Finite Sample Properties of the Maximum Likelihood Estimator and of Likelihood Ratio Tests in Hidden Markov Models

Hidden Markov models were successfully applied in various fields of time series analysis, especially for analyzing ion channel recordings. The maximum likelihood estimator (MLE) has recently been proven to be asymptotically normally distributed. Here, we investigate finite sample properties of the MLE and of different types of likelihood ratio tests (LRTs) by means of simulation studies. The MLE is shown to reach the asymptotic behavior within sample sizes that are common for various applications. Thus, reliable estimates and confidence intervals can be obtained. We give an approximative scaling function for the estimation error for finite samples, and investigate the power of different LRTs suitable for applications to ion channels, including tests for superimposed hidden Markov processes. Our results are applied to physiological sodium channel data.     

Maximum Likelihood Estimation of Exponential Polynomial Rate for Poisson Data

The heterogeneous Poisson process with discretized exponential quadratic rate function is considered. Maximum likelihood estimates of the parameters of the rate function are derived for the case when the data consists of numbers of occurrences in consecutive equal time periods. A likelihood ratio test of the null hypothesis of exponential quadratic rate is presented. Its power against exponential linear rate functions is estimated using Monte Carlo simulation. The maximum likelihood method is compared with a log-linear least squares techniques. An application of the technique to the analysis of mortality rates due to congenital malformations is presented.     

A Maximum Likelihood Estimator for Shape Parameters of S‐Distributions

The S‐distribution is a four‐parameter distribution that is defined in terms of a differential equation, in which the cumulative is represented as the dependent variable: The article proposes a maximum likelihood estimator for the shape parameters of this distribution.     

Small Sample Properties of the Likelihood Estimator in Proportional Hazard Regression Models with Omitted Variables

The paper deals with the effects of incorrectly omitted regressor variables in a parametric proportional hazard regression model. By studying conditions for equality between the estimators of correct and incorrect models it is demonstrated analytically that such cases are not to be expected in practise. A small sample Monte Carlo experiment indicates severe negative effects on the retained parameters both in terms of bias and mean square error.     

非线性再生散度随机效应模型的极大似然估计及EM算法

非线性再生散度随机效应模型是指数族非线性随机效应模型和非线性再生散度模型的推广和发展．通过视模型中的随机效应为假想的缺失数据和应用Metropolis-Hastings（MH）算法,提出了模型参数极大似然估计的Monte-Carlo EM（MCEM）算法,并用模拟研究和实例分析说明了该算法的可行性．     

A Note on Maximum Likelihood Ratio Test of No Outliers in Regression Models

A derivation of the maximum likelihood ratio test for testing no outliers in regression models is given using the method of WETHERILL (1981, pp. 106–107) for estimating the regression parameters. This method is essentially similar to the one outlined in BARNETT and LEWIS (1978, p. 263), although by our detailed derivation it is easier to see that the maximum likelihood estimate of θ of model (3) under the hypothesis that the i^th observation in an outlier is the same as that obtained from model (1) when the i^th observation is removed.     

Nonparametric Estimates of the Asymptotic Covariance Matrix Associated with BICKEL's Estimate for Multivariate Shift Parameters

Nonparametric estimates are proposed for the asymptotic covariance matrix associated with the vector of medians of averages of pairs when estimating the shift parameter in the multivariate one sample problem. These estimates are based on density estimation and are competitors to the “natural” estimate proposed by BICKEL (1965). Weak and strong consistency of the estimates as well as asymptotic normality are presented.     

非线性再生散度随机效应模型的极大似然估计及随机逼近算法

非线性再生散度随机效应模型包括了非线性随机效应模型和指数族非线性随机效应模型等．通过视模型中的随机效应为假想的缺失数据和应用Metropolis-Hastings(简称MH) 算法,提出了模型参数极大似然估计的随机逼近算法．模拟研究和实例分析表明了该算法的可行性．     

Asymptotic Normality of Parameter Estimators for Some Nearest Neighbour Models

For auto-regression models and first and second differences models it is shown that the REML estimators of unknown parameters are asymptotically normal when the number of observations tends to infinity.     

Log-linear Models in a Study of Gall Bladder Disease

Biliary drainage and X-rays have been used singly and together in the diagnosis of gall bladder disease. Data from two sets of patients were statistically analyzed using log-linear analysis of contingency tables. The results of the study show that “biliary drainage” is dependent of “X-ray” and “Pathology”; abnormal X-ray is associated with abnormal pathology. Thus, biliary drainage is not of such value as to base surgery on it alone. However, X-rays are of significant value in the diagnosis of the gall bladder disease and to decide whether a surgery is needed or not.     

相依模型参数估计的相对效率

本文考虑相依模型,对其未知参数向量给出了其最佳线性无偏估计相对于协方差改进估计的四种相对效率,同时,还给出了最小二乘估计相对于协方差改进估计的三种相对效率,在不同条件下,分别给出了相对效率的上界与下界。     

Maximum Likelihood Estimation of Long‐Term HIV Dynamic Models and Antiviral Response

Summary HIV dynamics studies, based on differential equations, have significantly improved the knowledge on HIV infection. While first studies used simplified short‐term dynamic models, recent works considered more complex long‐term models combined with a global analysis of whole patient data based on nonlinear mixed models, increasing the accuracy of the HIV dynamic analysis. However statistical issues remain, given the complexity of the problem. We proposed to use the SAEM (stochastic approximation expectation‐maximization) algorithm, a powerful maximum likelihood estimation algorithm, to analyze simultaneously the HIV viral load decrease and the CD4 increase in patients using a long‐term HIV dynamic system. We applied the proposed methodology to the prospective COPHAR2–ANRS 111 trial. Very satisfactory results were obtained with a model with latent CD4 cells defined with five differential equations. One parameter was fixed, the 10 remaining parameters (eight with between‐patient variability) of this model were well estimated. We showed that the efficacy of nelfinavir was reduced compared to indinavir and lopinavir.     

Maximum Likelihood Estimation of the Mortality Rate Function

Maximum likelihood estimator is obtained for the mortality rate function of a specific type appearing in survival data analysis. Strict consistency of this estimator is proved.     

Scheffy's Confidence Intervals in the Models of Analysis of Covariance

Scheffe's confidence intervals for linear functions of some subvectors of a vector of parameters are presented. The considered subvectors are such that covariance matrices of their estimators are known non-negative definite matrices multiplied by unknown positive constants. This property is characteristic of the least squares estimators of vectors of main and interaction effects in the analysis of covariance models of the following experimental designs: split-block, split-plot, completely randomized two-factor design and randomized complete block design. The formulas for confidence intervals for linear functions of vectors of main or interaction effects in the designs mentioned above are given in the paper. The practical example is given as an illustration.     

Maximum Likelihood Analysis of Population Differences in Allelic Frequencies

Statistical techniques are presented for the analysis of geographic variation in allelic frequencies. Likelihood ratio test criteria are derived from a multinominal sampling distribution, and are used to answer three questions. (1) Are there geographic differences in allelic frequencies? (2) Are population differences in allelic frequencies associated with environmental differences? (3) Is there any residual "lack of fit" variation among populations, after accounting for that variation associated with environmental differences? The two- and three-allele cases are explicitly treated, and the extension to more alleles is indicated.     

Maximum Likelihood Estimation of Genetic Parameters in Cell Populations

In this paper we consider a cell population such as bacteria consisting of two types of cells, mutant and nonmutant. Under the mutation and homogeneous pure birth processes, this paper derives a maximum likelihood estimation procedure for estimating mutation rate and birth rate. The method is applied to Newcombe's data; further some Monte Carlo studies are generated. The numerical results indicate that the method is quite efficient for estimating genetic parameters in cell populations.     

Conditional Maximum Likelihood Estimate and Exact Test of the Common Relative Difference in Combination of 2 × 2 Tables under Inverse Sampling

On the basis of the conditional distribution, given the marginal totals of non-cases fixed for each of independent 2 × 2 tables under inverse sampling, this paper develops the conditional maximum likelihood (CMLE) estimator of the underlying common relative difference (RD) and its asymptotic conditional variance. This paper further provides for the RD an exact interval calculation procedure, of which the coverage probability is always larger than or equal to the desired confidence level and for investigating whether the underlying common RD equals any specified value an exact test procedure, of which Type I error is always less than or equal to the nominal α-level. These exact interval estimation and exact hypothesis testing procedures are especially useful for the situation in which the number of index subjects in a study is small and the asymptotically approximate methods may not be appropriate for use. This paper also notes the condition under which the CMLE of RD uniquely exists and includes a simple example to illustrate use of these techniques.