Testing the hypothesis of a general linear model using nonparametric regression estimation

Summary Given the modelY _i =m(χ _i )+ɛi,whereE(ɛ _i) =0,X _i ≠Ci=1, ...,n, andC is ap-dimensional compact set, we have designed a new method for testing the hypothesis that the regression function follows a general linear model,m(·) ∈ {m _θ(·) =A ^t (·)θ}_{θ∈Θ⊂ℛq} , withA a function fromℜ ^p toℜ ^q. The statistic, denoted ΔASE, used fortesting the given hypothesis is defined to be the difference between the average squared errors (ASE) associated with the non-parametric estimator ofm and the minimum distance parametric estimator ofm. The asymptotic normality of both ΔASE and the minimum distance estimators is proved under general conditions. Alternative bootstrap versions of ΔASE are also considered.     

Intermittent estimation of stationary time series

Let {X _n } _n=0 ^∞ be a stationary real-valued time series with unknown distribution. Our goal is to estimate the conditional expectation ofX _n+1 based on the observations,X _i , 0≤i≤n in a strongly consistent way. Bailey and Ryabko proved that this is not possible even for ergodic binary time series if one estimates at all values ofn. We propose a very simple algorithm which will make prediction infinitely often at carefully selected stopping times chosen by our rule. We show that under certain conditions our procedure is strongly (pointwise) consistent, andL ₂ consistent without any condition. An upper bound on the growth of the stopping times is also presented in this paper.     

Improving predictive distributions

Summary Consider a sequence of decision problemsS ₁,S ₂, ... and suppose that in problemS _i the statistician must specify his predictive distributionF _i for some random variableX _i and then make a decision based on that distribution. For example,X _i might be the return on some particular investment and the statistician must decide whether or not to make that investment. The random variablesX ₁,X ₂ ... are assumed to be independent and completely unrelated. It is also assumed that each predictive distributionF _i assigned by the statistician is a subjective distribution based on his information and beliefs aboutX _i. In this context, the standard Bayesian approach provides no basis for evaluating whether the statistician’s subjective predictive distribution forX _i is good or bad, and does not even recognize this question as being meaningful. In this paper we describe models in which the statistician can study his process for specifying predictive distributions identify bad habits, and improve his predictions and decisions by gradually breaking these habits.     

New Insights on the Intrinsic Anisotropy of YBCO Films

Careful investigation of the angular dependence of resistivity ρ(θ) (θ is the angle between the magnetic field and the ab-planes) and the temperature dependence of resistivity ρ(T) within the superconducting transition in an applied magnetic field B up to 1 T for a series of YBa₂Cu₃O_7−δ (YBCO) thin films revealed a large variation of intrinsic anisotropy factor γ. The series of films studied included both optimally doped and underdoped samples of different T _c , critical current density J _c , film thickness, and preparation techniques. The variation in the shape and depth of the minimum measured for ρ(θ) near θ=0° could be directly correlated to the intrinsic anisotropy of the YBCO films. The results of fitting of ρ(θ) using Bardeen–Stephen theory allowed a quantitative determination of the value of γ which varies between 7 and 230, and is independent of T _c , film thickness, or J _c . The sharper the minimum in ρ(θ) around θ=0° the larger is the anisotropy. For highly anisotropic film, ρ(θ) showed an identical behavior for B ∥ J and B ⊥ J (i.e., ρ(θ) is independent of the angle θ between B and J for this film). The large variation in γ could be attributed to the “buckling” of the CuO₂ planes.     

Some multiple comparisons using sample quasi ranges on censored data

We considerk(k)≥2 independent populations (treatments or systems) and an solutely continuous member of location-scale family of distributions, index by the location parameter μ _i (-∞ < μ _i < ∞) and scale parameter θ _i (θ _i > 0), is used to model the observations from the ith population,i=1,...k. The problem of simultaneous selection of two subsets, one containing population associated with the smallest ϕ-value and other containing population with the largest ϕ-value with probability at least a pre-specified value is considered when the data are censored. We also construct 100P ^*% simultaneous upper and two-sided confidence intervals for where θ_[1] ≤ ... ≤ θ_[k] denotes the ordered values of ϕs. The proposed procedures, based on sample quasi ranges, are useful when the experimenter has smaller samples or censored samples or there is suspicion of outliers in the samples. The results are applied to exponential populations model and, for thes casi: (i) the constants have been computed to apply the proposed multiple comparisons; (ii) two members of the proposed class have been compared with the existing procedure. A numerical example is also given.     

Pseudo maximum likelihood estimates using ranked set sampling with applications to estimating correlation

Suppose θ is the parameter of interest and λ is the nuisance parameter. When obtaining the maximum likelihood estimator (MLE) of θ in the presence of λ requires intensive computation, the pseudo MLE of θ, based on a pseudo likelihood function, can be used. Gong and Samaniego (Ann. Stat. 9:861–869, 1981) proposed a pseudo MLE (PMLE) based on simple random samples. Ranked set sampling has been applied to the bivariate variables (X,Y) where measuring one of the variables is difficult or costly. In this paper, we obtain the pseudo MLE of the correlation coefficient from a bivariate normal distribution (X,Y) based on ranked set samples, assuming that Y is difficult or more expensive to measure and that the mean and variance of Y are the nuisance parameters. The PMLE is compared with three other estimators of the correlation coefficient. Simulations show that the PMLE is more (less) efficient than other estimators, depending on value of ρ. Testing of soil contamination provides an example of the use of the methods.     

A Bayesian approach to the multivariate Behrens-Fisher problem under the assumption of proportional covariance matrices

Summary Two independent random samples of sizesN ₁ andN ₂ from multivariate normal populationsN _p (θ₁,∑₁) andN _p (θ₂,∑₂) are considered. Under the null hypothesisH ₀: θ₁=θ₂, a single θ is generated from aN _p(μ, Σ) prior distribution, while underH ₁: θ₁≠θ₂ two means are generated from the exchangeable priorN _p(μ,σ). In both cases Σ will be assumed to have a vague prior distribution. For a simple covariance structure, the Bayes factorB and minimum Bayes factor in favour of the null hypotheses is derived. The Bayes risk for each hypothesis is derived and a strategy is discussed for using the Bayes factor and Bayes risks to test the hypothesis.     

Convergencia del vector de probabilidad a posteriori bajo una distribucion predictiva

Resumen La convergencia casi segura de una sucesión de variables aleatorias, con respecto aP _{X, Q} (distribución predictiva), se estudia en relación con la convergencia casi segura, con respecto aP _{X, θ} (para todo θ∈Θ), donde{P _{X, θ}}_θ∈Θ es una familia de modelos de probabilidad sobre el espacio muestralX. Como consecuencia, se estudia la convergencia casi segura del vector de probabilidad a posteriori con respecto aP _{X, Q}.      

Eliminating nuisance parameters: two characterizations

Given a statistical modelP = {P_θ : θ ∈ x} and a surjective functiong: ϑ→Λ the problem of transformingP into a new modelQ= {_λ : λ ∈ Λ} indexed by Λ is investigated. Two characterizations are given for those modelsQ of the form Q_λ = ∫ P_θ π_λ(dθ), where π_λ is some probability such that π_λ(g=λ)=1. The first is related to a geometric property ofQ, while the second rests on the inferential implications of adoptingQ. Also, in the first π_λ is allowed to be finitely additive, while in the second π_λ is σ-additive. Finally, integrated likelihoods are revisited in light of the second characterization.     

Superconductivity in ZrxNb1−2xMox alloys: Possible dominance of a soft phonon mode

Normal-state specific heat and superconducting transition temperature measurements were performed on pure polycrystalline Nb and isoelectronic bcc Zr_xNb_1−2xMo_x alloys (x = 0.02, 0.05, and 0.10). Measured T _c, γ and θ _D all decline nearly linearly from the pure Nb values with increasing x. The changes are 15, 10, and 6%, respectively, for x = 0.1. T _cH calculated by Hopfield's theory, using data and linearly interpolated values of η and θ _D /〈θ²〉_linear ^1/2, increases with x by 28% for x = 0.1. By relaxing the linear interpolation of θ_D/〈θ²〉^1/2, Hopfield's theory can be used to evaluate an average phonon frequency 〈θ²〉_emp ^1/2 from data. It is found to be nearly constant at about 229 K, in contrast to 〈θ²〉_linear ^1/2, which declines by 8% at x = 0.1. We suggest that one possible explanation for this behavior is the dominance of incipient soft-phonon longitudinal modes which occur in Nb and presumably in the alloys. Miedema's method has been used to calculate the variation of both γ and T _c with alloying. The calculated γ is nearly constant, while the calculated T _c declines more quickly than the measured variation. The band-structure electron density of states calculated with the aid of McMillan's equation for these isoelectronic alloys is found to decline by 8% at x = 0.1, in contrast to the prediction of the rigid band model. Normal-state low-temperature specific heat measurements on a pure annealed polycrystalline sample of Nb yield γ = 7.80 ± 0.02 mJ mole⁻¹ K⁻² and θ_D = 276 ± 2 K, in excellent agreement with previous measurements. The break in slope of C _p/T vs. T ² at 3.2 K is discussed. Work supported in part by a National Science Foundation Departmental Science Development Grant No. NSF-GU2603.     

Consistencia de un estimador no parametrico, recursivo, de la regresion bajo condiciones generales

Resumen Se define un estimador no paramétrico, recursivo, de la función de regresiónr(x)=E(Y/X=x), que se calcula a partir de un conjunto den observaciones {(X ₁,Y _i ):i=1,…n} del vector aleatorio (X, Y). Bajo la hipótesis de que los datos son idénticamente distribuidos pero no necesariamente independientes, lo que permite utilizar el estimador definido para estimar la función de autorregresión de una serie de tiempo, se obtienen resultados sobre la consistencia puntual débil (en probabilidad) y fuerte (completa) del estimador definido. Finalmente, se presentan ejemplos de utilización del estimador con datos simulados. Summary With a initial sample of sizen: {(X _i ,Y _i ):i=1,…n} of a radom vector (X,Y) a general recursive nonparametric estimator of a regression function,r(x)=E(Y/X=x) is obtained. Assume the data are identically distribuited but non necessarily independent. So, the defined estimator may be used to estimate the autoregression function of a time series. Weak and strong pointwise consistency are obtained for such estimator. Finally some examples of simulation are presented.

     

Esperanza condicionada para probabilidades finitamente aditivas

Let (Ω, θ,J) be a finitely additive probabilistic space formed by any set Ω, an algebra of subsets θ and a finitely additive probabilityJ. In these conditions ifF belongs toV ¹ (Ω, θ,J) there existsf, element of the completion ofL ¹ (Ω, θ,J), such thatF(E)=∫ _E fdJ for allE of θ and conversely. This integral representation gives sense to the following result, which is the objetive of this paper, in terms of the, point function: If β is a subalgebra of θ, for everyF ofV ¹ (Ω, θ,J) there exists a unique element ofV ¹ (Ω, β,J) which we note down byE(F/gb), conditional expetation ofF given β. E(F/β) is characterized by (E(F/β),G)=(F, G) for every ofV ^∞(Ω,β, J). Aside from this, the mappingE(./β):V ¹ (Ω, θ,J)→V ¹ (Ω, β,J) is linear, positive, contractive, idempotent andE(J/β)=J. IfF is ofV ^p (Ω, θ,J),p>1,E(F/β) is ofV ^p (Ω, β,J).

     

Effect of grain boundaries on dynamic recrystallization behaviour of copper isoaxial bicrystals with 〈001〉 tilt boundaries

Dynamic recrystallization behaviour of copper isoaxial bicrystals with various 〈001〉 tilt boundaries was investigated at 1023 and 1073 K at an initial strain rate of 3.33×10⁻⁴sec⁻¹, with special attention paid to the effect of grain boundaries on the recrystallization stress (strain) and the substructure developed during deformation. The grain boundaries in the bicrystals have almost no effect on the stress-strain curves. Flow stress was found to fall abruptly and significantly during deformation, being similar to the flow stress in the single crystals. The stressσ _R (ɛ _R), just before the stress fall indicates the stress (strain) at which dynamic recrystallization occurs. The growth of recrystallized grains is markedly fast in these bicrystals, suggesting that dynamic recrystallization is controlled by a nucleation process. The value ofσ _R in bicrystals with a tilt boundary of 9°,θ9 bicrystals, is almost equal to that in single crystals, but is larger thanσ _R inθ23,θ36 andθ43 bicrystals. These results indicate that theθ9 boundary has no effect on the dynamic recrystallization of bicrystals, whereas grain boundaries with tilt angles above 23° accelerate the dynamic recrystallization of these bicrystals. Theθ-dependence ofσ _R orɛ _R is discussed in connection with the stress concentration near the grain boundary.     

Rewriting Techniques and Degree Bounds for Higher Order Symmetric Polynomials

Any symmetric polynomial f∈R[X ₁, …, X _n] has a unique representation f = p(σ₁, …, σ_n) with p∈R[X ₁, …, X _n] in the elementary symmetric polynomials σ₁, …, σ_n. This paper investigates higher order symmetric polynomials; these are symmetric polynomials with a representation p, which is also symmetric. We present rewriting techniques for higher order symmetric polynomials and exact degree bounds for the generators of the corresponding invariant rings. Moreover, we point out how algorithms and degree bounds for these polynomials are related to Pascals triangle, Fibonacci numbers, Chebyshev polynomials, and cardinalities of finite distributive lattices of semi-ideals. Received: December 16, 1997     

On δ-record observations: asymptotic rates for the counting process and elements of maximum likelihood estimation

Given a sequence of random variables {X _n ,n≥1} and δ∈ℝ, an observation X _n is a δ-record if X _n >max {X ₁,…,X _n−1}+δ. We obtain, for δ≤0, weak and strong laws of large numbers for the counting process of δ-records among the first n observations from a sequence of independent identically distributed random variables, with common distribution F, possibly discontinuous. We provide examples of our results in the context of common probability distributions. Finally, we show how δ-records can be used for maximum likelihood estimation.     

Elastic properties of ZnF2-PbO-TeO2 glasses doped with certain rare earth ions

The elastic moduli (Y, η, σ andH) and some thermodynamical parameters, such as Debye temperatureθ _D, diffusion constantD _i and latent heat of melting ΔH _m, of ZnF₂-PbO-TeO₂ glasses doped with some rare earth (Pr³⁺, Nd³⁺, Eu³⁺ and Tb³⁺) ions are determined as functions of temperature using piezo electric composite oscillator technique. All the parameters are found to decrease with increase in the atomic number Z of the rare earth ions and with increase in the temperature of measurement. The results are explained on the basis of the density of localized bonding defect states within the band gap of the material.     

Growth behavior of compounds due to solid-state reactive diffusion between Cu and Al

To examine experimentally the kinetics of the reactive diffusion between solid-Cu and solid-Al, sandwich Al/Cu/Al diffusion couples were prepared by a diffusion-bonding technique and then isothermally annealed in the temperature range of T = 693–753 K for various times up to 336 h. Owing to annealing, compound layers of the γ ₁, δ, ζ ₂, η ₂, and θ phases are formed between the Cu and Al specimens. The γ ₁, δ, ζ ₂, η ₂, and θ phases are the only stable compounds at T = 693–753 K in the binary Cu–Al system. At each annealing time, the thickness of the θ phase is much greater than those of the δ, ζ ₂, and η ₂ phases but smaller than that of the γ ₁ phase. Hence, the overall growth of the compound layers is governed by the γ ₁ and θ phases. The mean thickness of each compound layer is proportional to a power function of the annealing time. For the γ ₁ phase, the exponent m of the power function is 0.5 at T = 753 K. Such a relationship is called a parabolic relationship. As the annealing temperature T decreases, however, m gradually increases and then reaches to 0.66 at T = 693 K. On the other hand, for the θ phase, m is close to 0.5 at T = 723–753 K and becomes 0.42 at T = 693 K. In the γ ₁ and θ phases, grain growth occurs at T = 693–753 K. Thus, the layer growth of the θ phase is controlled by volume diffusion at T = 723–753 K but partially by boundary diffusion at T = 693 K. On the other hand, for the γ ₁ phase, volume diffusion is the rate-controlling process of the layer growth at T = 753 K, but interface reaction contributes to the rate-controlling process at T = 693–723 K. Consequently, the rate-controlling process varies depending on the annealing temperature in a different manner for each compound.     

Acoustic investigations on PbO-Al2O3-B2O3 glasses doped with certain rare earth ions

Elastic moduli (Y, η), Poisson’s ratio (σ), microhardness (H) and some thermodynamical parameters such as Debye temperature (θ_D), diffusion constant (D _i),latent heat of melting (ΔH _m) etc of PbO-Al₂O₃-B₂O₃ glasses doped with rare earth ions viz. Pr³⁺, Nd³⁺, Sm³⁺, Eu³⁺, Tb³⁺, Dy³⁺, Ho³⁺, Er³⁺ and Yb³⁺, are studied as functions of temperatures (in the temperature range 30–200°C) by ultrasonic techniques. All these parameters are found to increase with increasing atomic numberZ of the rare earth ions and found to decrease with increasing temperature of measurement. From these results (together with IR spectra of these glasses), an attempt is made to throw some light on the mechanical strength of these glasses.     

The concept of directional binding energy and analysis of the phenomena involving surface and grain boundary: Part I Theory

The concept of directional binding energy (DBE) is proposed to describe the binding energy of the crystal and the ideal direction of the antiaction force against the surface tension, which derives the grain-boundary energy and also energy ratio (γ_gb/γ_s). It may provide a simple and lucid way to analyse the surface energy, grain-boundary energy and energy ratio from the correlative view point. It has been found that the grain-boundary energy can be derived only from the conceptual approach of DBE, irrespective of the dislocation model, which also makes it possible to determine the energy ratio with the misorientation angle. That is, the energy ratio (γ_gb/γ_s) is proportional to the misorientation angle, 2 sin (k′θ/2) where k′ is a constant and θ is the misorientation angle. This revised version was published online in November 2006 with corrections to the Cover Date.