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.     

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.     

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.     

Disorder and localization in bandtail hopping transport: experiments and concepts

This review describes the main transport experiments (including Ohmic and nonlinear transport in both dc and ac regimes) and related concepts which can be used to derive quantitative information on the localized electronic states above the Fermi level and in the bandtails, in low carrier mobility materials governed by hopping conductivity. For bandtail hopping, a model based on the filling rate of electronic states near the preferred transport path shows a sharp decrease of the hopping mobility with increasing bandtail disorder. While the slope T^1/40, derived from lnσ vs T^−1/4 plots, is weakly sensitive to the shape of the DOS distribution, the prefactor σ^∘;∘; values are extremely sensitive to bandtail disorder; uniform and exponential distributions can be discriminated experimentally (σ^∘;∘; ≫ 10² S.cm⁻¹ for bandtail hopping). As an illustration, the localized π states distribution in amorphous carbon nitride a-C_1−xN_x:H (x=0.23) is investigated in sandwich devices. Using a wide range of temperatures, electric fields and ac frequencies, several methods are proposed to derive some DOS parameters: density of states at the Fermi level, localization radius and disorder energy of the exponential bandtail distribution. In the field-enhanced bandtail hopping and in the activationless regimes, both apparent slope T^1/4₀ and prefactor σ^∘;∘; values are also properly described. An analytical model, based on the filling rate of localized states near the preferred transport energy, shows that the (F^S/T) scaling properties of nonlinear conductivity σ(F, T) (with S ≈; 2/3 in a-C_1−xN_x:H) are related to the field-dependent effective temperature describing the non-equilibrium distribution of π; ^∗; electrons in the bandtail. In ac transport, the characteristic response time τ(F, T) shows a linear lnτ vs T^−1/4 behaviour, with a smaller T-dependence as the field strength increases, closely related to the filling rate obtained from the dc hopping transport.     

Electrical-thermal switching effect in high-density polyethylene/graphite nanosheets conducting composites

The nonlinear response in high-density polyethylene/graphite nanosheets conducting composites under increasing applied voltage is investigated. Under sufficient applied constant voltage, the resistance increases initially due to Joule heating effect and then eventually reaches a steady value with a characteristic thermal relaxation time τ _h , which decreases as the applied field increases. The switch value, namely the ratio of the resistance under steady condition to the resistance of sample in linear regime, gradually increases with the increasing applied field. The threshold voltage (V ₀) at which the resistance start to increase with time scales with linear regime resistance (R ₀) of the sample as with the exponent x = 0.78 ± 0.05. All the curves of R/R ₀ vs. V/V ₀ collapse to a similar curve with the function R/R ₀ = 1 + α(V/V ₀) ^θ . The results reveal that the threshold voltage value decreases with increasing graphite nanosheets content in the composites.     

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.     

Sample size required to estimate a parameter in the power function distribution

The purpose of this paper is to establish a two step sampling procedure for estimating the parameter θ of the power function distribution to within givend units of its true value with a given probability 1—α;(0<α<1). The density of the power function distribution is a function of two parameters, the second of whichk is assumed known. Given a preliminary sample sizem, tables and formulas are presented by which one may establish the sizen of the second sample such thatP(|y _n —θ|<d)>1—α is true, wherey _n is the largest observation in the second sample. The method used is deriving the results of this paper is similar to that given by Graybill and Connell (1964) and since the power function density reduces to the uniform density whenk=0, their results can be derived from the formulas given here. Also a table of comparisons between the expected second sample size in this paper and two other solutions is given.

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Sumario El propósito de este escrito es el de establecer un procedimiento de muestreo de dos pasos, para estimar el parámetro de la distribución de funciones de potencia cerca ded unidades de su valor verdadero con una probabilidad 1—α;(0<0<α<1). la densidad de la distribución de funciones de potencia es una función de dos parámetros, el segundo de los cualesk se supone es conocido. Dado un tama?o de muestra preliminarm, se presentan tabulaciones y fórmulas con las cuales se puede establecer el tama?on de la segunda muestra, de tal manera queP(|y _n —θ|<d)>1—α sea cierto, dondey _n es la observación más grande en la segunda muestra. El método usado en la derivación de los resultados de este escrito es similar al dado por Graybill y Connell (1964, Journal of the American Statistical Association) y ya que la densidad de funciones de potencia se reduce a la densidad uniforme cuandok=0, sus resultados pueden ser derivados a partir de las fórmulas dadas aquí. Además, se presenta una tabulación de comparaciones entre el segundo tama?o de muestra dado en este escrito y otras dos soluciones. Lafdp de la distribución de funciones de potencia es de la formaf(u)=(k+1)θ ^−(k+1) u ^k ,0<u<θ, θ>0 y cero en cualquiera otra parte.

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Research supported under ONR contract N00014-68-A-0515.     

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 the location of roots of non-reciprocal integer polynomials

Let P be a polynomial of degree d with integer coefficients such that P(0) ≠ 0. Assuming that P has no reciprocal factors we obtain a lower bound on the modulus of the smallest root of P in terms of its degree d, its Mahler measure M(P) and the number of roots of P lying outside the unit circle, say, k. We derive from this that all d roots of P must lie in the annulus R ₀ < |z| < R ₁, where R ₀ = R ₀(d, k, M(P)) and R ₁ = R ₁(d, k, M(P)) are given explicitly. As an application, for non-reciprocal conjugate algebraic numbers α, α′ of degree d ≥ 2 and of Mahler’s measure M(α), we prove the inequality ${|\alpha\alpha'-1|\,{ > }\,(12M(\alpha)^2 \log M(\alpha))^{-d}}${|\alpha\alpha'-1|\,{ > }\,(12M(\alpha)^2 \log M(\alpha))^{-d}}. Some lower bounds on the moduli of the conjugates of a Pisot number are also given. In particular, it is shown that if α is a cubic Pisot number, then the disc |z| ≤ α ⁻¹ + 0.1999α ⁻² contains no conjugates of α. Here the constant 0.1999 cannot be replaced by the constant 0.2. We also show that if α is a Pisot number of degree at least 4 and α′ is its conjugate, then |α α′ − 1| > (19α ²)⁻¹.