On Prediction Intervals for Conditionally Heteroscedastic Processes

Kabaila (1999) argues that the standard 1−α prediction intervals for a broad class of conditionally heteroscedastic processes are justified by their possession of what he calls the 'relevance property'. He considers both the case that the parameters of the process are known and that these parameters are unknown. We consider the former case and ask whether these prediction intervals can, alternatively, be deduced from the requirements of both (a) unconditional coverage probability 1−α and (b) minimum unconditional expected length. We show that the answer to this question is no, by presenting a counterexample. This counterexample concerns the standard 95% one-step-ahead prediction interval in the context of a simple Markovian bilinear process.     

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.     

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.     

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.     

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.     

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.     

Integer‐Valued Autoregressive Models With Survival Probability Driven By A Stochastic Recurrence Equation

This paper proposes a new class of integer‐valued autoregressive models with a dynamic survival probability. The peculiarity of this class of models lies in the specification of the survival probability through a stochastic recurrence equation. The proposed models can effectively capture changing dependence over time and enhance both the in‐sample and out‐of‐sample performance of integer‐valued autoregressive models. This point is illustrated through an empirical application to a real‐time series of crime reports. Additionally, this paper discusses the reliability of likelihood‐based inference for the class of models. In particular, this study proves the consistency of the maximum likelihood estimator and a plug‐in estimator for the conditional probability mass function in a misspecified model setting.     

On the closed form of the covariance matrix and its inverse of the causal ARMA process

Abstract.  Derivation of the theoretical autocovariance function of a causal autoregressive moving-average process of order ( p ,  q ), ARMA( p ,  q ), when q  ≥ 1 is considered. A recursive relationship is established between the covariance matrices of an ARMA( p ,  q ) process and its associated ARMA( p ,  q −1) process. The obtained recursion is shown to produce the inverse of the covariance matrix and its determinant. Moreover, the introduced method can be easily implemented in any programming environment.     

A simple procedure for computing improved prediction intervals for autoregressive models

Abstract.  This article concerns the construction of prediction intervals for time series models. The estimative or plug-in solution is usually not entirely adequate, since the (conditional) coverage probability may differ substantially from the nominal value. Prediction intervals with improved (conditional) coverage probability can be defined by adjusting the estimative ones, using rather complicated asymptotic procedures or suitable simulation techniques. This article extends to Markov process models a recent result by Vidoni, which defines a relatively simple predictive distribution function, giving improved prediction limits as quantiles. This new solution is fruitfully considered in the challenging context of prediction for time-series models, with particular regard to AR and ARCH processes.     

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.     

Improved prediction intervals for stochastic process models

Abstract.  This paper reviews some recent results on the construction of improved prediction limits for time series models and presents a simple solution based on a fully conditional approach. A prediction limit, expressed as a modification of the estimative one, is obtained so that its conditional and unconditional coverage probability equals the target value to third-order accuracy. Although the specification of the ancillary statistic is not required, it respects the conditionality principle, to the relevant order of approximation. Moreover, the corresponding predictive density is specified in a relatively simple closed form. Simple examples show the usefulness of this conditional approach to prediction.     

Infinitely Divisible Distributions in Integer‐Valued Garch Models

We propose an integer‐valued stochastic process with conditional marginal distribution belonging to the class of infinitely divisible discrete probability laws. With this proposal, we introduce a wide class of models for count time series that includes the Poisson integer‐valued generalized autoregressive conditional heteroscedastic (INGARCH) model (Ferland et al., 2006) and the negative binomial and generalized Poisson INGARCH models (Zhu, 2011, 2012a). The main probabilistic analysis of this process is developed stating, in particular, first‐order and second‐order stationarity conditions. The existence of a strictly stationary and ergodic solution is established in a subclass including the Poisson and generalized Poisson INGARCH models.     

Improved prediction limits for a general class of Gaussian models

In this article we consider the problem of prediction for a general class of Gaussian models, which includes, among others, autoregressive moving average time‐series models, linear Gaussian state space models and Gaussian Markov random fields. Using an idea presented in Sjöstedt‐De Luna and Young (2003) , in the context of spatial statistics, we discuss a method for obtaining prediction limits for a future random variable of interest, taking into account the uncertainty introduced by estimating the unknown parameters. The proposed prediction limits can be viewed as a modification of the estimative prediction limit, with unconditional, and eventually conditional, coverage error of smaller asymptotic order. The modifying term has a quite simple form and it involves the bias and the mean square error of the plug‐in estimators for the conditional expectation and the conditional variance of the future observation. Applications of the results to Gaussian time‐series models are presented.     

QMLE for Quadratic ARCH Model with Long Memory

We discuss parametric quasi‐maximum likelihood estimation for quadratic autoregressive conditional heteroskedasticity (ARCH) process with long memory introduced in Doukhan emphet al. (2016) and Grublyt? and ?karnulis (2016) with conditional variance involving the square of inhomogeneous linear combination of observable sequence with square summable weights. The aforementioned model extends the quadratic ARCH model of Sentana ( 1995 ) and the linear ARCH model of Robinson ( 1991 ) to the case of strictly positive conditional variance. We prove consistency and asymptotic normality of the corresponding quasi‐maximum likelihood estimates, including the estimate of long memory parameter 0 < d < 1/2. A simulation study of empirical mean‐squared error is included.     

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.     

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.     

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.     

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).     

Thermal Hysteresis for the α'L↫β Transformations in Strontium Oxide-Doped Dicalcium Silicates

A series of Sr-bearing Ca₂SiO₄ solid solutions (C₂S( ss )), (Sr _x Ca_1−x)₂SiO₄ with 0 ≤ x ≤ 0.12, was prepared. They were examined by high-temperature powder X-ray diffractome-try to determine the start and finish temperatures of the α'_L-to-β and β-to-α'_L martensitic transformations. The thermal hysteresis was positive with x < 0.045, nearly equal to zero at x = 0.045, and negative with x > 0.045. The zero and negative hysteresis were consistent with the thermoelasticity of the transformations. With increasing x from 0.02 to 0.08, the hysteresis decreased steadily from positive to negative, suggesting a continuous increment in the stored elastic energy.