The Mantel-Haenszel estimator adapted for complex survey designs is not dually consistent

We show via simulation and counterexamples that the Mantel-Haenszel estimator of a common odds ratio, adapted for complex survey designs using survey weights, is inconsistent for sparse-data limiting models. We also propose an alternative estimator that is consistent for sparse-data limiting models satisfying a positivity condition, but not for large-strata limiting models.     

On exponential rates of likelihood ratio estimators for location parameters

In this paper the exponential rates, bounds, and local exponential rates for likelihood ratio estimators are studied. Under certain regularity conditions, a family of likelihood ratio estimators is shown to be admissible in exponential rate. It is also shown that the maximum likelihood estimator is the limit of this family of estimators.     

The Kullback-Leibler risk of the Stein estimator and the conditional MLE

The decomposition of the Kullback-Leibler risk of the maximum likelihood estimator (MLE) is discussed in relation to the Stein estimator and the conditional MLE. A notable correspondence between the decomposition in terms of the Stein estimator and that in terms of the conditional MLE is observed. This decomposition reflects that of the expected log-likelihood ratio. Accordingly, it is concluded that these modified estimators reduce the risk by reducing the expected log-likelihood ratio. The empirical Bayes method is discussed from this point of view.     

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     

Empirical likelihood estimation of discretely sampled processes of OU type

This paper presents an empirical likelihood estimation procedure for parameters of the discretely sampled process of Ornstein-Uhlenbeck type. The proposed procedure is based on the condi- tional characteristic function, and the maximum empirical likelihood estimator is proved to be consistent and asymptotically normal. Moreover, this estimator is shown to be asymptotically efficient under some mild conditions. When the background driving Lévy process is of type A or B, we show that the intensity parameter c...     

正态条件下带AR(1)-型方差结构GMANOVA-MANOVA模型极大似然估计的小样本特征

研究了一类带一阶自回归(AR(1))-型方差结构的广义多元方差分析-多元方差分析(GMANO VA-MANOVA)模型参数极大似然估计的小样本特征.对带AR(1)-型方差结构GMANOVA-MANOVA模型,文章在正态条件下给出了参数极大似然估计存在的一个充分必要条件,讨论了极大似然估计唯一的充分条件.在该充分条件下,文章证明了相关系数极大似然估计的精确分布只与相关系数有关,并依此给出了自相关系数简单假设H0:ρ=0v.s.H1:ρ≠0的一个不需要叠代计算估计的检验,同时模拟表明该检验为无偏检验且势函数与似然比检验势函数无太大差异.     

Information amount and higher-order efficiency in estimation

By means of second-order asymptotic approximation, the paper clarifies the relationship between the Fisher information of first-order asymptotically efficient estimators and their decision-theoretic performance. It shows that if the estimators are modified so that they have the same asymptotic bias, the information amount can be connected with the risk based on convex loss functions in such a way that the greater information loss of an estimator implies its greater risk. The information loss of the maximum likelihood estimator is shown to be minimal in a general set-up. A multinomial model is used for illustration.     

Second-order risk comparison of SLSE with GLSE and MLE in a regression with serial correlation

First, the second-order bias of the estimator of the autoregressive parameter based on the ordinary least squares residuals in a linear model with serial correlation is given. Second, the second-order expansion of the risk matrix of a generalized least squares estimator with the above estimated parameter is obtained. This expansion is the same as that based on a suitable estimator of the autoregressive parameter independent of the sample. Third, it is shown that the risk matrix of the generalized least squares estimator is asymptotically equivalent to that of the maximum likelihood estimator up to the second order. Last, a sufficient condition is given for the term due to the estimation of the autoregressive parameter in this expansion to vanish under Grenander's condition for the explanatory variates.     

Safe density ratio modeling

An important problem in logistic regression modeling is the existence of the maximum likelihood estimators. In particular, when the sample size is small, the maximum likelihood estimator of the regression parameters does not exist if the data are completely, or quasicompletely separated. Recognizing that this phenomenon has a serious impact on the fitting of the density ratio model–which is a semiparametric model whose profile empirical log-likelihood has the logistic form because of the equivalence between prospective and retrospective sampling–we suggest a linear programming methodology for examining whether the maximum likelihood estimators of the finite dimensional parameter vector of the model exist. It is shown that the methodology can be effectively utilized in the analysis of case–control gene expression data by identifying cases where the density ratio model cannot be applied. It is demonstrated that naive application of the density ratio model yields erroneous conclusions.     

纵向数据下部分非线性模型的广义经验似然推断

本文研究了纵向数据下部分非线性模型中未知参数的置信域的构造.利用经验似然方法,构造了非线性函数中未知参数的广义对数经验似然比统计量,证明了其渐近于卡方分布.同时,得到了未知参数的最大经验似然估计,并证明了其渐近正态性.     

Estimators for the binomial distribution that dominate the MLE in terms of Kullback–Leibler risk

Estimators based on the mode are introduced and shown empirically to have smaller Kullback–Leibler risk than the maximum likelihood estimator. For one of these, the midpoint modal estimator (MME), we prove the Kullback–Leibler risk is below \frac12{\frac{1}{2}} while for the MLE the risk is above \frac12{\frac{1}{2}} for a wide range of success probabilities that approaches the unit interval as the sample size grows to infinity. The MME is related to the mean of Fisher’s Fiducial estimator and to the rule of succession for Jefferey’s noninformative prior.     

Empirical likelihood based inference for semiparametric varying coefficient partially linear models with error-prone linear covariates

This paper considers statistical inference for semiparametric varying coefficient partially linear models with error-prone linear covariates. An empirical likelihood based statistic for parametric component is developed to construct confidence regions. The resulting statistic is shown to be asymptotically chi-square distributed. By the empirical likelihood ratio function, the maximum empirical likelihood estimator of the parameter is defined and the asymptotic normality is shown. A simulation experiment is conducted to compare the empirical likelihood, normal based and the naive empirical likelihood methods in terms of coverage accuracies of confidence regions.     

Empirical Euclidean likelihood for general estimating equations under association dependence

Empirical Euclidean likelihood for general estimating equations for association dependent processes is investigated. The strong consistency and asymptotic normality of the blockwise maximum empirical Euclidean likelihood estimator are presented. We show that it is more efficient than estimator without blocking. The blockwise empirical Euclidean log-likelihood ratio asymptotically follows a chi-square 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 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     

Likelihood-Based Inference for Extreme Value Models

Estimation of the extremal behavior of a process is often based on the fitting of asymptotic extreme value models to relatively short series of data. Maximum likelihood has emerged as a flexible and powerful modeling tool in such applications, but its performance with small samples has been shown to be poor relative to an alternative fitting procedure based on probability weighted moments. We argue here that the small-sample superiority of the probability weighted moments estimator is due to the assumption of a restricted parameter space, corresponding to finite population moments. To incorporate similar information in a likelihood-based analysis, we propose a penalized maximum likelihood estimator that retains the modeling flexibility and large-sample optimality of the maximum likelihood estimator, but improves on its small-sample properties. The properties of the penalized likelihood estimator are verified in a simulation study, and in application to sea-level data, which also enables the procedure to be evaluated in the context of structural models for extremes.     

Efficient estimation of stable Lévy process with symmetric jumps

Efficient estimation of a non-Gaussian stable Lévy process with drift and symmetric jumps observed at high frequency is considered. For this statistical experiment, the local asymptotic normality of the likelihood is proved with a non-singular Fisher information matrix through the use of a non-diagonal norming matrix. The asymptotic normality and efficiency of a sequence of roots of the associated likelihood equation are shown as well. Moreover, we show that a simple preliminary method of moments can be used as an initial estimator of a scoring procedure, thereby conveniently enabling us to bypass numerically demanding likelihood optimization. Our simulation results show that the one-step estimator can exhibit quite similar finite-sample performance as the maximum likelihood estimator.     

On the comparison of the pre-test and shrinkage phi-divergence test estimators for the symmetry model of categorical data

The estimation problem of the parameters in a symmetry model for categorical data has been considered for many authors in the statistical literature (for example, Bowker (1948) [1], Ireland et al. (1969) [2], Quade and Salama (1975) [3], Cressie and Read (1988) [4], Menéndez et al. (2005) [5]) without using uncertain prior information. It is well known that many new and interesting estimators, using uncertain prior information, have been studied by a host of researchers in different statistical models, and many papers have been published on this topic (see Saleh (2006) [9] and references therein). In this paper, we consider the symmetry model of categorical data and we study, for the first time, some new estimators when non-sample information about the symmetry of the probabilities is considered. The decision to use a “restricted” estimator or an “unrestricted” estimator is based on the outcome of a preliminary test, and then a shrinkage technique is used. It is interesting to note that we present a unified study in the sense that we consider not only the maximum likelihood estimator and likelihood ratio test or chi-square test statistic but we consider minimum phi-divergence estimators and phi-divergence test statistics. Families of minimum phi-divergence estimators and phi-divergence test statistics are wide classes of estimators and test statistics that contain as a particular case the maximum likelihood estimator, likelihood ratio test and chi-square test statistic. In an asymptotic set-up, the biases and the risk under the squared loss function for the proposed estimators are derived and compared. A numerical example clarifies the content of the paper.     

Estimation of the entropy of a multivariate normal distribution

Motivated by problems in molecular biosciences wherein the evaluation of entropy of a molecular system is important for understanding its thermodynamic properties, we consider the efficient estimation of entropy of a multivariate normal distribution having unknown mean vector and covariance matrix. Based on a random sample, we discuss the problem of estimating the entropy under the quadratic loss function. The best affine equivariant estimator is obtained and, interestingly, it also turns out to be an unbiased estimator and a generalized Bayes estimator. It is established that the best affine equivariant estimator is admissible in the class of estimators that depend on the determinant of the sample covariance matrix alone. The risk improvements of the best affine equivariant estimator over the maximum likelihood estimator (an estimator commonly used in molecular sciences) are obtained numerically and are found to be substantial in higher dimensions, which is commonly the case for atomic coordinates in macromolecules such as proteins. We further establish that even the best affine equivariant estimator is inadmissible and obtain Stein-type and Brewster–Zidek-type estimators dominating it. The Brewster–Zidek-type estimator is shown to be generalized Bayes.     

Minimax Hierarchical Empirical Bayes Estimation in Multivariate Regression

The multivariate normal regression model, in which a vector y of responses is to be predicted by a vector x of explanatory variables, is considered. A hierarchical framework is used to express prior information on both x and y. An empirical Bayes estimator is developed which shrinks the maximum likelihood estimator of the matrix of regression coefficients across rows and columns to nontrivial subspaces which reflect both types of prior information. The estimator is shown to be minimax and is applied to a set of chemometrics data for which it reduces the cross-validated predicted mean squared error of the maximum likelihood estimator by 38%.