A Unified Approach to Second Order Optimality Criteria in Nonlinear Design of Experiments

In the nonlinear regression model we consider the optimal design problem with a second order design D-criterion. Our purpose is to present a general approach to this problem, which includes the asymptotic second order bias and variance criterion of the least squares estimator and criteria using the volume of confidence regions based on different statistics. Under assumptions of regularity for these statistics a second order approximation of the volume of these regions is derived which is proposed as a quadratic optimality criterion. These criteria include volumes of confidence regions based on the u _n - representable statistics. An important difference between the criteria presented in this paper and the second order criteria commonly employed in the recent literature is that the former criteria are independent of the vector of residuals. Moreover, a refined version of the commonly applied criteria is obtained, which also includes effects of nonlinearity caused by third derivatives of the response function.     

Transformations with Improved Chi-Squared Approximations

Suppose that a nonnegative statistic T is asymptotically distributed as a chi-squared distribution with f degrees of freedom, χ²_f, as a positive number n tends to infinity. Bartlett correction T was originally proposed so that its mean is coincident with the one of χ²_f up to the order O(n⁻¹). For log-likelihood ratio statistics, many authors have shown that the Bartlett corrections are asymptotically distributed as χ²_f up to O(n⁻¹), or with errors of terms of O(n⁻²). Bartlett-type corrections are an extension of Bartlett corrections to other statistics than log-likelihood ratio statistics. These corrections have been constructed by using their asymptotic expansions up to O(n⁻¹). The purpose of the present paper is to propose some monotone transformations so that the first two moments of transformed statistics are coincident with the ones of χ²_f up to O(n⁻¹). It may be noted that the proposed transformations can be applied to a wide class of statistics whether their asymptotic expansions are available or not. A numerical study of some test statistics that are not a log-likelihood ratio statistic is discribed. It is shown that the proposed transformations of these statistics give a larger improvement to the chi-squared approximation than do the Bartlett corrections. Further, it is seen that the proposed approximations are comparable with the approximation based on an Edgeworth expansion.     

Asymptotic Expansions for Large Deviation Probabilities of Noncentral Generalized Chi-Square Distributions

Asymptotic expansions for large deviation probabilities are used to approximate the cumulative distribution functions of noncentral generalized chi-square distributions, preferably in the far tails. The basic idea of how to deal with the tail probabilities consists in first rewriting these probabilities as large parameter values of the Laplace transform of a suitably defined function f_k; second making a series expansion of this function, and third applying a certain modification of Watson's lemma. The function f_k is deduced by applying a geometric representation formula for spherical measures to the multivariate domain of large deviations under consideration. At the so-called dominating point, the largest main curvature of the boundary of this domain tends to one as the large deviation parameter approaches infinity. Therefore, the dominating point degenerates asymptotically. For this reason the recent multivariate asymptotic expansion for large deviations in Breitung and Richter (1996, J. Multivariate Anal.58, 1–20) does not apply. Assuming a suitably parametrized expansion for the inverse g⁻¹ of the negative logarithm of the density-generating function, we derive a series expansion for the function f_k. Note that low-order coefficients from the expansion of g⁻¹ influence practically all coefficients in the expansion of the tail probabilities. As an application, classification probabilities when using the quadratic discriminant function are discussed.     

THE SINGULAR PERTURBATION OF NONLOCAL BOUNDARY VALUE PROBLEMS FOR NONLINEAR ELLIPTIC EQUATION OF FOURTH ORDER

A class of singularly perturbed nonlocal problems for nonlinear elliptic equation of fourth order are considered. The uniformly valid asymptotic expansion of the solutionis obtained.     

A polynomial-time complexity bound for the computation of the singular part of a Puiseux expansion of an algebraic function

In this paper we present a refined version of the Newton polygon process to compute the Puiseux expansions of an algebraic function defined over the rational function field. We determine an upper bound for the bit-complexity of computing the singular part of a Puiseux expansion by this algorithm, and use a recent quantitative version of Eisenstein's theorem on power series expansions of algebraic functions to show that this computational complexity is polynomial in the degrees and the logarithm of the height of the polynomial defining the algebraic function.

     

径向膨胀流的同位旋效应

利用同位旋相关的量子分子动力学模型,研究了¹²⁴Ba+¹²⁴Ba和¹²⁴Sn+¹²⁴Sn两个系统在不同入射能量下对心碰撞的径向膨胀流.发现丰中子系统¹²⁴Sn+¹²⁴Sn的径向膨胀流系统性地小于系统¹²⁴Ba+¹²⁴Ba的径向膨胀流.同时发现这种差别主要来自于核子–核子碰撞截面的同位旋相关性而几乎与对称能无关.     

Oxygen ionic and electronic transport in apatite-type La10−x(Si,Al)6O26±δ

The main factor governing the oxygen ionic conductivity in apatite-type La_10−xSi_6−yAl_yO_{27−3x/2−y/2} (x=0-0.33; y=0.5-1.5) is the concentration of mobile interstitials determined by the total oxygen content. The ion transference numbers, measured by modified faradaic efficiency technique, vary in the range 0.9949-0.9997 in air and increase on reducing oxygen partial pressure due to decreasing p-type electronic conduction. The activation energies for ionic and hole transport are (56-67)±3 kJ/mol and (57-100)±8 kJ/mol, respectively. Increasing oxygen content leads to higher hole conduction in oxidizing atmospheres and promotes minor oxygen losses from the lattice when the oxygen pressure decreases, although the overall level of ionic conductivity is almost constant in the p(O₂) range from 50 kPa down to 10⁻¹⁶ Pa. Under reducing conditions at temperatures above 1100 K, silicon oxide volatilization from the surface layers of apatite ceramics results in a moderate decrease of the conductivity with time. This suggests that the operation of electrochemical cells with silicate-based solid electrolytes should be limited to the intermediate-temperature range, such as 800-1000 K, where the ionic transport in most-conductive apatite phases containing 26.50-26.75 oxygen atoms per unit formula is higher than that in stabilized zirconia. The average thermal expansion coefficients of apatite ceramics, calculated from dilatometric data in air, are (8.7-10.8)×10⁻⁶ K⁻¹ at 300-1300 K.     

Extended Classical Asymptotic Expansions in the Case of Gaussian Limit Distribution

Let X, X ₁, X ₂,... be a sequence of independent and identically distributed random variables with common distribution function F. Denote by F _n the distribution function of centered and normed sum S _n . Let F belong to the domain of attraction of the standard normal law , that is, lim F _n (x)= (x), as n , uniformly in x . We obtain extended asymptotic expansions for the particular case where the distribution function F has the density p(x) = cx ^––1 ln(x), x > r, where 2, , c > 0, and r > 1. We write the classical asymptotic expansion (in powers of n ^–1/2) and then add new terms of orders n ^–/2 ln n, n ^–/2 ln^-1 n, etc., where 0.     

The oscillatory behavior of the high-temperature expansion of Dyson's hierarchical model: A renormalization group analysis

We calculate 800 coefficients of the high-temperature expansion of the magnetic susceptibility of Dyson's hierarchical model with a Landau-Ginzburg measure. Log-periodic corrections to the scaling laws appear as in the case of an Ising measure. The period of oscillation appears to be a universal quantity given in good approximation by the logarithm of the largest eigenvalue of the linearized RG transformation, in agreement with a possibility suggested by Wilson and developed by Niemeijer and van Leeuwen. We estimate to be 1.300 (with a systematic error of the order of 0.002), in good agreement with the results obtained with other methods, such as the -expansion. We briefly discuss the relationship between the oscillations and the zeros of the partition function near the critical point in the complex temperature plane.     

On large deviation expansion of distribution of maximum likelihood estimator and its application in large sample estimation

For estimating an unknown parameter , the likelihood principle yields the maximum likelihood estimator. It is often favoured especially by the applied statistician, for its good properties in the large sample case. In this paper, a large deviation expansion for the distribution of the maximum likelihood estimator is obtained. The asymptotic expansion provides a useful tool to approximate the tail probability of the maximum likelihood estimator and to make statistical inference. Theoretical and numerical examples are given. Numerical results show that the large deviation approximation performs much better than the classical normal approximation.This work is supported in part by the Natural Science and Engineering Research Council of Canada under grant NSERC A-9216.This author is also partially supported by the National Science Foundation of China.