr, O) layer and a two-phase α-Ti(O)+β'-Ti(Zr, O) layer were found in the titanium side after annealing at T ≥1300°C and T ≥1400°C, respectively. A three-phase layer, consisting of Ti₂ZrO+α-Ti(O, Zr)+β'-Ti (O, Zr), was formed after annealing at 1550°C. In the zirconia side near the original interface, β'-Ti coexisted with fine spherical c- ZrO_{2− x} , which dissolved a significant amount of Y₂O₃ in solid solution at T ≥1300°C. Further into the ceramic side, the α-Zr was formed due to the exsolution of Zr out of the metastable ZrO_{2− x} after annealing at T ≥1300°C: the α-Zr was very fine and dense at 1300°C, continuously distributed along grain boundaries at 1400°C, and became coarsened at 1550°C. Zirconia grains grew significantly at T ≥1400°C, with the lenticular t -ZrO_{2− x} being precipitated in c -ZrO_{2− x} . Finally, the microstructural development and diffusion paths in the Ti/ZrO₂ diffusion couples annealed at various temperatures were also described with the aid of the Ti–Zr–O ternary phase diagram.     

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.     

Self-Reinforced Nitrogen-Rich Calcium–α-SiAlON Ceramics

Nitrogen-rich Ca–α-SiAlON ceramics with nominal compositions Ca _x Si_{12−2 x} Al_{2 x} N₁₆ and 0.2≤ x ≤2.6, extending along the Si₃N₄–1/2Ca₃N₂:3AlN tie line, were prepared from Si₃N₄, AlN, and CaH₂ precursors by hot pressing at 1800°C. The x values attained were determined by energy-dispersive X-ray (EDX) microanalysis and X-ray powder diffraction (XRPD) data using the Rietveld method. The results show that Ca–α-SiAlONs form continuously within the compositional range x =0 to at least x =1.82. Phase assemblages, lattice parameters, Vickers hardness, and fracture toughness were determined and correlated to the calcium content, x . Owing to a high sintering temperature and the use of CaH₂ as a precursor, grain growth was kinetically enhanced, resulting in self-reinforced microstructures with elongated grains. The obtained Ca–α-SiAlON ceramics demonstrate a combination of both high hardness ∼21 GPa, and high fracture toughness ∼5.5 MPa·m^1/2.     

Thermal Expansion Anisotropy of Ceria-Stabilized Tetragonal Zirconia

Ceria-stabilized tetragonal zirconia polycrystals were obtained by thermal treatment of amorphous powder prepared by the sol–gel method. Detailed XRD profile analysis was employed to study microstructural disorder and crystallite size and shape; in particular, no fluctuation of stoichiometry was found, the main cause of disorder being attributable to dislocations. Thermal expansion measurements were carried out by high-temperature XRD at 294, 473, 673, 873, and 1073 K using silicon as an internal standard. Thermal expansion coefficients are anisotropic and changes in the stabilizer content have little effect on them. A mean value, α _a = 10.6 × 10⁻⁶ (K⁻¹) and α _c = 13.5 × 10⁻⁶ (K⁻¹), can be assumed for Zr_{1− x} Ce _x O₂ with x in the range 0.12–0.18.     

Temperature Dependent Luminescence of Yellow-Emitting α-Sialon:Eu2+ Oxynitride Phosphors for White Light-Emitting Diodes

Yellow-emitting α-sialon:Eu²⁺ phosphors have been reported as interesting down-conversion luminescent materials in white LEDs. In this work, the thermal quenching of α-sialon:Eu²⁺ with the compositions of M^val+_{( m /val+)}Si_{12−( m + n )}Al _{m + n} O _n N_{16− n} (M=Ca, Mg, Lu) is studied by investigating the effects of chemical composition, activator concentration, and substitution cation on the temperature-dependent luminescence. The chemical composition of α-sialon:Eu²⁺ was varied in a wide range (0.5≤ m ≤2.0, 1≤ n ≤1.8). It shows that the m value significantly affects the thermal quenching of α-sialon, whereas the n value hardly does. This difference is ascribed to the obvious lattice expansion and the increase of absolute activator concentration as m increases. The thermal quenching increases with increasing the Eu²⁺ concentration, which is due to enhanced Stokes shift. The type of substitution cation also has an influence on thermal quenching. Among the substitution cations in this work, Lu-α-sialon:Eu²⁺ exhibits largest thermal quenching. Photoionization is considered as the mechanism for the thermal quenching of Lu-α-sialon: Eu²⁺.     

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.     

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.