ON STATIONARITY OF THE SOLUTION OF A DOUBLY STOCHASTIC MODEL

Abstract. Consider the discrete parameter process {X_I} satisfying the doubly stochastic model X_t =ø_tX_t-1+ε_t where {ø} and {ε_t} are also stochastic processes. Necessary and sufficient conditions on {ø} are given for { X₁ } to be a second order process. When {ø_t} is a strictly stationary process, some sufficient conditions in terms of {ø} are given which guarantee the wide sense stationarity of {X_t} . It turns out that for these problems the distribution and dependence structure of the process {log |ø|} play an important role.     

STATIONARY PROCESSES WITH A FINITE NUMBER OF NON-ZERO CANONICAL CORRELATIONS BETWEEN FUTURE AND PAST

Abstract. The spectral densities f (γ) are determined for stationary random processes X (t) with continuous time which have the property that the number of non-zero canonical correlations between the past X(t) ( t ≤ 0) (more accurately the present and the past) and the future X(t) ( t ≥θ) is finite (equal to N ) at any θ≥τ for some r > 0. A method for finding all the corresponding canonical correlations P₁, …, P_N and the canonical variables X ₁^-, …, X _N^- and X ₁⁺, …, X _N⁺ is given. Similar results related to processes X(t) with discrete (integral) time are briefly considered.     

Polyvariograms and their Asymptotes

A definition of a polyvariogram (PV) γ _b ( h )( h = 1, 2, ...) of order b ( b ≥ max(0, d − 1) is suggested for time series { Z ( t })} satisfying {∇ ^d ( Z ( t ) = W ( t ) (where d is a non-negative integer and { W ( t )} is a second-order stationary time series and is not over-differenced). When b = 0, 1 and 2, this definition corresponds to Cressie's ( J. Am. Stat. Assoc. 83 (1988), 1108–16; 85 (1990), 272) semivariogram linvariogram and quadvariogram respectively and is simpler. Under very general conditions about { W ( t )}, we obtain the relationship between γ _b ( h ) and the autocovariance function of { W ( t )} and show that the asymptote of γ _b ( h ) is a straight line having a positive slope when b = d − 1 and levelling out when b ≥ d .
A definition of a sample polyvariogram (SPV) of order b is given and is shown to be an unbiased and consistent estimate of the PV; and further, some uniformly (in h ) almost sure convergence rates are obtained. These properties provide theoretical support for using the SPV to replace the practically unknown PV and generalize the guidelines for identifying d given by Cressie, where { W ( t )} was restricted to a white noise and b ≤ 2. Some further asymptotic theorems and avenues for using them for statistically testing d and parameters of models for { W ( t )} are briefly introduced.     

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.     

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.     

COMPARISON OF SOME NON-LINEAR AUTOREGRESSIVE PROCESSES

Abstract. We investigate autoregressive processes {X_n} ({Y_"}) satisfying X_n+1=Λ(X_n) + Z_n+1( Y_n+1=μ(Y_n) + Z_n+1) , n = 0, 1, 2, …, where Λ and μ are odd increasing functions and the Z_n's are i.i.d. with unimodal symmetric distribution. The process {|X_n|} is then stochastically monotone on [0, ∞). If |Λ| ≤ |μ| then |X_n is stochastically dominated by |Y_n| for n = 1, 2, …. Some related asymptotic properties, e.g., stationary distributions, are also discussed.     

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 .     

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         

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 and Electrical Properties of New Oxide Ion Conductors Ce6−xGdxMoO15−δ (0.0≤x≤1.8)

A series of oxide ion conductors Ce_{6− x} Gd _x MoO_15−δ (0.0≤ x ≤1.8) have been prepared by the sol–gel method. Their properties were characterized by differential thermal analysis/thermogravimetry (DTA/TG), X-ray diffraction (XRD), Raman, IR, X-ray photoelectron spectroscopy (XPS), and AC impedance spectroscopy. The XRD patterns showed that the materials were single phase with a cubic fluorite structure. The conductivity of Ce_{6− x} Gd _x MoO_15−δ increases as x increases and reaches the maximum at x =0.15. The conductivity of Ce_4.5Gd_1.5MoO_15−δ is σ_t=3.6 × 10⁻³ S/cm at 700°C, which is higher than that of Ce_4.5/6Gd_1.5/6O_2−δ (σ_t=2.6 × 10⁻³ S/cm), and the corresponding activation energy of Ce_4.5Gd_1.5MoO_15−δ (0.92 eV) is lower than that of Ce_4.5/6Gd_1.5/6O_2−δ (1.18 eV).     

Alkoxide–Hydroxide Route for the Preparation of γ-LiAlO2-Based Ceramics

An alkoxide-hydroxide route has been developed to prepare Li_{4 + x} Al_{4 − 3 x} Si_{2 x} O₈ (0 ≤ x ≤ 0.25) powders by taking into account fundamental aspects of the sol-gel process. This technique allows one to prepare powders which exhibit the β-LiAlO₂ type of structure after drying at 150°C. The β→γ-LiAlO₂ topotactic transformation spreads over a large temperature range (746–839°C for x = 0.125) with no significant dilatometric and enthalpic change. Stoichiometric γ-LiAlO₂-based ceramics with a large variety of uniform microstructures are fabricated by a direct sintering of β-LiAlO₂ powders in the temperature range of 900–1100°C.     

STATIONARITY OF THE SOLUTION OF Xt= AtXt-1+εt AND ANALYSIS OF NON-GAUSSIAN DEPENDENT RANDOM VARIABLES

Abstract. We give general and concrete conditions in terms of the coefficient (stochastic) process {A_t} so that the (doubly) stochastic difference equation X_t= A_tX_t-1+ε_t has a second-order strictly stationary solution. It turns out that by choosing {A_t} and the "innovation" process {ε_t} properly, a host of stationary processes with non-Gaussian marginals and long-range dependence can be generated using this difference equation. Examples of such nowGaussian marginals include exponential, mixed exponential, gamma, geometric, etc. When {A_t} is a binary time series, the conditional least-squares estimator of the parameters of this model is the same as those of the parameters of a Galton-Watson branching process with immigration.     

Topotaxy between BaTiO3 and Ba6Ti17O40

The orientational relationship between intergrown lamellae of tetragonal (t) BaTiO₃ and monoclinic (m) Ba₆Ti₁₇O₄₀ was found to be ( 1 1 1 )_t||(001)_m, (001)_t||( 1 0 1 )_m, and [110]_t||[0 1 0]_m. On the basis of this relationship a model of a topotactic boundary between these two phases was constructed.     

Densification Energy during Nanoindentation of Silica Glass

Load/unload displacement curves at room temperature (humidity 49%) for silica glass have been measured in the penetration range of 0.5–1.2 μm using a Vickers nanoindentation technique (load/unload speed 50 mN/s). Deformation energies have been estimated for the first time. The universal (dynamic) hardness, H _u, and elastic recovery, E _R, at the penetration depth, h _t, of 1.0 μm are H _u= 4.1 GPa and E _R= 0.7. The following energies for total deformation, U _t, elastic deformation, U _e, and plastic deformation (i.e., densification during loading), U _p, are obtained: U _t=190, U _e=135 and U _p= 55 kJ/mol at h _t= 0.5 μm and U _t= 139, U _e= 96 and U _p= 43 kJ/mol at h _t=1.0 μm. All these deformation energies increase with decreasing penetration depth. It is found that plastic deformation energies of 38–55 kJ/mol for 0.5 < h _t < 1.2 μm are very close to the activation energy (46–54 kJ/mol) for the recovery of densification in silica glass, but are very small compared with the single bond strength (443 kJ/mol for Si—O bond) of SiO₂.     

A TIME SERIES MODEL WITH SUDDENLY CHANGING PARAMETERS

Abstract. A process X_t = θ_t + e , is investigated where e_t is a strict white noise and θ_t is a Markov chain with two real states. A realization of X_t fluctuates around two levels which correspond to the two states of θ_t . Formulae for the extrapolation of the process X_t and for the mean square error of the extrapolation are derived. The moments of X_t and its covariance function are calculated. The results are used to derive moment estimators for the parameters of the model.     

Pressure Filtration Model of Ceramic Nanoparticles

The consolidation behavior of nanometer-sized particles at 20–800 nm was examined using a pressure filtration apparatus at a constant compressive rate. The relation of applied pressure (Δ P _t)–volume of dehydrated filtrate ( V _f) was compared with the established filtration theory for the well-dispersed suspension. The theory was effective in the early stage of the filtration but deviation between the experiment and the theory started when Δ P _t exceeded a critical pressure (Δ P _tc). It was found that this deviation is associated with the phase transition from a dispersed suspension to a flocculated suspension at Δ P _tc. The factors affecting Δ P _tc are the ζ potential, concentration, and size of the particles. Based on the colloidal phase transition, a new filtration theory was developed to explain the Δ P _t– h _t (height of suspension) relation for a flocculated suspension. Good agreement was shown between the developed theory and experimental results.     

SEMIPARAMETRIC TIME SERIES REGRESSION

Abstract. Let ( X _i, Y _i), i = 0, pL 1,… denote a bivariate stationary time series with X _i being R^d-valued and Y _i being real-valued. We consider the regression model Y _i=θ( X _i) + Z _i, where θ(·) is an unknown function and Z_i is an autoregressive process. Given a realization of length n , we examine the problem of estimating the nonparametric function θ(·) and the parametric component Z _i. Under appropriate regularity conditions, it is shown that both components can be optimally estimated.     

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.     

The System Bi2O3-SiO2

The Bi₂O₃-rich side of the system Bi₂O₃-SiO₂ was studied with powder X-ray diffraction and differential thermal analysis. In the composition 6Bi₂O₃. x SiO₂, the metastable γ phase (bcc) was observed to exist over the range of 0 < x ≤ 1. In most of the compositions studied, metastable phases of water-quenched melts transformed into another metastable phase before reaching stable phases. A modification of the phase diagram is proposed.     

A note on L1 density estimation for linear processes

In this paper, we consider the L ₁ performance of a kernel estimator, f^_n of the density of a linear process X_t ∑^∞ _{k =0} a _k Z _t−k , a ₀ = 1, where { Z _t } is a sequence of independent and identically distributed (i.i.d.) random variables with E | Z ₁|^ε< ∞, for some ε > 1, and { a_k } is a sequence of reals converging to zero at a certain rate. Asymptotic minimizations of the integrated L ₁ risk of f_n and its upper bounds are considered. This paper extends the earlier results for the i.i.d. case by Devroye and Gyorfi ( Nonparametric Density Estimation: The L₁ View. New York: Wiley, 1985) and by Hall and Wand (Minimizing the L ₁ distance in nonparametric density estimation, J. Multivariate Anal. 26 (1988), 59–88) to the linear process case. Numerical examples to illustrate the performance of f_n are also presented.