Likelihood estimation for longitudinal zero-inflated power series regression models

In this paper, a zero-inflated power series regression model for longitudinal count data with excess zeros is presented. We demonstrate how to calculate the likelihood for such data when it is assumed that the increment in the cumulative total follows a discrete distribution with a location parameter that depends on a linear function of explanatory variables. Simulation studies indicate that this method can provide improvements in obtaining standard errors of the estimates. We also calculate the dispersion index for this model. The influence of a small perturbation of the dispersion index of the zero-inflated model on likelihood displacement is also studied. The zero-inflated negative binomial regression model is illustrated on data regarding joint damage in psoriatic arthritis.     

A Novel Bayesian Parameter Mapping Method for Estimating the Parameters of an Underlying Scientific Model

Population-parameter mapping (PPM) is a method for estimating the parameters of latent scientific models that describe the statistical likelihood function. The PPM method involves a Bayesian inference in terms of the statistical parameters and the mapping from the statistical parameter space to the parameter space of the latent scientific parameters, and obtains a model coherence estimate, P(coh). The P(coh) statistic can be valuable for designing experiments, comparing competing models, and can be helpful in redesigning flawed models. Examples are provided where greater estimation precision was found for small sample sizes for the PPM point estimates relative to the maximum likelihood estimator (MLE).     

Using BBPSO Algorithm to Estimate the Weibull Parameters with Censored Data

This article proposes the maximum likelihood estimates based on bare bones particle swarm optimization (BBPSO) algorithm for estimating the parameters of Weibull distribution with censored data, which is widely used in lifetime data analysis. This approach can produce more accuracy of the parameter estimation for the Weibull distribution. Additionally, the confidence intervals for the estimators are obtained. The simulation results show that the BB PSO algorithm outperforms the Newton–Raphson method in most cases in terms of bias, root mean square of errors, and coverage rate. Two examples are used to demonstrate the performance of the proposed approach. The results show that the maximum likelihood estimates via BBPSO algorithm perform well for estimating the Weibull parameters with censored data.     

Optimal cutpoints for random observations

For the discretisation of a continuous random variable into different categories the choice of cutpoints is essential. A popular application is the contingent valuation method. In a parametric approach, the choice of cutpoints directly effects the quality of the estimates. Therefore, optimal cutpoints are desirable in order to estimate the parameters most accurately. We consider an arbitrary number of cutpoints and determine optimal cutpoints for the exponential and Gumbel distribution and prove that the c-optimal cutpoints for the location parameter of the logistic distribution have corresponding equal category probabilities. Furthermore, we show that in the limiting case for infinitely many cutpoints there is no loss of information.     

Bayesian likelihood methods for estimating the end point of a distribution

Summary.  We consider maximum likelihood methods for estimating the end point of a distribution. The likelihood function is modified by a prior distribution that is imposed on the location parameter. The prior is explicit and meaningful, and has a general form that adapts itself to different settings. Results on convergence rates and limiting distributions are given. In particular, it is shown that the limiting distribution is non-normal in non-regular cases. Parametric bootstrap techniques are suggested for quantifying the accuracy of the estimator. We illustrate performance by applying the method to multiparameter Weibull and gamma distributions.     

Optimal B-robust estimators for the parameters of the Burr XII distribution

The Burr XII distribution offers a flexible alternative to the distributions that play important role for modelling data in reliability, risk and process capability. However, estimating the shape parameters of the Burr XII distribution is a challenging problem. The classical estimation methods such as maximum likelihood and least squares are often used to estimate the parameters of the Burr XII distribution, but these methods are very sensitive to the outliers in the data. Thus, a robust estimation method alternative to the classical methods is needed to find robust estimators that are less sensitive to the outliers in the data. The purpose of this paper is to use the optimal B-robust estimation method [Hampel FR, Ronchetti EM, Rousseeuw PJ, Stahel WA. Robust statistics: the approach based on influence functions. New York: Wiley; 1986] to obtain robust estimators for the shape parameters of the Burr XII distribution. The simulation results show that the optimal B-robust estimators generally outperform the classical estimators in terms of the bias and root mean square errors when there are outliers in data.     

测量误差模型稳健方法及应用研究

回归模型在经济学、生物医学、流行病学、工农业生产等众多领域有着广泛的应用,而在实际数据收集时常常出现无法获得变量的精确数据或全部数据的情况,即常碰到测量误差数据、缺失数据等复杂数据情形。对于回归模型中存在测量误差的情况,如在参数估计时不加以修正,则易产生估计偏差,使得估计精度下降。对于数据缺失情形,如不采取合理的处理方法也会导致模型分析结果不佳。故此,本文研究含有测量误差数据时,解释变量具有随机缺失时的线性测量误差模型和部分线性测量误差模型的稳健参数估计问题。本文提出了一种在测量误差服从拉普拉斯分布时参数的损失修正估计,通过蒙特卡洛模拟和医学研究中的实证分析,显示本文所提的估计方法具有偏差小、精度高、稳健性强的优势。     

Jackknife Empirical Likelihood for the Accelerated Failure Time Model with Censored Data

Kendall and Gehan estimating functions are commonly used to estimate the regression parameter in accelerated failure time model with censored observations in survival analysis. In this paper, we apply the jackknife empirical likelihood method to overcome the computation difficulty about interval estimation. A Wilks’ theorem of jackknife empirical likelihood for U-statistic type estimating equations is established, which is used to construct the confidence intervals for the regression parameter. We carry out an extensive simulation study to compare the Wald-type procedure, the empirical likelihood method, and the jackknife empirical likelihood method. The proposed jackknife empirical likelihood method has a better performance than the existing methods. We also use a real data set to compare the proposed methods.     

Comparison of methods of estimation of parameters of the weibull distribution

The generalised least squares, maximum likelihood, Bain-Antle 1 and 2, and two mixed methods of estimating the parameters of the two-parameter Weibull distribution are compared. The comparison is made using (a) the observed relative efficiency of parameter estimates and (b) themean squared relative error in estimated quantiles, to summarize the results of 1000 simulated samples of sizes 10 and 25. The results are that: generalised least squares is the best method of estimating the shape parameter ß the best method of estimating the scale parameter a depends onthe size of ß for quantile estimation maximum likelihood is best Bain-Antle 2 is uniformly the worst of the methods.     

Bayesian Estimation and Model Choice in Item Response Models

Item response models are essential tools for analyzing results from many educational and psychological tests. Such models are used to quantify the probability of correct response as a function of unobserved examinee ability and other parameters explaining the difficulty and the discriminatory power of the questions in the test. Some of these models also incorporate a threshold parameter for the probability of the correct response to account for the effect of guessing the correct answer in multiple choice type tests. In this article we consider fitting of such models using the Gibbs sampler. A data augmentation method to analyze a normal-ogive model incorporating a threshold guessing parameter is introduced and compared with a Metropolis-Hastings sampling method. The proposed method is an order of magnitude more efficient than the existing method. Another objective of this paper is to develop Bayesian model choice techniques for model discrimination. A predictive approach based on a variant of the Bayes factor is used and compared with another decision theoretic method which minimizes an expected loss function on the predictive space. A classical model choice technique based on a modified likelihood ratio test statistic is shown as one component of the second criterion. As a consequence the Bayesian methods proposed in this paper are contrasted with the classical approach based on the likelihood ratio test. Several examples are given to illustrate the methods.     

部分函数型线性变系数模型的序列相关检验

部分函数型线性变系数模型(PFLVCM)是近几年出现的一个比较灵活、应用广泛的新模型。在实际应用中,搜集到的经济和金融数据往往存在序列相关性。如果不考虑数据间的相关性直接对其进行建模,会影响模型中参数估计的精度和有效性。本文主要研究了PFLVCM中误差的序列相关性的检验问题,基于经验似然,把标量时间序列数据相关性检验的方法拓展到函数型数据中,提出了经验对数似然比检验统计量,并在零假设下得到了检验统计量的近似分布。通过蒙特卡洛数值模拟说明该统计量在有限样本下有良好的水平和功效。最后,把该方法用于检验美国商业用电消费数据是否有序列相关性,证明该统计量的有效性和实用性。     

Modeling Longitudinal Obesity Data with Intermittent Missingness Using a New Latent Variable Model

We propose a latent variable model for informative missingness in longitudinal studies which is an extension of latent dropout class model. In our model, the value of the latent variable is affected by the missingness pattern and it is also used as a covariate in modeling the longitudinal response. So the latent variable links the longitudinal response and the missingness process. In our model, the latent variable is continuous instead of categorical and we assume that it is from a normal distribution. The EM algorithm is used to obtain the estimates of the parameter we are interested in and Gauss–Hermite quadrature is used to approximate the integration of the latent variable. The standard errors of the parameter estimates can be obtained from the bootstrap method or from the inverse of the Fisher information matrix of the final marginal likelihood. Comparisons are made to the mixed model and complete-case analysis in terms of a clinical trial dataset, which is Weight Gain Prevention among Women (WGPW) study. We use the generalized Pearson residuals to assess the fit of the proposed latent variable model.     

Comparison of two methods in estimating standard error of the method of simulated moments estimators for generalized linear mixed models

ABSTRACT

The likelihood of a generalized linear mixed model (GLMM) often involves high-dimensional integrals, which in general cannot be computed explicitly. When direct computation is not available, method of simulated moments (MSM) is a fairly simple way to estimate the parameters of interest. In this research, we compared parametric bootstrap (PB) and nonparametric bootstrap methods (NPB) in estimating the standard errors of MSM estimators for GLMM. Simulation results show that when the group size is large, the PB and NPB perform similarly; when group size is medium, NPB performs better than PB in estimating standard errors of the mean.     

非线性回归模型参数估计方法研究——以C-D生产函数为例

通过理论分析和蒙特卡罗模拟,对C-D生产函数模型参数的估计方法进行比较研究的结果表明：当误差项满足经典假设时,非线性最小二乘估计量具有与线性最小二乘估计类似的、近似BLUE的特性,且当误差项存在异方差时,用加权非线性最小二乘法也能大大改善估计量的性质。     

Semiparametric estimation for the dispersion parameter in the analysis of over- or underdispersed count data

This paper investigates several semiparametric estimators of the dispersion parameter in the analysis of over- or underdispersed count data when there is no likelihood available. In the context of estimating the dispersion parameter, we consider the double-extended quasi-likelihood (DEQL), the pseudo-likelihood and the optimal quadratic estimating (OQE) equations method and compare them with the maximum likelihood method, the method of moments and the extended quasi-likelihood through simulation study. The simulation study shows that the estimator based on the DEQL has superior bias and efficiency property for moderate and large sample size, and for small sample size the estimator based on the OQE equations outperforms the other estimators. Three real-life data sets arising in biostatistical practices are analyzed, and the findings from these analyses are quite similar to what are found from the simulation study.     

Estimation of parameters in incomplete data models defined by dynamical systems

Parametric incomplete data models defined by ordinary differential equations (ODEs) are widely used in biostatistics to describe biological processes accurately. Their parameters are estimated on approximate models, whose regression functions are evaluated by a numerical integration method. Accurate and efficient estimations of these parameters are critical issues. This paper proposes parameter estimation methods involving either a stochastic approximation EM algorithm (SAEM) in the maximum likelihood estimation, or a Gibbs sampler in the Bayesian approach. Both algorithms involve the simulation of non-observed data with conditional distributions using Hastings–Metropolis (H–M) algorithms. A modified H–M algorithm, including an original local linearization scheme to solve the ODEs, is proposed to reduce the computational time significantly. The convergence on the approximate model of all these algorithms is proved. The errors induced by the numerical solving method on the conditional distribution, the likelihood and the posterior distribution are bounded. The Bayesian and maximum likelihood estimation methods are illustrated on a simulated pharmacokinetic nonlinear mixed-effects model defined by an ODE. Simulation results illustrate the ability of these algorithms to provide accurate estimates.     

零膨胀计数数据的联合建模及变量选择

零膨胀计数数据破坏了泊松分布的方差-均值关系，可由取值服从泊松分布的数据和取值为零(退化分布)的数据各占一定比例所构成的混合分布所解释。本文基于自适应弹性网技术， 研究了零膨胀计数数据的联合建模及变量选择问题.对于零膨胀泊松分布，引入潜变量，构造出零膨胀泊松模型的完全似然, 其中由零膨胀部分和泊松部分两项组成.考虑到协变量可能存在共线性和稀疏性，通过对似然函数加自适应弹性网惩罚得到目标函数,然后利用EM算法得到回归系数的稀疏估计量，并用贝叶斯信息准则BIC来确定最优调节参数.本文也给出了估计量的大样本性质的理论证明和模拟研究，最后把所提出的方法应用到实际问题中。     

Fitting and comparing seed germination models with a focus on the inverse normal distribution

This paper reviews current methods for fitting a range of models to censored seed germination data and recommends adoption of a probability‐based model for the time to germination. It shows that, provided the probability of a seed eventually germinating is not on the boundary, maximum likelihood estimates, their standard errors and the resultant deviances are identical whether only those seeds which have germinated are used or all seeds (including seeds ungerminated at the end of the experiment). The paper recommends analysis of deviance when exploring whether replicate data are consistent with a hypothesis that the underlying distributions are identical, and when assessing whether data from different treatments have underlying distributions with common parameters. The inverse normal distribution, otherwise known as the inverse Gaussian distribution, is discussed, as a natural distribution for the time to germination (including a parameter to measure the lag time to germination). The paper explores some of the properties of this distribution, evaluates the standard errors of the maximum likelihood estimates of the parameters and suggests an accurate approximation to the cumulative distribution function and the median time to germination. Additional material is on the web, at http://www.agric.usyd.edu.au/staff/oneill/ .     

Estimating the parameters of a bathtub-shaped distribution under progressive type-II censoring

We consider the problem of estimating unknown parameters, reliability function and hazard function of a two parameter bathtub-shaped distribution on the basis of progressive type-II censored sample. The maximum likelihood estimators and Bayes estimators are derived for two unknown parameters, reliability function and hazard function. The Bayes estimators are obtained against squared error, LINEX and entropy loss functions. Also, using the Lindley approximation method we have obtained approximate Bayes estimators against these loss functions. Some numerical comparisons are made among various proposed estimators in terms of their mean square error values and some specific recommendations are given. Finally, two data sets are analyzed to illustrate the proposed methods.     

THE ANALYSIS OF DISCRETE CHOICE EXPERIMENTS WITH CORRELATED ERROR STRUCTURE

In a stated preference discrete choice experiment each subject is typically presented with several choice sets, and each choice set contains a number of alternatives. The alternatives are defined in terms of their name (brand) and their attributes at specified levels. The task for the subject is to choose from each choice set the alternative with highest utility for them. The multinomial is an appropriate distribution for the responses to each choice set since each subject chooses one alternative, and the multinomial logit is a common model. If the responses to the several choice sets are independent, the likelihood function is simply the product of multinomials. The most common and generally preferred method of estimating the parameters of the model is maximum likelihood (that is, selecting as estimates those values that maximize the likelihood function). If the assumption of within-subject independence to successive choice tasks is violated (it is almost surely violated), the likelihood function is incorrect and maximum likelihood estimation is inappropriate. The most serious errors involve the estimation of the variance-covariance matrix of the model parameter estimates, and the corresponding variances of market shares and changes in market shares.

In this paper we present an alternative method of estimation of the model parameter coefficients that incorporates a first-order within-subject covariance structure. The method involves the familiar log-odds transformation and application of the multivariate delta method. Estimation of the model coefficients after the transformation is a straightforward generalized least squares regression, and the corresponding improved estimate of the variance-covariance matrix is in closed form. Estimates of market share (and change in market share) follow from a second application of the multivariate delta method. The method and comparison with maximum likelihood estimation are illustrated with several simulated and actual data examples.

Advantages of the proposed method are: 1) it incorporates the within-subject covariance structure; 2) it is completely data driven; 3) it requires no additional model assumptions; 4) assuming asymptotic normality, it provides a simple procedure for computing confidence regions on market shares and changes in market shares; and 5) it produces results that are asymptotically equivalent to those produced by maximum likelihood when the data are independent.