SEM Modeling with Singular Moment Matrices Part I: ML-Estimation of Time Series

A structural equation model (SEM) with deterministic intercepts is introduced. The Gaussian likelihood function does not contain determinants of sample moment matrices and is thus well-defined for only one statistical unit. The SEM is applied to the dynamic state space model and compared with the Kalman filter (KF) approach. The likelihood of both methods are shown to be equivalent, but for long time series numerical problems occur in the SEM approach, which are traced to the inversion of the latent state covariance matrix. Both approaches are compared on several aspects. The SEM approach is now open for idiographic (N = 1) analysis and estimation of panel data with correlated units.     

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.     

Statistical Inference for the Parameter of Rayleigh Distribution Based on Progressively Typ e-I Interval Censored Sample

In this paper, the estimation of parameters based on a progressively type-I interval censored sample from a Rayleigh distribution is studied. Different methods of estimation are discussed. They include...     

On Gaussian HJM framework for Eurodollar Futures

One of the standard tools for the theoretical analysis of fixed income securities and their associated derivatives is the term structure model of Heath, Jarrow and Morton. In this paper the question, what specific HJM model is consistent with the observed price of an Eurodollar Futures contract? is discussed. Eurodollar Futures, apart from being the most heavily traded futures are connected to London Inter Bank Offered Rate (LIBOR) and to domestic monetary conditions. The answer to the above question will help in pricing any new derivative on Eurodollar Futures or the one that is not heavily traded. A simple tool to measure the adequacy of different HJM structures that may be used to model Eurodollar Futures price process is suggested. Moreover, the question of estimation of parameters of these models by different methods—method of realized volatility, method of maximum likelihood (ML) and a two‐stage method that combines both the realized volatility and ML—is addressed. Although it sounds like a typical statistical procedure, one must be careful in applying standard statistical techniques that are not suitable under arbitrage theory, in particular, ML method. Copyright © 2010 John Wiley & Sons, Ltd.     

Asymptotic Behavior of Poisson Kernels on NA Groups

On a Lie group S = NA, that is a split extension of a nilpotent Lie group N by a one-parameter group of automorphisms A, a probability measure μ is considered and treated as a distribution according to which transformations s ∈ S acting on N = S/A are sampled. Under natural conditions, formulated some over thirty years ago, there is a μ-invariant measure m on N. Properties of m have been intensively studied by a number of authors. The present article deals with the situation when μ(A) = ?(s _t  ∈ A), where ?⁺ ? t → s _t  ∈ S is the diffusion on S generated by a second order subelliptic, hypoelliptic, left-invariant operator on S. In this article the most general operators of this kind are considered. Precise asymptotic for m at infinity and for the Green function of the operator are given. To achieve this goal a pseudodifferential calculus for operators with coefficients of finite smoothness is formulated and applied.     

指数分布冷贮备系统产品的统计分析——转换开关指数型的情形

给出了全样本场合下指数分布冷贮备系统产品寿命分布中参数θ≠λ时的矩估计和极大似然估计,通过Monte-Carlo给出了参数矩估计的精度,考察了1000次满足条件时所需要的模拟次数,随着样本量的增大,矩估计存在的比率逐渐增大,而极大似然估计的结果与样本有关.同时给出了参数θ=λ时的矩估计、极大似然估计和逆矩估计,通过Monte-Carlo模拟考察了参数点估计精度,认为矩估计比较优.文章还给出了求参数区间估计的两种方法——精确方法和近似方法,通过Monte-Carlo模拟认为精确方法精度较高.     

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．     

Optimal designs for parameter estimation of the Ornstein–Uhlenbeck process

This paper deals with optimal designs for Gaussian random fields with constant trend and exponential correlation structure, widely known as the Ornstein–Uhlenbeck process. Assuming the maximum likelihood approach, we study the optimal design problem for the estimation of the trend µ and the correlation parameter θ using a criterion based on the Fisher information matrix. For the problem of trend estimation, we give a new proof of the optimality of the equispaced design for any sample size (see Statist. Probab. Lett. 2008; 78 :1388–1396). We also show that for the estimation of the correlation parameter, an optimal design does not exist. Furthermore, we show that the optimal strategy for µ conflicts with the one for θ, since the equispaced design is the worst solution for estimating the correlation. Hence, when the inferential purpose concerns both the unknown parameters we propose the geometric progression design, namely a flexible class of procedures that allow the experimenter to choose a suitable compromise regarding the estimation's precision of the two unknown parameters guaranteeing, at the same time, high efficiency for both. Copyright © 2008 John Wiley & Sons, Ltd.     

线性混合模型中方差分量的估计与QR分解

在线性混合模型中, 极大似然估计是一种很重要的估计方法, 但是它常常需要通过迭代求解. 应用设计阵的QR分解, 可以把设计阵变换成上三角矩阵. 这样可以降低参与迭代运算的矩阵的阶数, 还可以减少参与运算的数据量, 从而提高运算的速度. 本文讨论了QR分解在EM算法中的应用, 并用模拟的方法验证了QR分解可以极大的提高运算的速度. 本文同时讨论了QR分解在另外一种估计方法, 即ANOVA估计中的应用.     

Moment Estimation for Multivariate Extreme Value Distribution in a Nested Logistic Model

This paper considers multivariate extreme value distribution in a nested logistic model. The dependence structure for this model is discussed. We find a useful transformation that transformed variables possess the mixed independence. Thus, the explicit algebraic formulae for a characteristic function and moments may be given. We use the method of moments to derive estimators of the dependence parameters and investigate the properties of these estimators in large samples via asymptotic theory and in finite samples via computer simulation. We also compare moment estimation with a maximum likelihood estimation in finite sample sizes. The results indicate that moment estimation is good for all practical purposes.     

Kullback-Leibler Information Consistent Estimation for Censored Data

This paper is intended as an investigation of parametric estimation for the randomly right censored data. In parametric estimation, the Kullback-Leibler information is used as a measure of the divergence of a true distribution generating a data relative to a distribution in an assumed parametric model M. When the data is uncensored, maximum likelihood estimator (MLE) is a consistent estimator of minimizing the Kullback-Leibler information, even if the assumed model M does not contain the true distribution. We call this property minimum Kullback-Leibler information consistency (MKLI-consistency). However, the MLE obtained by maximizing the likelihood function based on the censored data is not MKLI-consistent. As an alternative to the MLE, Oakes (1986, Biometrics, 42, 177–182) proposed an estimator termed approximate maximum likelihood estimator (AMLE) due to its computational advantage and potential for robustness. We show MKLI-consistency and asymptotic normality of the AMLE under the misspecification of the parametric model. In a simulation study, we investigate mean square errors of these two estimators and an estimator which is obtained by treating a jackknife corrected Kaplan-Meier integral as the log-likelihood. On the basis of the simulation results and the asymptotic results, we discuss comparison among these estimators. We also derive information criteria for the MLE and the AMLE under censorship, and which can be used not only for selecting models but also for selecting estimation procedures.     

Approximate Maximum Likelihood Estimation of Circle Parameters

The estimation of a circles centre and radius from a set of noisy measurements of its circumference has many applications. It is a problem of fitting a circle to the measurements and the fit can be in algebraic or geometric distances. The former gives linear equations, while the latter yields nonlinear equations. Starting from estimation theory, this paper first proves that the maximum likelihood (ML), i.e., the optimal estimation of the circle parameters, is equivalent to the minimization of the geometric distances. It then derives a pseudolinear set of ML equations whose coefficients are functions of the unknowns. An approximate ML algorithm updates the coefficients from the previous solution and selects the solution that gives the minimum cost. Simulation results show that the ML algorithm attains the Cramer-Rao lower bound (CRLB) for arc sizes as small as 90°. For arc sizes of 15° and 5° the ML algorithm errors are slightly above the CRLB, but lower than those of other linear estimators.Communicated by L. C. W. Dixon     

On the Maximum Likelihood estimation of a linear structural relationship when the intercept is known

This paper considers the Maximum Likelihood (ML) estimation of the five parameters of a linear structural relationship y = α + βx when α is known. The parameters are β, the two variances of observation errors on x and y, the mean and variance of x. When the ML estimates of the parameters cannot be obtained by solving a simple simultaneous system of five equations, they are found by maximizing the likelihood function directly. Some asymptotic properties of the estimates are also obtained.     

Strong consistency of MLE for finite uniform mixtures when the scale parameters are exponentially small

We consider maximum likelihood estimation of finite mixture of uniform distributions. We prove that maximum likelihood estimator is strongly consistent, if the scale parameters of the component uniform distributions are restricted from below by exp(−n ^d ), 0<d<1, wheren is the sample size.     

On (Semi)Regular Morphisms

Let M and N be right R-modules. Hom(M, N) is called regular if for each f ∈ Hom(M, N), there exists g ∈ Hom(N, M) such that f = fgf. Let [M, N] = Hom _R (M, N). We prove that if M is finitely generated, then [M, N] is regular if and only if every homomorphism M → N is locally split. In this article, we also study the substructures of Hom(M, N) such as the Jacobson radical J[M, N], the singular ideal Δ[M, N], and the co-singular ideal ?[M, N]. We prove several new results. The question is to characterize when the Jacobson radical is equal to the singular ideal Δ[M, N] or the co-singular ideal ?[M, N] under injectivity and projectivity.     

On ADS Modules and Rings

A right module M over a ring R is said to be ADS if for every decomposition M = S ⊕ T and every complement T′ of S, we have M = S ⊕ T′. In this article, we study and provide several new characterizations of this new class of modules. We prove that M is semisimple if and only if every module in σ[M] is ADS. SC and SI rings also characterized by the ADS notion. A ring R is right SC-ring if and only if every 2-generated singular R-module is ADS.     

Regular Action in Rings

Let R be a ring with identity, X(R) the set of all nonzero non-units of R and G(R) the group of all units of R. By considering left and right regular actions of G(R) on X(R), the following are investigated: (1) For a local ring R such that X(R) is a union of n distinct orbits under the left (or right) regular action of G(R) on X(R), if J ⁿ  ≠ 0 = J ⁿ⁺¹ where J is the Jacobson radical of R, then the set of all the distinct ideals of R is exactly {R, J, J ²,…, J ⁿ , 0}, and each orbit under the left regular action is equal to the one under the right regular action. (2) Such a ring R is left (and right) duo ring. (3) For the full matrix ring S of n × n matrices over a commutative ring R, the number of orbits under left regular action of G(S) on X(S) is equal to the number of orbits under right regular action of G(S) on X(S); the result also holds for the ring of n × n upper triangular matrices over R.     

混合双参数广义Pareto分布的参数估计

Pareto分布族因其厚尾特点,在金融分析、寿命分析中都是非常重要的统计模型.但是对于混合双参广义Pareto分布,在模型参数估计时,传统的矩法估计和极大似然估计在理论上可以实现,实践时比较困难.本文应用EM算法之ECM算法,研究了混合广义Pareto分布在完全数据场合下的参数估计问题,并模拟说明EM算法来估计混合广义Pareto分布是一种容易实现又非常有效的方法.     

Estimating complex covariance by observing two variables at a time

The estimation of covariance matrices is central in array signal processing systems. This note addresses complex covariance estimation for the situation, where the complex data are available only as independent pairwise sets (observations) corresponding to individual elements of the matrix. The formulation for the empirical estimate and the normal maximum likelihood estimate is developed for the general case of different sample sizes for each observation. The approach allows, for example, the estimate of the p by p covariance matrix of a p-port sensor array from a two-port measurement instrument.