浅谈似然函数的两种不同定义

本文通过两个具体例题表明当连续型总体可能的取值范围不是(-∞,+∞)时,利用第一种似然函数的定义解决点估计问题,学生不仅能够很容易地掌握最大似然估计法,同时对样本来自总体且估计离不开样本这一统计思想加深了理解。     

Estimating the Derivative of a Convex Density

We obtain a relation between the time between two bird-catchings and the total resting period of a bird, leading to the problem of estimating the derivative of a convex density. We state a fundamental result on the nonparametric maximum likelihood estimator of a convex density. Further, we derive the optimal rate in the minimax risk sense for estimating the derivative of a convex density.     

Nonparametric estimators of the bivariate survival function under random censoring

A large number of proposals for estimating the bivariate survival function under random censoring have been made. In this paper we discuss the most prominent estimators, where prominent is meant in the sense that they are best for practical use; Dabrowska's estimator, the Prentice–Cai estimator, Pruitt's modified EM-estimator, and the reduced data NPMLE of van der Laan. We show how these estimators are computed and present their intuitive background. The asymptotic results are summarized. Furthermore, we give a summary of the practical performance of the estimators under different levels of dependence and censoring based on extensive simulation results. This leads also to a practical advise.     

Asymptotic normality of the NPMLE of linear functionals for interval censored data, case 1

We give a new proof of the asymptotic normality of a class of linear functionals of the nonparametric maximum likelihood estimator (NPMLE) of a distribution function with "case 1" interval censored data. In particular our proof simplifies the proof of asymptotic normality of the mean given in Groeneboom and Wellner (1992). The proof relies strongly on a rate of convergence result due to van de Geer (1993), and methods from empirical process theory.     

Estimation of variances in orthogonal finite discrete spectrum linear regression models

The Invariant Quadratic Estimators, the Maximum Likelihood Estimator (MLE) and Restricted Maximum Likelihood Estimator (REML) of variances in an orthogonal Finite Discrete Spectrum Linear Regression Model (FDSLRM) are derived and the problems of unbiasedness and consistency of these estimators are investigated.Acknowledgement. The research was supported by the grants 1/0272/03, 1/0264/03 and 2/4026/04 of the Slovak Scientific Grant Agency VEGA.     

Maximum Likelihood Estimation of a General Unbalanced Spatial Random Effects Model: a Monte Carlo Study

Abstract

This paper discusses the maximum likelihood estimator of a general unbalanced spatial random effects model with normal disturbances, assuming that some observations are missing at random. Monte Carlo simulations show that the maximum likelihood estimator for unbalanced panels performs well and that missing observations affect mainly the root mean square error. As expected, these estimates are less efficient than those based on the unobserved balanced model, especially if the share of missing observations is large or spatial autocorrelation in the error terms is pronounced.

Estimation de vraisemblance maximale d'un modèle général d'effets aléatoires spatiaux déséquilibré: une étude Monte Carlo

RÉSUMÉ La présente communication se penche sur l'estimateur du maximum de vraisemblance d'un modèle général d'effets aléatoires spatiaux déséquilibré avec des perturbations normales, en supposant l'absence aléatoire de certaines observations. Des simulations de Monte Carlo montrent que des groupes déséquilibrés se comporte bien, et que les observations manquantes affectent principalement l'erreur de la moyenne quadratique. Comme prévu, ces évaluations sont moins efficaces que celles qui sont basées sur le modèle équilibré non observé, notamment si la part des observations manquantes est importantes, ou l'on déclare une autocorrélation spatiale dans les termes d'erreur.

Estimación de la probabilidad máxima de un modelo espacial general desequilibrado de efectos al azar: un estudio de Monte Carlo

RÉSUMÉN Este trabajo discute el estimador de probabilidad máxima de un modelo espacial general desequilibrado de efectos al azar con alteraciones normales, suponiendo que faltan algunas observaciones al azar. Las simulaciones de Monte Carlo muestran que el estimador de probabilidad máxima para los paneles desequilibrados funciona satisfactoriamente, y que las observaciones omisas afectan principalmente al error de la media cuadrática. Como se suponía, estas estimaciones son menos eficientes que las basadas en el modelo equilibrado inadvertido, especialmente si la cantidad de omisiones es grande/o la autocorrelación en los términos de error es pronunciada.

     

半参数变系数空间滞后模型的截面极大似然估计

在线性参数空间滞后模型中,解释变量的系数一般假设为固定常数,本文首先放松了这种假设,将解释变量的系数设定为某一变量的未知函数,提出一类全新的半参数变系数空间滞后模型;其次导出了该模型的截面极大似然估计,并证明了该估计的一致性;最后用蒙特卡洛数值模拟方法考察了该估计在小样本条件下的性质,数值模拟结果显示我们提出的估计方法在小样本条件下依然有优良的表现。     

Constructive identification of the mixed proportional hazards model

We give a new proof of the identifiably of the MPH model. This proof is constructive: it is a recipe for constructing the triple—regression function, base-line hazard, and distribution of the individual effect—from the observed cumulative distribution functions.
We then prove that the triples do not depend continuously on the observed cumulative distribution functions. Uniformly consistent estimators do not exist.
Finally we show that the MPH model is even identifiable from two-sided censored observations. This proof is constructive, too.     

样本数据缺失情形下泊松过程的强度估计和检验

文章讨论了样本数据缺失情形下泊松过程的强度估计和检验问题。用极大似然估计、矩估计法和最小二乘估计法对强度进行估计,分别得出了极大似然估计强度的迭代方法,矩估计值及最小二乘估计值。证明了矩估计值和最小二乘估计值的无偏性和相合性,导出了其统计量的极限分布。最后,对两个Poisson过程的差异进行了假设检验同时给出渐近置信区间。     

Maximum likelihood methods for fitting the burr type XII distribution to multiply (progressively) censored life test data

Summary This paper develops mathematical and computational methodology for fitting, by the method of maximum likelihood (ML), the Burr Type XII distribution to multiply (or progressively) censored life test data. Mathematical expressions are given for approximating the asymptotic variances and covariances of the ML estimates (MLEs) of the parameters of the Burr Type XII distribution. A rigorous mathematical analysis is undertaken to investigate the existence and uniqueness of the MLEs for arbitrary sample data. The methodology of this paper is applied to progressively censored sample data arising in a life test experiment.     

Exponential deconvolution: two asymptotically equivalent estimators

Two isotonic estimators for the distribution function in a specific deconvolution model, the exponential deconvolution model, are considered. The first estimator is a least squares projection of a naive estimator for the distribution function on the set of distribution functions. The second estimator is the well known maximum likelihood estimator. The two estimators are shown to be first order asymptotically equivalent at a fixed point.     

Asymptotically optimal estimation of smooth functionals for interval censoring, part 2

Estimation in the interval censoring model is considered. A class of smooth functionals is introduced, of which the mean is an example. We consider case 2, with two observation times for each unobservable event time, in the situation that the observation times cannot become arbitrarily close to each other. It is proved that the nonparametric maximum likelihood estimator of the functional asymptotically reaches the information lower bound.     

基于定数截尾样本的指数分布的环境因子估计

讨论了定数截尾样本下,指数分布环境因子的极大似然估计和区间估计,为研究估计的精度,运用随机模拟方法,对环境因子的置信区间的精度进行了讨论。     

Response to the Letter to the Editor on ‘On Quantile-based Asymmetric Family of Distributions: Properties and Inference’

Rubio (2020) points out an identification problem for the four-parameter family of two-piece asymmetric densities introduced by Nassiri & Loris (2013). This implies that statistical inference for that family is problematic. Establishing probabilistic properties for this four-parameter family however still makes sense. For the three-parameter family, there is no identification problem. The main contribution in Gijbels et al. (2019a) is to provide asymptotic results for maximum likelihood and method-of-moments estimators for all members of the three-parameter quantile-based asymmetric family of distributions.