正态AR(1)模型参数极大似然估计的精确解

本文对正态ＡＲ（１）模型，当Ｒ０已知，且时，证明了极大似然估计存在，但不唯一，这与Ｒ０，Ｒ１两个参数整体求极大似然估计的结果有本质上的不同，同时还研究了极大似然估计（）的数学特性与解析表达式．     

Testing Departures from Gamma, Rayleigh and Truncated Normal Distributions

This paper provides necessary and sufficient conditions for a solution to likelihood equations for an exponential family of distributions, which includes Gamma, Rayleigh and singly truncated normal distributions. Furthermore, the maximum likelihood estimator is obtained as a limit case when the equations have no solution. These results provide a way to test departures from Rayleigh and singly truncated normal distributions using the likelihood ratio test. A new easy way to test departures from a Gamma distribution is also introduced.     

The efficiency criteria problem for stochastic processes

This paper is concerned with the problem of finding a suitable (asymptotic) efficiency criterion for inference concerning parameters of stochastic processes. Special attention is aid to conditional exponential families of stochastic processes and to three tests based on the maximum likelihood estimate as well as to the likelihood ratio test. A contiguity calculation is used to show that a previously suggested criterion is inadequate and itself provides a partial solution to the problem. A heuristic argument is also put forward to support a proposition implying the optimality of the maximum likelihood estimate in a certain sense. Two examples which illustrate the theory are discussed.     

A Note on the Estimation of Semiparametric Two-Sample Density Ratio Models

In this paper,a semiparametric two-sample density ratio model is considered and the empirical likelihood method is applied to obtain the parameters estimation.A commonly occurring problem in computing is that the empirical likelihood function may be a concaveconvex function.Here a simple Lagrange saddle point algorithm is presented for computing the saddle point of the empirical likelihood function when the Lagrange multiplier has no explicit solution.So we can obtain the maximum empirical likelihood estimation (MELE) of parameters.Monte Carlo simulations are presented to illustrate the Lagrange saddle point algorithm.     

半参数两样本密度比率模型估计问题的一点注记

In this paper,a semiparametric two-sample density ratio model is considered and the empirical likelihood method is applied to obtain the parameters estimation.A commonly occurring problem in computing is that the empirical likelihood function may be a concaveconvex function.Here a simple Lagrange saddle point algorithm is presented for computing the saddle point of the empirical likelihood function when the Lagrange multiplier has no explicit solution.So we can obtain the maximum empirical likelihood estimation (MELE) of parameters.Monte Carlo simulations are presented to illustrate the Lagrange saddle point algorithm.     

An extension of the method of maximum likelihood and the Stein's problem

Summary An extension of the method of maximum likelihood leads to a natural solution of the problem raised by Stein, the inadmissibility of the ordinary maximum likelihood estimator for the mean of a multivariate normal distribution. Part of this paper was presented at an invited session of the Fall Meeting of the Mathematical Society of Japan, Tokyo, October 1975. The Institute of Statistical Mathematics     

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.     

Existence and consistency of the maximum likelihood estimator for the extreme value index

The paper is about the asymptotic properties of the maximum likelihood estimator for the extreme value index. Under the second order condition, Drees et al. [H. Drees, A. Ferreira, L. de Haan, On maximum likelihood estimation of the extreme value index, Ann. Appl. Probab. 14 (2004) 1179-1201] proved asymptotic normality for any solution of the likelihood equations (with shape parameter γ>−1/2) that is not too far off the real value. But they did not prove that there is a solution of the equations satisfying the restrictions.In this paper, the existence is proved, even for γ>−1. The proof just uses the domain of attraction condition (first order condition), not the second order condition. It is also proved that the estimator is consistent. When the second order condition is valid, following the current proof, the existence of a solution satisfying the restrictions in the above-cited reference is a direct consequence.     

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.     

Nonlinear Smoothing and the EM Algorithm for Positive Integral Equations of the First Kind

We study a modification of the EMS algorithm in which each step of the EMS algorithm is preceded by a nonlinear smoothing step of the form , where S is the smoothing operator of the EMS algorithm. In the context of positive integral equations (à la positron emission tomography) the resulting algorithm is related to a convex minimization problem which always admits a unique smooth solution, in contrast to the unmodified maximum likelihood setup. The new algorithm has slightly stronger monotonicity properties than the original EM algorithm. This suggests that the modified EMS algorithm is actually an EM algorithm for the modified problem. The existence of a smooth solution to the modified maximum likelihood problem and the monotonicity together imply the strong convergence of the new algorithm. We also present some simulation results for the integral equation of stereology, which suggests that the new algorithm behaves roughly like the EMS algorithm. Accepted 1 April 1997     

A notion of an obstructive residual likelihood

Summary A new notion of an obstructive residual likelihood is proposed and explored. Examples where the conditional maximum likelihood estimator is preferable to the unconditional maximum likelihood estimator are discussed. In these examples the residual likelihood can be obstructive in deriving a preferable estimator, when the maximum likelihood criterion is applied. This notion is different from a similar notion ancillarity, which simply emphasizes that a residual likelihood is un-informative. The Institute of Statistical Mathematics     

The maximum full and partial likelihood estimators in the proportional hazard model

Summary The maximum full likelihood estimator in the proportional hazard model is explored in relation to the maximum partial likelihood estimator. In the scalar parameter case both the estimators have a common sign, and the absolute value of the former is strictly greater than that of the latter except for trivial cases. We point out also that the maximum full likelihood estimator after a simple modification of the likelihood equation provides a good approximation to the maximum partial likelihood estimator. Similar results are valid for the likelihood ratio tests. The Institute of Statistical Mathematics     

A recognition problem for a vector alphabet generating a sequence with a quasiperiodic structure

An off-line recognition problem is analyzed for a vector alphabet generating sequences with quasiperiodic vector fragments that coincide with alphabet vectors. It is shown that the solution of this problem reduces to solving a special optimization problem. It is proved that the problem considered is solvable in polynomial time, and an algorithm for its exact solution is justified. The algorithm ensures the maximum likelihood recognition of a vector alphabet for the case of additive noise which is a Gaussian sequence of independent random values having an identical distribution.     

Asymptotic properties of conditional maximum likelihood estimator in a certain exponential model

The conditional maximum likelihood estimator is suggested as an alternative to the maximum likelihood estimator and is favorable for an estimator of a dispersion parameter in the normal distribution, the inverse-Gaussian distribution, and so on. However, it is not clear whether the conditional maximum likelihood estimator is asymptotically efficient in general. Consider the case where it is asymptotically efficient and its asymptotic covariance depends only on an objective parameter in an exponential model. This remand implies that the exponential model possesses a certain parallel foliation. In this situation, this paper investigates asymptotic properties of the conditional maximum likelihood estimator and compares the conditional maximum likelihood estimator with the maximum likelihood estimator. We see that the bias of the former is more robust than that of the latter and that two estimators are very close, especially in the sense of bias-corrected version. The mean Pythagorean relation is also discussed.     

A Note on Convergence of the Equi-Energy Sampler

This paper studies maximum likelihood estimation for a parameterised elliptic diffusion in a manifold. The focus is on asymptotic properties of maximum likelihood estimates obtained from continuous time observation. These are well known when the underlying manifold is a Euclidean space. However, no systematic study exists in the case of a general manifold. The starting point is to write down the likelihood function and equation. This is achieved using the tools of stochastic differential geometry. Consistency, asymptotic normality and asymptotic optimality of maximum likelihood estimates are then proved, under regularity assumptions. Numerical computation of maximum likelihood estimates is briefly discussed.     

Pairwise likelihood inference for multivariate ordinal responses with applications to customer satisfaction

A common practice in customer satisfaction analysis is to administer surveys where subjects are asked to express opinions on a number of statements, or satisfaction scales, by use of ordered categorical responses. Motivated by this application, we propose a pseudo‐likelihood approach to estimate the dependence structure among multivariate categorical variables. As it is commonly carried out in this area, we assume that the responses are related to latent continuous variables that are truncated to induce categorical responses. A Gaussian likelihood is assumed for the latent variables leading to the so‐called ordered probit model. Because the calculation of the exact likelihood is computationally demanding, we adopt an approximate solution based on pairwise likelihood. To asses the performance of the approach, simulation studies are conducted comparing the proposed method with standard likelihood methods. A parametric bootstrap approach to evaluate the variance of the maximum pairwise likelihood estimator is proposed and discussed. An application to customer satisfaction survey is performed showing the effectiveness of the approach in the presence of covariates and under other generalizations of the model. Copyright © 2015 John Wiley & Sons, Ltd.     

Incorporating scatter search and threshold accepting in finding maximum likelihood estimates for the multinomial probit model

This paper presents a procedure that incorporates scatter search and threshold accepting to find the maximum likelihood estimates for the multinomial probit (MNP) model. Scatter search, widely used in optimization-related studies, is a type of evolutionary algorithm that uses a small set of solutions as the selection pool for mating and generating new solutions to search for a globally optimal solution. Threshold accepting is applied to the scatter search to improve computational efficiency while maintaining the same level of solution quality. A set of numerical experiments, based on synthetic data sets with known model specifications and error structures, were conducted to test the effectiveness and efficiency of the proposed framework. The results indicated that the proposed procedure enhanced performance in terms of likelihood function value and computational efficiency for MNP model estimation as compared to the original scatter search framework.     

Maximum likelihood estimation of covariance matrices under simple tree ordering

The closed-form maximum likelihood estimators for the completely balanced multivariate one-way random effect model are obtained by Anderson et al. (Ann. Statist. 14 (1986) 405). It remains open whether there exist the closed-form maximum likelihood estimators for the more general completely balanced multivariate multi-way random effects models. In this paper, a new parameterization technique for covariance matrices is used to grasp the inside structure of likelihood function so that the maximum likelihood equations can be dramatically simplified. As such we obtain the closed-form maximum likelihood estimators of covariance matrices for Wishart density functions over the simple tree ordering set, which can then be applied to get the maximum likelihood estimators for the completely balanced multivariate multi-way random effects models without interactions.     

Maximum likelihood estimation of the Heston stochastic volatility model using asset and option prices: an application of nonlinear filtering theory

Let us suppose that the dynamics of the stock prices and of their stochastic variance is described by the Heston model, that is by a system of two stochastic differential equations with a suitable initial condition. The aim of this paper is to estimate the parameters of the Heston model and one component of the initial condition, that is the initial stochastic variance, from the knowledge of the stock and option prices observed at discrete times. The option prices considered refer to an European call on the stock whose prices are described by the Heston model. The method proposed to solve this problem is based on a filtering technique to construct a likelihood function and on the maximization of the likelihood function obtained. The estimated parameters and initial value component are characterized as being a maximizer of the likelihood function subject to some constraints. The solution of the filtering problem, used to construct the likelihood function, is based on an integral representation of the fundamental solution of the Fokker–Planck equation associated to the Heston model, on the use of the wavelet expansions presented in (Fatone et al. in High performance algorithms based on a new wavelet expansion for time dependent acoustic obstacle scattering. Commun. Computat. Phys. (2007), Research Developments in Acoustics, vol. 2, pp. 39–69. Transworld Research Network, Kerala (2005), New wavelet bases made of piecewise polynomial functions: approximation theory, quadrature rules and applications to kernel sparsification and image compression. SIAM J. Sci. Comput. (submitted)) to approximate the integral kernel appearing in the representation formula of the fundamental solution, on a simple truncation procedure to exploit the sparsifying properties of the wavelet expansions and on the use of the fast Fourier transform (FFT). The use of these techniques generates a very efficient and fully parallelizable numerical procedure to solve the filtering problem, this last fact makes possible to evaluate very efficiently the likelihood function and its gradient. As a byproduct of the solution of the filtering problem we have developed a stochastic variance tracking technique that gives very good results in numerical experiments. The maximum likelihood problem used in the estimation procedure is a low dimensional constrained optimization problem, its solution with ad hoc techniques is justified by the computational cost of evaluating the likelihood function and its gradient. We use parallel computing and a variable metric steepest ascent method to solve the maximum likelihood problem. Some numerical examples of the estimation problem using synthetic and real data, that is data relative to an index of the Milano stock exchange (S&PMIB30), obtained with a parallel implementation of the previous numerical method are presented. Very impressive speed up factors are obtained in the numerical examples using the parallel implementation of the numerical method proposed. The website: http://www.econ.univpm.it/pacelli/mariani/finance/w1 contains animations and some auxiliary material that helps the understanding of this paper and makes available to the interested users the computer programs used to produce the numerical experience presented. The numerical experience reported in this paper has been obtained using the computing grid of ENEA (Rome, Italy). The support and sponsorship of ENEA are gratefully acknowledged.     

Pareto分布环境因子的估计及其应用

给出了Pareto分布环境因子的定义,讨论了在定数截尾样本下Pareto分布环境因子的极大似然估计和修正极大似然估计,并尝试把环境因子用于可靠性评估中.最后运用Monte Carlo方法对极大似然估计,修正极大似然估计和可靠性指标的均方误差(MSE),进行了模拟比较,结果表明修正极大似然估计优于极大似然估计且考虑环境因子的可靠性评估结果较好.