On the Kaplan–Meier Estimator of Long-Range Dependent Sequences

The limiting distributions are obtained for the Kaplan–Meier estimator of unknown distribution function of stationary time series of the form G(X _j), where X _j is stationary Gaussian process with long-range dependence and G(·) is non-random function.     

Harmonic and refined Rayleigh–Ritz for the polynomial eigenvalue problem

After reviewing the harmonic Rayleigh–Ritz approach for the standard and generalized eigenvalue problem, we discuss several extraction processes for subspace methods for the polynomial eigenvalue problem. We generalize the harmonic and refined Rayleigh–Ritz approaches which lead to new approaches to extract promising approximate eigenpairs from a search space. We give theoretical as well as numerical results of the methods. In addition, we study the convergence of the Jacobi–Davidson method for polynomial eigenvalue problems with exact and inexact linear solves and discuss several algorithmic details. Copyright © 2008 John Wiley & Sons, Ltd.     

Validation of a Structural Optimization Algorithm Transforming Dynamic Loads into Equivalent Static Loads

Generally, structural optimization is carried out based on external static loads. However, all forces have dynamic characteristics in the real world. Mathematical optimization with dynamic loads is almost impossible in a large-scale problem. Therefore, in engineering practice, dynamic loads are often transformed into static loads via dynamic factors, design codes, and so on. Recently, a systematic transformation of dynamic loads into equivalent static loads has been proposed in Refs. 1–3. Equivalent static loads are made to generate at each time step the same displacement field as the one generated by the dynamic loads. In this research, it is verified that the solution obtained via the algorithm of Refs. 1–3 satisfies the Karush–Kuhn–Tucker necessary conditions. Application of the algorithm is discussed.     

The admissibility of the empirical mean location for the matrix von Mises–Fisher family

In this note we consider von Mises–Fisher families of probability densities on spheres and more generally on Stiefel manifolds, which include the orthogonal groups. It addresses the estimation of the mean direction or the mean location by empirical mean location, which for the von Mises–Fisher family coincides with the maximum likelihood estimator. It is shown that (with a few exceptions) the empirical mean location of a sample is almost surely uniquely defined and that it is unbiased in the sense that its mean location coincides with the mean location of the von Mises–Fisher distribution. The main goal, however, is to show that empirical mean location is an admissible estimator for the mean location of the von Mises–Fisher distribution.     

Quantum-behaved particle swarm optimization with Gaussian distributed local attractor point

This paper proposes a novel variant of quantum-behaved particle swarm optimization (QPSO) algorithm with the local attractor point subject to a Gaussian probability distribution (GAQPSO). The local attractor point in QPSO plays an important in that determining the convergence behavior of an individual particle. As such, the mean value and standard deviation of the proposed Gaussian probability distribution in GAQPSO are carefully selected. The distributions and diversities of the local attractor points in GAQPSO and QPSO are evaluated and compared. For the purpose of comparison, two variants of the GAQPSO algorithm are proposed by using a mutation probability and other types of probability distribution. The GAQPSO has been comprehensively evaluated on the suite of CEC2005 benchmark functions, and the experimental results are compared with those of the PSO and QPSO algorithms based on different probability distributions. It is shown by the results that the GAQPSO algorithm is an effective approach that can improve the QPSO performance considerably, that is, the GAQPSO algorithm is less likely to be stuck in local optima and hence it can achieve better solutions in most cases.     

反冲塞效应的热弹性理论分析(Ⅰ)──温度场分析*

本文对我们新近发现的长脉冲激光束对金属薄片材料的一种新的破坏方式──“反冲塞”效应进行了温度场分析,用Hankel变换和Laplace变换,得到了温度场的精确解,以激光束的空间分布为均匀分布和高斯分布为例对温度场进行了计算,结果表明激光束的空间均匀分布是产生反冲塞效应的主要原因。     

Edge scaling limits for a family of non-Hermitian random matrix ensembles

A family of random matrix ensembles interpolating between the Ginibre ensemble of n × n matrices with iid centered complex Gaussian entries and the Gaussian unitary ensemble (GUE) is considered. The asymptotic spectral distribution in these models is uniform in an ellipse in the complex plane, which collapses to an interval of the real line as the degree of non-Hermiticity diminishes. Scaling limit theorems are proven for the eigenvalue point process at the rightmost edge of the spectrum, and it is shown that a non-trivial transition occurs between Poisson and Airy point process statistics when the ratio of the axes of the supporting ellipse is of order n ^?1/3. In this regime, the family of limiting probability distributions of the maximum of the real parts of the eigenvalues interpolates between the Gumbel and Tracy–Widom distributions.     

A probabilistic model for the Solar System planetary distribution

The discovery of several extrasolar systems, each one characterized by its own planetary distribution around a central star, made the scientific interest addressed to the analysis of models permitting to predict, or at least estimate, the orbital features of the extrasolar planets. The main purpose of this work is to describe a mathematical model, inspired by quantum mechanics, able to provide a probability distribution of planets placing in a star system, mainly driven by the central star mass. More in detail, for any given eigenvalue of the model discrete spectrum, a distinct probability distribution with respect to the central star distance can be built. As per the Solar System, it has been possible to prove that both inner and outer planets belongs to two different spectral sequences, each one originated by the minimum angular momentum owned by silicate/carbonate and icy planetesimals respectively. In both sequences, the peak of the probability distributions almost precisely coincided with the average planets distance from Sun; furthermore, the eigenvalue spectrum of the inner planets thickens in an accumulation point corresponding to the asteroids belt, thus showing a striking similarity to the real matter distribution in the Solar System. From this point of view, the Titius–Bode law for the Solar System planets distribution is nothing but an exponential interpolation of the eigenvalues of both inner and outer sequences.     

Estimating a Distribution Function for Censored Time Series Data

Consider a long term study, where a series of dependent and possibly censored failure times is observed. Suppose that the failure times have a common marginal distribution function, but they exhibit a mode of time series structure such as α-mixing. The inference on the marginal distribution function is of interest to us. The main results of this article show that, under some regularity conditions, the Kaplan–Meier estimator enjoys uniform consistency with rates, and a stochastic process generated by the Kaplan–Meier estimator converges weakly to a certain Gaussian process with a specified covariance structure. Finally, an estimator of the limiting variance of the Kaplan–Meier estimator is proposed and its consistency is established.     

光滑平稳高斯过程极值几乎处处极限定理的一个推广

本文研究了连续均方可微的平稳高斯过程的极限性态. 通过选择一个不同于Tan (2013) 的权重函数, 在较弱的条件下得到了连续均方可微平稳高斯过程极值的一个几乎必然仅限定理, 推广了Tan(2013) 的结论.     

The evolution of a truncated gaussian probability density through time—modelling animal liveweights after selection

The form of the probability density derived from the evolution in time of a previously truncated frequency distribution of animal Liveweights is of interest in animal husbandry. Truncated frequency distributions arise when the heavier animals are sold for slaughter and the lighter animals retained. The demands of modern quality assurance schemes require that, given information on animal growth, the farmer is able to estimate the number of animals that would meet the specifications at some time in the future after truncation. Assuming that animal growth can be described by a linear stochastic differential equation, we derive an explicit expression for the probability density of animal Liveweights at any time after the truncation of an initial Gaussian density. It is shown that this probability density converges rapidly to a Gaussian density, so that after about 20 days of typical growth rates for lambs, the resulting density is practically indistinguishable from Gaussian.     

More comparisons of MLE with UMVUE for exponential families

Under some regularity conditions, the asymptotic expected deficiency (AED) of the maximum likelihood estimator (MLE) relative to the uniformly minimum variance unbiased estimator (UMVUE) for a given one-parameter estimable function of an exponential family is obtained. The exact expressions of the AED for normal, lognormal, inverse Gaussian, exponential (or gamma), Pareto, hyperbolic secant, Bernoulli, Poisson and geometric (or negative binomial) distributions are also derived.     

A global optimization algorithm using adaptive random search

A new random-search global optimization is described in which the variance of the step-size distribution is periodically optimized. By searching over a variance range of 8 to 10 decades, the algorithm finds the step-size distribution that yields the best local improvement in the criterion function. The variance search is then followed by a specified number of iterations of local random search where the step-size variance remains fixed. Periodic wide-range searches are introduced to ensure that the process does not stop at a local minimum. The sensitivity of the complete algorithm to various search parameters is investigated experimentally for a specific test problem. The ability of the method to locate global minima is illustrated by an example. The method also displays considerable problem independence, as demonstrated by two large and realistic example problems: (1) the identification of 25 parameters in a nonlinear model of a five-degrees-of-freedom mechanical dynamic system and (2) solution of a 24-parameter inverse problem required to identify a pulse train whose frequency spectrum matched a desired reference spectrum.     

On spectral estimation for a homogeneous random process on the circle

A homogeneous random process on the circle {X(P): P∈S} is a process whose mean is zero and whose covariance function depends only on the angular distance θ between the points, i.e. E{X(P)}≡0 and E{X(P)X(Q)}=R(θ). We assume that the homogeneous process X(P) is observed at a finite number of points, equally spaced on the circle. Given independent realizations of the process, we first propose unbiased estimates for the parameters of the aliased spectrum and for the covariance function. We assume further that the process is Gaussian. The exact distribution of the spectral estimates and the asymptotic distribution of the estimates of the covariance function are derived. Finally, it is shown that the estimates proposed are in fact the maximum likelihood estimates and that they have minimum variance in the class of unbiased estimates.     

Non‐homogeneous infinitely many sites discrete‐time model with exact coalescent

Kingman's coalescent is among the most fertile concepts in mathematical population genetics. However, it only approximates the exact coalescent process associated with the Wright–Fisher model, in which the ancestry of a sample does not have to be a binary tree. The distinction between the approximate and exact coalescent becomes important when population size is small and time has to be measured in discrete units (generations). In the present paper, we explore the exact coalescent, with mutations following the infinitely many sites model. The methods used involve random point processes and generating functionals. This allows obtaining joint distributions of segregating sites in arbitrary intervals or collections of intervals, and generally in arbitrary Borel subsets of two or more chromosomes. Using this framework it is possible to find the moments of the numbers of segregating sites on pairs of chromosomes, as well as the moments of the average of the number of pairwise differences, in the form that is more general than usually. In addition, we demonstrate limit properties of the first two moments under a range of demographic scenarios, including different patterns of population growth. This latter part complements results obtained earlier for Kingman's coalescent. Finally, we discuss various applications, including the analysis of fluctuation experiments, from which mutation rates of biological cells can be inferred. Copyright © 2009 John Wiley & Sons, Ltd.     

Optimal Scaling of Random Walk Metropolis Algorithms with Non-Gaussian Proposals

The asymptotic optimal scaling of random walk Metropolis (RWM) algorithms with Gaussian proposal distributions is well understood for certain specific classes of target distributions. These asymptotic results easily extend to any light tailed proposal distribution with finite fourth moment. However, heavy tailed proposal distributions such as the Cauchy distribution are known to have a number of desirable properties, and in many situations are preferable to light tailed proposal distributions. Therefore we consider the question of scaling for Cauchy distributed proposals for a wide range of independent and identically distributed (iid) product densities. The results are somewhat surprising as to when and when not Cauchy distributed proposals are preferable to Gaussian proposal distributions. This provides motivation for finding proposal distributions which improve on both Gaussian and Cauchy proposals, both for finite dimensional target distributions and asymptotically as the dimension of the target density, d → ∞. Throughout we seek the scaling of the proposal distribution which maximizes the expected squared jumping distance (ESJD).     

Strong local nondeterminism and exact modulus of continuity for spherical Gaussian fields

In this paper, we are concerned with sample path properties of isotropic spherical Gaussian fields on ${\mathbb{S}}^2$. In particular, we establish the property of strong local nondeterminism of an isotropic spherical Gaussian field based on the high-frequency behaviour of its angular power spectrum; we then exploit this result to establish an exact uniform modulus of continuity for its sample paths. We also discuss the range of values of the spectral index for which the sample functions exhibit fractal or smooth behaviour.     

Generalized z-Distributions and Related Stochastic Processes

The class of generalized z–distributions is defined and their properties are investigated. Ornstein–Uhlenbeck–type and self–similar generalized z–processes are constructed and described. Esscher transforms of the generalized z–processes and the mixed generalized z–processes are characterized. Finally, construction and some properties of generalized z–diffusions are also discussed.     

An economic interpretation of exact penalty functions using sensitivity analysis

It has already been demonstrated that under some assumptions, a local minimum of a constrained problem is also a local unconstrained minimum of a function which is called an exact penalty function. Here, we present the same result with a new demonstration. By using sensitivity analysis, we give an economic interpretation for exact penalty functions.     

Effect of phase-lags on Rayleigh wave propagation in initially stressed magneto-thermoelastic orthotropic medium

The present article deals with Rayleigh surface wave propagation in homogeneous magneto-thermoelastic orthotropic medium. Effect of initial stress and magnetic field on Rayleigh waves is studied in the context of three-phase-lag model of generalized thermoelasticity. The normal mode analysis is used to obtain the exact expressions for the displacement components, stresses and temperature distribution. Various frequency equations are derived and compared with the existing literature. The path of surface particles is elliptical during Rayleigh wave propagation. Effect of phase-lags on Rayleigh wave velocity, attenuation coefficient and specific loss are presented graphically. It is observed from graphical presentation that the effect of magnetic field and initial stress on different wave characteristics is pronounced.