Approximate Inference for the Factor Loading of a Simple Factor Analysis Model

We study a factor analysis model with two normally distributed observations and one factor. Two approximate conditional inference procedures for the factor loading are developed. The first proposal is a very simple procedure but it is not very accurate. The second proposal gives extremely accurate results even for very small sample size. Moreover, the calculations require only the signed log-likelihood ratio statistic and a measure of the standardized maximum likelihood departure. Simulations are used to study the accuracy of the proposed procedures.     

分层线性模型的最大后验估计

最大后验估计（MAPE）和最大似然估计（MLE）都是重要的参数点估计方法。在介绍一般分层线性模型（HLM）MAPE方法的基础上,给出这种方法的期望最大化算法（EM）的具体步骤,运用对数似然函数的二阶导数推导了MAPE估计的方差估计量。同时运用数据模拟比较了EM算法下的MAPE和MLE。对于固定效应的估计,两种方法得到的估计量是一致的。当组数较少时,EM计算的MAPE的方差协方差成分比MLE的更靠近真实值,而且MAPE的迭代次数明显小于MLE。     

Uniqueness of the Maximum Likelihood Estimate of the Weibull Distribution Tampered Failure Rate Model

Abstract

Bhattacharyya and Soejoeti (Bhattacharyya, G. K., Soejoeti, Z. A. (1989 Bhattacharyya, G. K. and Soejoeti, Z. A. 1989. Tampered failure rate model for step-stress accelerated life test. Commun. Statist.—Theory Meth., 18(5): 1627–1643. [Taylor & Francis Online], [Web of Science ®] , [Google Scholar]). Tampered failure rate model for step-stress accelerated life test. Commun. Statist.—Theory Meth. 18(5):1627–1643.) pro- posed the TFR model for step-stress accelerated life tests. Under the TFR model, this article proves that the maximum likelihood estimate of the shape parameters is unique for the Weibull distribution in a multiple step-stress accelerated life test, and investigates the accuracy of the maximum likelihood estimate using the Monte-Carlo simulation.     

Comparison of Maximum Likelihood with Conditional Pairwise Likelihood Estimation of Person Parameters in the Rasch Model

This paper is concerned with person parameter estimation in the binary Rasch model. The loss of efficiency of a pseudo, quasi, or composite likelihood approach investigated. By means of a Monte Carlo study, two quasi likelihood estimators are compared to two well-established maximum likelihood approaches, one of which being a weighted likelihood procedure. The results show that the observed values of the root mean squared error are practically equivalent for the compared estimators in the case of a sufficiently large number of items.     

Maximum Likelihood Estimation of the Log-Binomial Model

Maximum likelihood estimation of prevalence ratios using the log-binomial model is problematic when the estimates are on the boundary of the parameter space. When the model is correct, maximum likelihood is often the method of choice. The authors provide a theorem, formulas, and methodology for obtaining maximum likelihood estimators of the log-binomial model and their estimated standard errors when the solution is on the boundary of the parameter space. Examples are given to illustrate the method.     

Maximum Likelihood Estimation in Gaussian Chain Graph Models under the Alternative Markov Property

Abstract.  The Andersson–Madigan–Perlman (AMP) Markov property is a recently proposed alternative Markov property (AMP) for chain graphs. In the case of continuous variables with a joint multivariate Gaussian distribution, it is the AMP rather than the earlier introduced Lauritzen–Wermuth–Frydenberg Markov property that is coherent with data-generation by natural block-recursive regressions. In this paper, we show that maximum likelihood estimates in Gaussian AMP chain graph models can be obtained by combining generalized least squares and iterative proportional fitting to an iterative algorithm. In an appendix, we give useful convergence results for iterative partial maximization algorithms that apply in particular to the described algorithm.     

Bias Corrected Maximum Likelihood Estimator Under the Generalized Linear Model for a Binary Variable

Under the generalized linear models for a binary variable, an approximate bias of the maximum likelihood estimator of the coefficient, that is a special case of linear parameter in Cordeiro and McCullagh (1991), is derived without a calculation of the third-order derivative of the log likelihood function. Using the obtained approximate bias of the maximum likelihood estimator, a bias-corrected maximum likelihood estimator is defined. Through a simulation study, we show that the bias-corrected maximum likelihood estimator and its variance estimator have a better performance than the maximum likelihood estimator and its variance estimator.     

Maximum Likelihood Estimation of a Poisson Parameter in a Model of Equioverlapping Samples:A Simulation Study

In a model of equioverlapping samples maximum likelihood estimation of a Poisson parameter is examined and compared with two linear unbiased estimations by mean squared error. Since a likelihood estimator is not explicitly available in general, a simulation study has been performed and the results are illustrated     

Semiparametric Likelihood Estimation in the Clayton–Oakes Failure Time Model

Multivariate failure time data arise when the sample consists of clusters and each cluster contains several possibly dependent failure times. The Clayton–Oakes model (Clayton, 1978; Oakes, 1982) for multivariate failure times characterizes the intracluster dependence parametrically but allows arbitrary specification of the marginal distributions. In this paper, we discuss estimation in the Clayton–Oakes model when the marginal distributions are modeled to follow the Cox (1972) proportional hazards regression model. Parameter estimation is based on an approximate generalized maximum likelihood estimator. We illustrate the model's application with example datasets.     

Bias Reduction for the Maximum Likelihood Estimator of the Parameters of the Generalized Rayleigh Family of Distributions

We derive analytic expressions for the biases, to O(n^{? 1}), of the maximum likelihood estimators of the parameters of the generalized Rayleigh distribution family. Using these expressions to bias-correct the estimators is found to be extremely effective in terms of bias reduction, and generally results in a small reduction in relative mean squared error. In general, the analytic bias-corrected estimators are also found to be superior to the alternative of bias-correction via the bootstrap.     

A Likelihood Ratio Test for Idiosyncratic Unit Roots in the Exact Factor Model with Integrated Factors

We consider an exact factor model with unobservable common stochastic trends imposed by non-stationary factors, and study, by simulation, the power of the likelihood ratio test for unit roots in the idiosyncratic components. The power of the test is compared with the analogous Lagrange multiplier test and the Fisher-type test proposed by Bai and Ng. The results suggest that the benefit of the likelihood ratio test is in panels with a small cross-section.     

On the Bias of the Maximum Likelihood Estimator for the Two-Parameter Lomax Distribution

The Lomax (Pareto II) distribution has found wide application in a variety of fields. We analyze the second-order bias of the maximum likelihood estimators of its parameters for finite sample sizes, and show that this bias is positive. We derive an analytic bias correction which reduces the percentage bias of these estimators by one or two orders of magnitude, while simultaneously reducing relative mean squared error. Our simulations show that this performance is very similar to that of a parametric bootstrap correction based on a linear bias function. Three examples with actual data illustrate the application of our bias correction.     

Estimation for the Generalized Pareto Distribution Using Maximum Likelihood and Goodness of Fit

In this article, we introduce a new estimator for the generalized Pareto distribution, which is based on the maximum likelihood estimation and the goodness of fit. The asymptotic normality of the new estimator is shown and a small simulation. From the simulation, the performance of the new estimator is roughly comparable with maximum likelihood for positive values of the shape parameter and often much better than maximum likelihood for negative values.     

Maximum Likelihood Estimation of Life-Span Based on Censored and Passively Registered Historical Data

We consider the estimation of life length of people who were born in the seventeenth or eighteenth century in England. The data consist of a sequence of times of life events that is either ended by a time of death or is right-censored by an unobserved time of migration. We propose a semi parametric model for the data and use a maximum likelihood method to estimate the unknown parameters in this model. We prove the consistency of the maximum likelihood estimators and describe an algorithm to obtain the estimates numerically. We have applied the algorithm to data and the estimates found are presented.     

Guaranteed Local Maximum Likelihood Detection of a Change Point in Nonparametric Logistic Regression

We consider nonparametric logistic regression and propose a generalized likelihood test for detecting a threshold effect that indicates a relationship between some risk factor and a defined outcome above the threshold but none below it. One important field of application is occupational medicine and in particular, epidemiological studies. In epidemiological studies, segmented fully parametric logistic regression models are often threshold models, where it is assumed that the exposure has no influence on a response up to a possible unknown threshold, and has an effect beyond that threshold. Finding efficient methods for detection and estimation of a threshold is a very important task in these studies. This article proposes such methods in a context of nonparametric logistic regression. We use a local version of unknown likelihood functions and show that under rather common assumptions the asymptotic power of our test is one. We present a guaranteed non asymptotic upper bound for the significance level of the proposed test. If applying the test yields the acceptance of the conclusion that there was a change point (and hence a threshold limit value), we suggest using the local maximum likelihood estimator of the change point and consider the asymptotic properties of this estimator.     

Empirical Likelihood Confidence Region for the Parameter in a Partially Linear Errors-in-Variables Model

In this article, a partially linear errors-in-variables model is considered, and empirical log-likelihood ratio statistic for the unknown parameter in the model is suggested. It is proved that the proposed statistic is asymptotically standard chi-square distribution under some suitable conditions, and hence it can be used to construct the confidence region of the parameter. A simulation study indicates that, in terms of coverage probabilities and average lengths of the confidence intervals, the proposed method performs better than the least-squares method.     

On the Product of Inverse Wishart and Normal Distributions with Applications to Discriminant Analysis and Portfolio Theory

Abstract. In this article we analyse the product of the inverse Wishart matrix and a normal vector. We derive the explicit joint distribution of the components of the product. Furthermore, we suggest several exact tests of general linear hypothesis about the elements of the product. We illustrate the developed techniques on examples from discriminant analysis and from portfolio theory.     

Simple Computation of Maximum Likelihood Estimates in Latent Class Model with Equality and Constant Constraints

A simple computational method for estimation of parameters via a type of EM algorithm is proposed in restricted latent class analysis, where equality and constant constraints are considered. These constraints create difficulty in estimation. In order to simply and stably estimate parameters in restricted latent class analysis, a simple computational method using only first-order differentials is proposed, where the step-halving method is adopted. A simulation study shows that in almost all cases the new method gives parameter sequences monotonously increasing the Q-function in the EM algorithm. Analysis of real data is provided.     

Exact Inference for a Simple Step-stress Model with Generalized Type-I Hybrid Censored Data from the Exponential Distribution

In this article, the simple step-stress model is considered based on generalized Type-I hybrid censored data from the exponential distribution. The maximum likelihood estimators (MLEs) of the unknown parameters are derived assuming a cumulative exposure model. We then derive the exact distributions of the MLEs of the parameters using conditional moment generating functions. The Bayesian estimators of the parameters are derived and then compared with the MLEs. We also derive confidence intervals for the parameters using these exact distributions, asymptotic distributions of the MLEs, Bayesian, and the parametric bootstrap methods. The problem of determining the optimal stress-changing point is discussed and the MLEs of the pth quantile and reliability functions at the use condition are obtained. Finally, Monte Carlo simulation and some numerical results are presented for illustrating all the inferential methods developed here.