Semilocal convergence of an eighth-order method in Banach spaces and its computational efficiency

The aim of this paper is to study the semilocal convergence of the eighth-order iterative method by using the recurrence relations for solving nonlinear equations in Banach spaces. The existence and uniqueness theorem has been proved along with priori error bounds. We have also presented the comparative study of the computational efficiency in case of R^m with some existing methods whose semilocal convergence analysis has been already discussed. Finally, numerical application on nonlinear integral equations is given to show our approach.     

Alternating Minimization as Sequential Unconstrained Minimization: A Survey

Sequential unconstrained minimization is a general iterative method for minimizing a function over a given set. At each step of the iteration we minimize the sum of the objective function and an auxiliary function. The aim is to select the auxiliary functions so that, at least, we get convergence in function value to the constrained minimum. The SUMMA is a broad class of these methods for which such convergence holds. Included in the SUMMA class are the barrier-function methods, entropic and other proximal minimization algorithms, the simultaneous multiplicative algebraic reconstruction technique, and, after some reformulation, penalty-function methods. The alternating minimization method of Csiszár and Tusnády also falls within the SUMMA class, whenever their five-point property holds. Therefore, the expectation maximization maximum likelihood algorithm for the Poisson case is also in the SUMMA class.     

Iterative continuous maximum-likelihood reconstruction method

In this paper we continue our studying of the iterative maximum-likelihood reconstruction method. We consider only the continuous case and show some convergence properties of the algorithm. In the discrete case convergence has already been proved. An example demonstrating divergence of the iterative method shows how to handle the continuous case.     

指数分布抽样基本定理及在三参数一般指数分布参数估计中的应用

讨论三参数一般指数分布的参数估计,首先讨论了三参数一般指数分布参数的最大似然估计的求解问题,当其中参数α=1时,应用指数分布抽样基本定理,得到了三参数一般指数分布其它参数的一致最小方差无偏估计;并且由此给出求解三参数一般指数分布参数最大似然估计的迭代方法,得到了三参数一般指数分布参数最大似然估计的近似值,给出了模拟结果以说明迭代方法的收敛性;并以相关文献的观察数据作为样本,得到了三参数一般指数分布的参数估计,从而说明了迭代方法的有效性.     

Landweber iterative methods for angle-limited image reconstruction

We introduce a general iterative scheme for angle-limited image reconstruction based on Landwebet＇s method. We derive a representation formula for this scheme and consequently establish its convergence conditions. Our results suggest certain relaxation strategies for an accelerated convergence for angle-limited image reconstruction in L^2-norm comparing with alternative projection methods. The convolution-backprojection algorithm is given for this iterative process.     

Perturbed recurrence relations II the general case

In this paper, we give some new explicit relations between two families of polynomials defined by recurrence relations of all order. These relations allow us to analyze, even in the Sobolev case, how some properties of a family of orthogonal polynomials are affected when the coefficients of the recurrence relation and the order are perturbed. In a paper we have already given a method which allows us to study the polynomials defined by a three-term recurrence relation. Also here some generalizations are given.     

Generalized least squares and maximum likelihood estimations of multivariate polychoric correlations

This paper investigates the generalized least squares estimation and the maximum likelihood estimation of the parameters in a multivariate polychoric correlations model, based on data from a multidimensional contingency table. Asymptotic properties of the estimators are discussed. An iterative procedure based on the Gauss-Newton algorithm is implemented to produce the generalized least squares estimates and the standard errors estimates. It is shown that via an iteratively reweighted method, the algorithm produces the maximum likelihood estimates as well. Numerical results on the finite sample behaviors of the methods are reported.     

Linear mixed models and penalized least squares

Linear mixed-effects models are an important class of statistical models that are used directly in many fields of applications and also are used as iterative steps in fitting other types of mixed-effects models, such as generalized linear mixed models. The parameters in these models are typically estimated by maximum likelihood or restricted maximum likelihood. In general, there is no closed-form solution for these estimates and they must be determined by iterative algorithms such as EM iterations or general nonlinear optimization. Many of the intermediate calculations for such iterations have been expressed as generalized least squares problems. We show that an alternative representation as a penalized least squares problem has many advantageous computational properties including the ability to evaluate explicitly a profiled log-likelihood or log-restricted likelihood, the gradient and Hessian of this profiled objective, and an ECME update to refine this objective.     

交替迭代最大似然估计量的偏倚校正

In this paper, we give a definition of the alternating iterative maximum likelihood estimator (AIMLE) which is a biased estimator. Furthermore we adjust the AIMLE to result in asymptotically unbiased and consistent estimators by using a bootstrap iterative bias correction method as in Kuk (1995). Two examples and simulation results reported illustrate the performance of the bias correction for AIMLE.     

Binary clustering with missing data

A clustering method is presented for analysing multivariate binary data with missing values. When not all values are observed, Govaert³ has studied the relations between clustering methods and statistical models. The author has shown how the identification of a mixture of Bernoulli distributions with the same parameter for all clusters and for all variables corresponds to a clustering criterion which uses L₁ distance characterizing the MNDBIN method (Marchetti⁸). He first generalized this model by selecting parameters which can depend on variables and finally by selecting parameters which can depend both on variables and on clusters. We use the previous models to derive a clustering method adapted to missing data. This method optimizes a criterion by a standard iterative partitioning algorithm which removes the necessity either to ignore objects or to substitute the missing data. We study several versions of this algorithm and, finally, a brief account is given of the application of this method to some simulated data.     

Likelihood estimation of the extremal index

The article develops the approach of Ferro and Segers (J.R. Stat. Soc., Ser. B 65:545, 2003) to the estimation of the extremal index, and proposes the use of a new variable decreasing the bias of the likelihood based on the point process character of the exceedances. Two estimators are discussed: a maximum likelihood estimator and an iterative least squares estimator based on the normalized gaps between clusters. The first provides a flexible tool for use with smoothing methods. A diagnostic is given for condition , under which maximum likelihood is valid. The performance of the new estimators were tested by extensive simulations. An application to the Central England temperature series demonstrates the use of the maximum likelihood estimator together with smoothing methods.      

Strong Consistency of the Maximum Product of Spacings Estimates with Applications in Nonparametrics and in Estimation of Unimodal Densities

Distributions with unimodal densities are among the most commonly used in practice. However, for many unimodal distribution families the likelihood functions may be unbounded, thereby leading to inconsistent estimates. The maximum product of spacings (MPS) method, introduced by Cheng and Amin and independently by Ranneby, has been known to give consistent and asymptotically normal estimators in many parametric situations where the maximum likelihood method fails. In this paper, strong consistency theorems for the MPS method are obtained under general conditions which are comparable to the conditions of Bahadur and Wang for the maximum likelihood method. The consistency theorems obtained here apply to both parametric models and some nonparametric models. In particular, in any unimodal distribution family the asymptotic MPS estimator of the underlying unimodal density is shown to be universally L¹ consistent without any further conditions (in parametric or nonparametric settings).     

On some properties of the “apparent length” distribution and a compound distribution

This paper extends and supplements some of the properties of the apparent length distribution introduced by Baker (J. Appl. Statist. 27(1) (2000) 2–21). Recurrence relations for moments and formulae for moments derived via difference and differential operators will be given. We also establish maximum likelihood estimator and Bayesian estimators of a parameter of this distribution. Compound distributions based on the apparent length distribution are also discussed.     

Linear precision for parametric patches

We give a precise mathematical formulation for the notions of a parametric patch and linear precision, and establish their elementary properties. We relate linear precision to the geometry of a particular linear projection, giving necessary (and quite restrictive) conditions for a patch to possess linear precision. A main focus is on linear precision for Krasauskas’ toric patches, which we show is equivalent to a certain rational map on \mathbb C\mathbb P^d{\mathbb C}{\mathbb P}^d being a birational isomorphism. Lastly, we establish the connection between linear precision for toric surface patches and maximum likelihood degree for discrete exponential families in algebraic statistics, and show how iterative proportional fitting may be used to compute toric patches.     

Bivariate Distributions with Pearson Type VII Conditionals

In this article the most general class of bivariate distributions such that both conditional densities are Pearson Type VII, with fixed shape parameter, is fully characterized. Some of its properties and relations with other distributions are explored. The estimation of parameters is considered by the methods of maximum likelihood and pseudolikelihood and a method for random variate generation is presented along with a simulation experiment. Bivariate and multivariate extensions of the Pearson Type VII conditionals distribution are also discussed.     

含参变量的三阶方向牛顿法及其收敛性

通过递推关系归纳迭代公式的讨论,研究含多个未知数的非光滑方程组及其收敛性,并以此证明希尔伯特空间上的含参变量的实系数非线性方程组的三阶方向牛顿法的半局部收敛性,给出解的存在性以及先验误差界.     

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本文综述混合效应模型参数估计方面的若干新进展. 平衡混合效应方差分析模型的协方差阵具有一定结构. 对这类模型, 文献[1]提出了参数估计的一种新方法, 称为谱分解法. 新方法的突出特点是, 能同时给出固定效应和方差分量的估计, 前者是线性的, 后者是二次的,且相互独立. 而后, 文献[2--9]证明了谱分解估计的进一步的统计性质, 同时给出了协方差阵对应的估计, 它不仅是正定阵, 而且可获得它的风险函数, 这些文献还研究了谱分解估计与方差分析估计, 极大似然估计, 限制极大似然估计以及最小范数二次无偏估计的关系. 本文综述这一方向的部分研究成果, 并提出一些待进一步研究的问题.     

Parametric estimation of discretely sampled Gamma-OU processes

The stationary Gamma-OU processes are recommended to be the volatility of the financial assets. A parametric estimation for the Gamma-OU processes based on the discrete observations is considered in this paper. The estimator of an intensity parameter A and its convergence result are given, and the simulations show that the estimation is quite accurate. Assuming that the parameter A is estimated, the maximum likelihood estimation of shape parameter c and scale parameter a, whose likelihood function is not explicitly computable, is considered. By means of the Gaver-Stehfest algorithm, we construct an explicit sequence of approximations to the likelihood function and show that it converges the true (but unkown) one. Maximizing the sequence results in an estimator that converges to the true maximum likelihood estimator and the approximation shares the asymptotic properties of the true maximum likelihood estimator. Some simulation experiments reveal that this method is still quite accurate in most of rational situations for the background of volatility.     

Numerical approximation of one phase quadrature domains

In this work, we present two numerical schemes for a free boundary problem that is called one phase quadrature domain. In the first method, using the properties of a given free boundary problem, we derive a method that leads us to a fast iterative solver. The iteration procedure is adapted to work in the case when topology changes. The second method is based on shape reconstruction to establish an efficient shape Quasi‐Newton method. Various numerical experiments confirm the efficiency of the derived numerical methods. © 2013 Wiley Periodicals, Inc. Numer Methods Partial Differential Eq, 2013     

Direct and inverse problems for prestressed functionally graded plates in the framework of the Timoshenko model

In this study, modelling and identification of prestress state in functionally graded plate within the framework of the Timoshenko theory are discussed. With the help of variational principles, statements of boundary problems for stationary vibration of inhomogeneous prestressed plates have been derived taking into account various factors of prestress state. The comparative analysis of classical and nonclassical models has been conducted. The effect of the prestress state factors on the solution characteristics has been estimated. New approaches to solving the inverse problems on a reconstruction of inhomogeneous prestress functions in a functionally graded plate have been proposed on the basis of derivation of reciprocity relations and iterative regularization. The results of numerical reconstruction experiments are presented; practical recommendations on a selection of frequency range for the purpose of getting the highest reconstruction accuracy are given.