Finite mixtures of multivariate Poisson distributions with application

In the present paper we examine finite mixtures of multivariate Poisson distributions as an alternative class of models for multivariate count data. The proposed models allow for both overdispersion in the marginal distributions and negative correlation, while they are computationally tractable using standard ideas from finite mixture modelling. An EM type algorithm for maximum likelihood (ML) estimation of the parameters is developed. The identifiability of this class of mixtures is proved. Properties of ML estimators are derived. A real data application concerning model based clustering for multivariate count data related to different types of crime is presented to illustrate the practical potential of the proposed class of models.     

Estimation of the Mean and Standard Deviation of the Logistic Distribution Based on Multiply Type-II Censored Samples

In this paper, we discuss the problem of estimating the mean and standard deviation of a logistic population based on multiply Type-II censored samples. First, we discuss the best linear unbiased estimation and the maximum likelihood estimation methods. Next, by appropriately approximating the likelihood equations we derive approximate maximum likelihood estimators for the two parameters and show that these estimators are quite useful as they do not need the construction of any special tables (as required for the best linear unbiased estimators) and are explicit estimators (unlike the maximum likelihood estimators which need to be determined by numerical methods). We show that these estimators are also quite efficient, and derive the asymptotic variances and covariance of the estimators. Finally, we present an example to illustrate the methods of estimation discussed in this paper.     

Target estimation for the logistic regression model

The method of target estimation developed by Cabrera and Fernholz [(1999). Target estimation for bias and mean square error reduction. The Annals of Statistics, 27(3), 1080–1104.] to reduce bias and variance is applied to logistic regression models of several parameters. The expectation functions of the maximum likelihood estimators for the coefficients in the logistic regression models of one and two parameters are analyzed and simulations are given to show a reduction in both bias and variability after targeting the maximum likelihood estimators. In addition to bias and variance reduction, it is found that targeting can also correct the skewness of the original statistic. An example based on real data is given to show the advantage of using target estimators for obtaining better confidence intervals of the corresponding parameters. The notion of the target median is also presented with some applications to the logistic models.     

MAXIMUM LIKELIHOOD ESTIMATION IN LINEAR MODELS WITH EQUI-CORRELATED RANDOM ERRORS

Necessary and sufficient conditions for the existence of maximum likelihood estimators of unknown parameters in linear models with equi‐correlated random errors are presented. The basic technique we use is that these models are, first, orthogonally transformed into linear models with two variances, and then the maximum likelihood estimation problem is solved in the environment of transformed models. Our results generalize a result of Arnold, S. F. (1981) [The theory of linear models and multivariate analysis. Wiley, New York]. In addition, we give necessary and sufficient conditions for the existence of restricted maximum likelihood estimators of the parameters. The results of Birkes, D. & Wulff, S. (2003) [Existence of maximum likelihood estimates in normal variance‐components models. J Statist Plann. Inference. 113 , 35–47] are compared with our results and differences are pointed out.     

Heteroskedasticity in the Tobit model

The paper deals with parameter estimation and the testing of individual parameters in heteroskedastic Tobit models. The statistical properties of semiparametric and maximum likelihood estimators are evaluated. Correspondingt-test statistics are compared. Results from a Monte Carlo experiment indicate that the semiparametric estimator performs relatively better than the maximum likelihood estimator. The associatedt-test statistics appear to perform better than the corresponding maximum likelihood test statistics. *** DIRECT SUPPORT *** A06GP002 00008     

Estimation of the location and scale parameters of the extreme value distmbution based on multiply type-II censored samples

In this paper, we consider the problem of estimating the location and scale parameters of an extreme value distribution based on multiply Type-II censored samples. We first describe the best linear unbiased estimators and the maximum likelihood estimators of these parameters. After observing that the best linear unbiased estimators need the construction of some tables for its coefficients and that the maximum likelihood estimators do not exist in an explicit algebraic form and hence need to be found by numerical methods, we develop approximate maximum likelihood estimators by appropriately approximating the likelihood equations. In addition to being simple explicit estimators, these estimators turn out to be nearly as efficient as the best linear unbiased estimators and the maximum likelihood estimators. Next, we derive the asymptotic variances and covariance of these estimators in terms of the first two single moments and the product moments of order statistics from the standard extreme value distribution. Finally, we present an example in order to illustrate all the methods of estimation of parameters discussed in this paper.     

SURVIVAL ANALYSIS WITH TIME DEPENDENT FRAILTY USING A LONGITUDINAL MODEL

A method of estimation for generalised mixed models is applied to the estimation of regression parameters in a proportional hazards model with time dependent frailty. A parameter representing change over time is introduced and is modelled in turn into a fixed effect, a normally distributed random effect and a longitudinal effect in which the random component relates to the patient characteristics. Both maximum likelihood and residual maximum likelihood estimators are given.     

ESTIMATION OF EXPONENTIATED WEIBULL SHAPE PARAMETERS UNDER LINEX LOSS FUNCTION

ABSTRACT

The paper deals with Bayes estimation of the exponentiated Weibull shape parameters under linex loss function when independent non-informative type of priors are available for the parameters. Generalized maximum likelihood estimators have also been obtained. Performances of the proposed Bayes estimator, generalized maximum likelihood estimators, posterior mean (i.e., Bayes estimator under squared error loss function) and maximum likelihood estimators have been studied on the basis of their risks under linex loss function. The comparison is based on a simulation study because the expressions for risk functions of these estimators cannot be obtained in nice closed forms.     

一类近似因子模型的GMM估计及其统计性质研究

对于一类异质性误差项存在截面相关性的近似因子模型,本文首先提出了估计共同因子向量和因子载荷矩阵的广义矩估计方法（GMM）,该方法推广了Doz等（2012）的极大似然估计方法;其次,分别研究了模型参数广义矩估计的渐近性质和有限样本的统计性质,在适当的条件下,证明了参数的GMM估计是具有渐近正态分布的一致估计;最后,利用近似因子模型对我国各类上市公司增长性的共同驱动因素及其差异性进行了实证分析。     

The Bernstein–von Mises theorem in semiparametric competing risks models

Semiparametric Bayesian models are nowadays a popular tool in event history analysis. An important area of research concerns the investigation of frequentist properties of posterior inference. In this paper, we propose novel semiparametric Bayesian models for the analysis of competing risks data and investigate the Bernstein–von Mises theorem for differentiable functionals of model parameters. The model is specified by expressing the cause-specific hazard as the product of the conditional probability of a failure type and the overall hazard rate. We take the conditional probability as a smooth function of time and leave the cumulative overall hazard unspecified. A prior distribution is defined on the joint parameter space, which includes a beta process prior for the cumulative overall hazard. We first develop the large-sample properties of maximum likelihood estimators by giving simple sufficient conditions for them to hold. Then, we show that, under the chosen priors, the posterior distribution for any differentiable functional of interest is asymptotically equivalent to the sampling distribution derived from maximum likelihood estimation. A simulation study is provided to illustrate the coverage properties of credible intervals on cumulative incidence functions.     

LASSO and shrinkage estimation in Weibull censored regression models

In this paper we address the problem of estimating a vector of regression parameters in the Weibull censored regression model. Our main objective is to provide natural adaptive estimators that significantly improve upon the classical procedures in the situation where some of the predictors may or may not be associated with the response. In the context of two competing Weibull censored regression models (full model and candidate submodel), we consider an adaptive shrinkage estimation strategy that shrinks the full model maximum likelihood estimate in the direction of the submodel maximum likelihood estimate. We develop the properties of these estimators using the notion of asymptotic distributional risk. The shrinkage estimators are shown to have higher efficiency than the classical estimators for a wide class of models. Further, we consider a LASSO type estimation strategy and compare the relative performance with the shrinkage estimators. Monte Carlo simulations reveal that when the true model is close to the candidate submodel, the shrinkage strategy performs better than the LASSO strategy when, and only when, there are many inactive predictors in the model. Shrinkage and LASSO strategies are applied to a real data set from Veteran's administration (VA) lung cancer study to illustrate the usefulness of the procedures in practice.     

Estimating the parameters of a doubly truncated normal distribution

This paper deals with the maximum likelihood estimation of parameters for a doubly truncated normal distribution when the truncation points are known. We prove, in this case, that the MLEs are nonexistent (become infinite) with positive probability. For estimators that exist with probability one, the class of Bayes modal estimators or modified maximum likelihood estimators is explored. Another useful estimating procedure, called mixed estimation, is proposed. Simulations compare the behavior of the MLEs, the modified MLEs, and the mixed estimators which reveal that the MLE, in addition to being nonexistent with positive probability, behaves poorly near the upper boundary of the interval of its existence. The modified MLEs and the mixed estimators are seen to be remarkably better than the MLE     

Near universal consistency of the maximum pseudolikelihood estimator for discrete models

Maximum pseudolikelihood (MPL) estimators are useful alternatives to maximum likelihood (ML) estimators when likelihood functions are more difficult to manipulate than their marginal and conditional components. Furthermore, MPL estimators subsume a large number of estimation techniques including ML estimators, maximum composite marginal likelihood estimators, and maximum pairwise likelihood estimators. When considering only the estimation of discrete models (on a possibly countably infinite support), we show that a simple finiteness assumption on an entropy-based measure is sufficient for assessing the consistency of the MPL estimator. As a consequence, we demonstrate that the MPL estimator of any discrete model on a bounded support will be consistent. Our result is valid in parametric, semiparametric, and nonparametric settings.     

Efficient closed-form maximum a posteriori estimators for the gamma distribution

We proposed a new class of maximum a posteriori estimators for the parameters of the Gamma distribution. These estimators have simple closed-form expressions and can be rewritten as a bias-corrected maximum likelihood estimators presented by Ye and Chen [Closed-form estimators for the gamma distribution derived from likelihood equations. Am Statist. 2017;71(2):177–181]. A simulation study was carried out to compare different estimation procedures. Numerical results revels that our new estimation scheme outperforms the existing closed-form estimators and produces extremely efficient estimates for both parameters, even for small sample sizes.     

Generalised quasi‐likelihood inference in a semi‐parametric binary dynamic mixed logit model

There exists a recent study where dynamic mixed‐effects regression models for count data have been extended to a semi‐parametric context. However, when one deals with other discrete data such as binary responses, the results based on count data models are not directly applicable. In this paper, we therefore begin with existing binary dynamic mixed models and generalise them to the semi‐parametric context. For inference, we use a new semi‐parametric conditional quasi‐likelihood (SCQL) approach for the estimation of the non‐parametric function involved in the semi‐parametric model, and a semi‐parametric generalised quasi‐likelihood (SGQL) approach for the estimation of the main regression, dynamic dependence and random effects variance parameters. A semi‐parametric maximum likelihood (SML) approach is also used as a comparison to the SGQL approach. The properties of the estimators are examined both asymptotically and empirically. More specifically, the consistency of the estimators is established and finite sample performances of the estimators are examined through an intensive simulation study.     

Estimation for Location-Scale Models with Censored Data

This article considers a class of estimators for the location and scale parameters in the location-scale model based on ‘synthetic data’ when the observations are randomly censored on the right. The asymptotic normality of the estimators is established using counting process and martingale techniques when the censoring distribution is known and unknown, respectively. In the case when the censoring distribution is known, we show that the asymptotic variances of this class of estimators depend on the data transformation and have a lower bound which is not achievable by this class of estimators. However, in the case that the censoring distribution is unknown and estimated by the Kaplan–Meier estimator, this class of estimators has the same asymptotic variance and attains the lower bound for variance for the case of known censoring distribution. This is different from censored regression analysis, where asymptotic variances depend on the data transformation. Our method has three valuable advantages over the method of maximum likelihood estimation. First, our estimators are available in a closed form and do not require an iterative algorithm. Second, simulation studies show that our estimators being moment-based are comparable to maximum likelihood estimators and outperform them when sample size is small and censoring rate is high. Third, our estimators are more robust to model misspecification than maximum likelihood estimators. Therefore, our method can serve as a competitive alternative to the method of maximum likelihood in estimation for location-scale models with censored data. A numerical example is presented to illustrate the proposed method.     

On probit versus logit dynamic mixed models for binary panel data

In this paper, we consider inferences in a binary dynamic mixed model. The existing estimation approaches mainly estimate the regression effects and the dynamic dependence parameters either through the estimation of the random effects or by avoiding the random effects technically. Under the assumption that the random effects follow a Gaussian distribution, we propose a generalized quasilikelihood (GQL) approach for the estimation of the parameters of the dynamic mixed models. The proposed approach is computationally less cumbersome than the exact maximum likelihood (ML) approach. We also carry out the GQL estimation under two competitive, namely, probit and logit mixed models, and discuss both the asymptotic and small-sample behaviour of their estimators.     

Maximum likelihood estimation foe two parameter exponentials under type I censoring

The paper considers maximum likelihood estimation of the location parameters and the failure rates of two parameter exponentials under type I censoring. Both the one and two sample cases are considered. It turns out that by debiasing the maximum likelihood estimators of the location parameters, one can achieve asymptotically 50% mean squared error reduction.     

On second-order improved estimation of a gamma scale parameter

This paper concludes our comprehensive study on point estimation of model parameters of a gamma distribution from a second-order decision theoretic point of view. It should be noted that efficient estimation of gamma model parameters for samples ‘not large’ is a challenging task since the exact sampling distributions of the maximum likelihood estimators and its variants are not known. Estimation of a gamma scale parameter has received less attention from the earlier researchers compared to shape parameter estimation. What we have observed here is that improved estimation of the shape parameter does not necessarily lead to improved scale estimation if a natural moment condition (which is also the maximum likelihood restriction) is satisfied. Therefore, this work deals with the gamma scale parameter estimation as a separate new problem, not as a by-product of the shape parameter estimation, and studies several estimators in terms of second-order risk.     

Parametric estimation for ARFIMA models via spectral methods

Given a fractional integrated, autoregressive, moving average,ARFIMA (p, d, q) process, the simultaneous estimation of the short and long memory parameters can be achieved by maximum likelihood estimators. In this paper, following a two-step algorithm, the coefficients are estimated combining the maximum likelihood estimators with the general orthogonal decomposition of stochastic processes. In particular, the principal component analysis of stochastic processes is exploited to estimate the short memory parameters, which are plugged into the maximum likelihood function to obtain the fractional differencingd.