Generalized variance estimators in the multivariate gamma models

It is already known that the uniformly minimum variance unbiased (UMVU) estimator of the generalized variance always exists for any natural exponential family. However, in practice, this estimator is often difficult to obtain. This paper provides explicit forms of the UMVU estimators for the bivariate and symmetric multivariate gamma models, which are diagonal quadratic exponential families. For the non-independent multivariate gamma models, it is shown that the UMVU and the maximum likelihood estimators are not proportional.      

Linear relations in time series models. II

In this paper an asymptotic theory is developed for a new time series model which was introduced in a previous paper [5]. An algorithm for computing estimates of the parameters of this time series model is given, and it is shown that these estimators are asymptotically efficient in the sense that they have the same asymptotic distribution as the maximum likelihood estimators.     

Simultaneous estimation of parameters in exponential families

Summary The paper considers estimation of the natural parameter vector or the mean vector from independent distributions each belonging to the one-parameter discrete or absolutely continuous exponential family. The usual estimators (maximum likelihood, minimum variance unbiased or best invariant) are improved simultaneously under various weighted squared error losses. Research supported by the NSF Grant Number MCS-8202116.     

Asymptotic optimal inference for multivariate branching-Markov processes via martingale estimating functions and mixed normality

Multivariate tree-indexed Markov processes are discussed with applications. A Galton-Watson super-critical branching process is used to model the random tree-indexed process. Martingale estimating functions are used as a basic framework to discuss asymptotic properties and optimality of estimators and tests. The limit distributions of the estimators turn out to be mixtures of normals rather than normal. Also, the non-null limit distributions of standard test statistics such as Wald, Rao’s score, and likelihood ratio statistics are shown to have mixtures of non-central chi-square distributions. The models discussed in this paper belong to the local asymptotic mixed normal family. Consequently, non-standard limit results are obtained.     

Minimum f-divergence estimators and quasi-likelihood functions

Maximum quasi-likelihood estimators have several nice asymptotic properties. We show that, in many situations, a family of estimators, called the minimum f-divergence estimators, can be defined such that each estimator has the same asymptotic properties as the maximum quasi-likelihood estimator. The family of minimum f-divergence estimators include the maximum quasi-likelihood estimators as a special case. When a quasi-likelihood is the log likelihood from some exponential family, Amari's dual geometries can be used to study the maximum likelihood estimator. A dual geometric structure can also be defined for more general quasi-likelihood functions as well as for the larger family of minimum f-divergence estimators. The relationship between the f-divergence and the quasi-likelihood function and the relationship between the f-divergence and the power divergence is discussed.This work was supported by National Science Foundation grant DMS 88-03584.     

Maximum integrated likelihood estimator of the interest parameter when the nuisance parameter is location or scale

The problem of estimation of an interest parameter in the presence of a nuisance parameter, which is either location or scale, is studied. Two estimators are considered: the usual maximum likelihood estimator and the estimator based on maximization of the integrated likelihood function. The estimators are compared, asymptotically, with respect to the bias and with respect to the mean squared error. The examples are given.     

Improving on the maximum likelihood estimators of the means in Poisson decomposable graphical models

In this article we study the simultaneous estimation of the means in Poisson decomposable graphical models. We derive some classes of estimators which improve on the maximum likelihood estimator under the normalized squared losses. Our estimators are based on the argument in Chou [Simultaneous estimation in discrete multivariate exponential families, Ann. Statist. 19 (1991) 314-328.] and shrink the maximum likelihood estimator depending on the marginal frequencies of variables forming a complete subgraph of the conditional independence graph.     

Stochastic Order and MLE of the Mean of the Exponential Distribution

The maximum likelihood estimator of the mean of the exponential distribution, based on various data structures has been studied extensively. However, the order preserving property of these estimators is not found in the literature. This article discusses this property. Suppose that two samples of the same size are drawn from two independent exponential populations that have different means. It is shown in this article that the regular stochastic ordering holds between the two MLEs corresponding to the two exponential means, based on various censored data. In particular, conditions are given on inspection times so that the result is also true for grouped data.     

A New Class of Minimum Power Divergence Estimators with Applications to Cancer Surveillance

The annual percent change (APC) has been adopted as a useful measure for analyzing the changing trends of cancer mortality and incidence rates by the NCI SEER program. Difficulties, however, arise when comparing the sample APCs between two overlapping regions because of induced dependence (e.g., comparing the cancer mortality change rate of California with that of the national level). This paper deals with a new perspective for understanding the sample distribution of the test-statistics for comparing the APCs between overlapping regions. Our proposal allows for computational readiness and easy interpretability. We further propose a more general family of estimators, namely, the so-called minimum power divergence estimators, including the maximum likelihood estimators as a special case. Our simulation experiments support the superiority of the proposed estimator to the conventional maximum likelihood estimator. The proposed method is illustrated by the analysis of the SEER cancer mortality rates observed from 1991 to 2006.     

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.     

Convergence Rate of the Distributions of Normalized Maximum Likelihood Estimators for Irregular Parametric Families

We study the convergence rate of the distributions of normalized maximum likelihood estimators defined by a parametric family of discontinuous multidimensional densities in the case of a vector parameter.     

Maximum likelihood estimation of a multivariate linear functional relationship

Many authors have discussed maximum likelihood estimation in the simple linear functional relationship model. In this paper, we derive maximum likelihood estimators (MLEs) for parameters in a much more general model. Several special cases including the multivariate linear functional relationship model are discussed. Estimators of some of the parameters are shown to be inconsistent.     

A Cautionary Note on Likelihood Ratio Tests in Mixture Models

We show that iterative methods for maximizing the likelihood in a mixture of exponentials model depend strongly on their particular implementation. Different starting strategies and stopping rules yield completely different estimators of the parameters. This is demonstrated for the likelihood ratio test of homogeneity against two-component exponential mixtures, when the test statistic is calculated by the EM algorithm.     

A note on the simple structural regression model

In this paper we investigate some aspects like estimation and hypothesis testing in the simple structural regression model with measurement errors. Use is made of orthogonal parametrizations obtained in the literature. Emphasis is placed on some properties of the maximum likelihood estimators and also on the distribution of the likelihood ratio statistics.     

Construction of improved estimators in multiparameter estimation for continuous exponential families

A unified theory of simultaneous estimation of parameters for the continuous exponential family is presented. Estimators are constructed that improve on the standard ones (the maximum likelihood, UMVUE or best invariant estimator). These improved estimators shift the standard ones towards possibly non-zero points or data based points.     

带协变量的负二项回归模型中离散参数估计的研究

针对带协变量的负二项回归模型中离散参数估计问题,推广了极大似然估计和Bootstrap极大似然估计方法,并在绝对偏差的意义下,通过模拟研究和实际数据分析研究了估计的优良性.研究结果表明协变量和样本量均对离散参数估计有影响.     

Minimum divergence estimators,maximum likelihood and exponential families

The dual representation formula of the divergence between two distributions in a parametric model is presented. Resulting estimators do not make use of any grouping or smoothing. For smooth divergences they all coincide with the MLE on any regular exponential family.     

Nonparametric likelihood ratio goodness-of-fit tests for survival data

Berk and Jones (Z. Wahrsch. Verw. Gebiete 47 (1979) 47) described a nonparametric likelihood test of uniformity that is more efficient, in Bahadur's sense, than any weighted Kolmogorov-Smirnov test at any alternative. This article shows how to obtain a nonparametric likelihood test of a general parametric family for incomplete survival data. A nonparametric likelihood ratio test process is employed to measure the discrepancy between a parametric family and the observed data. Large sample properties of the likelihood ratio test process are studied under both the null and alternative hypotheses. A Monte Carlo simulation method is proposed to estimate its null distribution. We show how to produce a likelihood ratio graphical check as well as a formal test of a parametric family based on the developed theory. Our method is developed for the right-censorship model, but can be easily extended to some other survival models. Illustrations are given using both real and simulated data.     

A modified power series distribution

Summary A modification of the usual power series distribution is proposed and a statistical treatment of data generated by this probability model is indicated. It is shown that this modified power series distribution belongs to a multiparameter discrete exponential family and standard theory is then applied to obtain complete sufficient statistics of the parameters, sampling distribution of these statistics, maximum likelihood and minimum variance unbiased estimators, uniformly most powerful unbiased tests, uniformly most accurate confidence intervals and expected cover tolerance regions. CSIRO, Division of Mathematical Statistics     

On the distribution of penalized maximum likelihood estimators: The LASSO, SCAD, and thresholding

We study the distributions of the LASSO, SCAD, and thresholding estimators, in finite samples and in the large-sample limit. The asymptotic distributions are derived for both the case where the estimators are tuned to perform consistent model selection and for the case where the estimators are tuned to perform conservative model selection. Our findings complement those of Knight and Fu [K. Knight, W. Fu, Asymptotics for lasso-type estimators, Annals of Statistics 28 (2000) 1356–1378] and Fan and Li [J. Fan, R. Li, Variable selection via non-concave penalized likelihood and its oracle properties, Journal of the American Statistical Association 96 (2001) 1348–1360]. We show that the distributions are typically highly non-normal regardless of how the estimator is tuned, and that this property persists in large samples. The uniform convergence rate of these estimators is also obtained, and is shown to be slower than n^−1/2 in case the estimator is tuned to perform consistent model selection. An impossibility result regarding estimation of the estimators’ distribution function is also provided.