Beta Regression for Modelling Rates and Proportions

This paper proposes a regression model where the response is beta distributed using a parameterization of the beta law that is indexed by mean and dispersion parameters. The proposed model is useful for situations where the variable of interest is continuous and restricted to the interval (0, 1) and is related to other variables through a regression structure. The regression parameters of the beta regression model are interpretable in terms of the mean of the response and, when the logit link is used, of an odds ratio, unlike the parameters of a linear regression that employs a transformed response. Estimation is performed by maximum likelihood. We provide closed-form expressions for the score function, for Fisher's information matrix and its inverse. Hypothesis testing is performed using approximations obtained from the asymptotic normality of the maximum likelihood estimator. Some diagnostic measures are introduced. Finally, practical applications that employ real data are presented and discussed.     

Ordinal ridge regression with categorical predictors

In multi-category response models, categories are often ordered. In the case of ordinal response models, the usual likelihood approach becomes unstable with ill-conditioned predictor space or when the number of parameters to be estimated is large relative to the sample size. The likelihood estimates do not exist when the number of observations is less than the number of parameters. The same problem arises if constraint on the order of intercept values is not met during the iterative procedure. Proportional odds models (POMs) are most commonly used for ordinal responses. In this paper, penalized likelihood with quadratic penalty is used to address these issues with a special focus on POMs. To avoid large differences between two parameter values corresponding to the consecutive categories of an ordinal predictor, the differences between the parameters of two adjacent categories should be penalized. The considered penalized-likelihood function penalizes the parameter estimates or differences between the parameter estimates according to the type of predictors. Mean-squared error for parameter estimates, deviance of fitted probabilities and prediction error for ridge regression are compared with usual likelihood estimates in a simulation study and an application.     

The Cox Proportional Hazards Model with a Partly Linear Relative Risk Function

The conventional Cox proportional hazards regression model contains a loglinear relative risk function, linking the covariate information to the hazard ratio with a finite number of parameters. A generalization, termed the partly linear Cox model, allows for both finite dimensional parameters and an infinite dimensional parameter in the relative risk function, providing a more robust specification of the relative risk function. In this work, a likelihood based inference procedure is developed for the finite dimensional parameters of the partly linear Cox model. To alleviate the problems associated with a likelihood approach in the presence of an infinite dimensional parameter, the relative risk is reparameterized such that the finite dimensional parameters of interest are orthogonal to the infinite dimensional parameter. Inference on the finite dimensional parameters is accomplished through maximization of the profile partial likelihood, profiling out the infinite dimensional nuisance parameter using a kernel function. The asymptotic distribution theory for the maximum profile partial likelihood estimate is established. It is determined that this estimate is asymptotically efficient; the orthogonal reparameterization enables employment of profile likelihood inference procedures without adjustment for estimation of the nuisance parameter. An example from a retrospective analysis in cancer demonstrates the methodology.     

Semiparametric Regression with Kernel Error Model

Abstract.  We propose and study a class of regression models, in which the mean function is specified parametrically as in the existing regression methods, but the residual distribution is modelled non-parametrically by a kernel estimator, without imposing any assumption on its distribution. This specification is different from the existing semiparametric regression models. The asymptotic properties of such likelihood and the maximum likelihood estimate (MLE) under this semiparametric model are studied. We show that under some regularity conditions, the MLE under this model is consistent (when compared with the possibly pseudo-consistency of the parameter estimation under the existing parametric regression model), is asymptotically normal with rate and efficient. The non-parametric pseudo-likelihood ratio has the Wilks property as the true likelihood ratio does. Simulated examples are presented to evaluate the accuracy of the proposed semiparametric MLE method.     

Estimation of two order restricted normal means with unknown and possibly unequal variances

Estimation of each of and linear functions of two order restricted normal means is considered when variances are unknown and possibly unequal. We replace unknown variances with sample variances and construct isotonic regression estimators, which we call in our paper the plug-in estimators, to estimate ordered normal means. Under squared error loss, a necessary and sufficient condition is given for the plug-in estimators to improve upon the unrestricted maximum likelihood estimators uniformly. As for the estimation of linear functions of ordered normal means, we also show that when variances are known, the restricted maximum likelihood estimator always improves upon the unrestricted maximum likelihood estimator uniformly, but when variances are unknown, the plug-in estimator does not always improve upon the unrestricted maximum likelihood estimator uniformly.     

Extending conditional likelihood in models for stratified binary data

The conditional likelihood is widely used in logistic regression models with stratified binary data. In particular, it leads to accurate inference for the parameters of interest, which are common to all strata, eliminating stratum-specific nuisance parameters. The modified profile likelihood is an accurate approximation to the conditional likelihood, but has the advantage of being available for general parametric models. Here, we propose the modified profile likelihood as an ideal extension of the conditional likelihood in generalized linear models for binary data, with generic link function. An important feature is that for the implementation we only need standard outputs of routines for generalized linear models. The accuracy of the method is supported by theoretical properties and is confirmed by simulation results.This research was supported by MIUR COFIN 2001-2003.     

Parameter estimation in semi-linear models using a maximal invariant likelihood function

In this paper, we consider the problem of estimation of semi-linear regression models. Using invariance arguments, Bhowmik and King [2007. Maximal invariant likelihood based testing of semi-linear models. Statist. Papers 48, 357–383] derived the probability density function of the maximal invariant statistic for the non-linear component of these models. Using this density function as a likelihood function allows us to estimate these models in a two-step process. First the non-linear component parameters are estimated by maximising the maximal invariant likelihood function. Then the non-linear component, with the parameter values replaced by estimates, is treated as a regressor and ordinary least squares is used to estimate the remaining parameters. We report the results of a simulation study conducted to compare the accuracy of this approach with full maximum likelihood and maximum profile-marginal likelihood estimation. We find maximising the maximal invariant likelihood function typically results in less biased and lower variance estimates than those from full maximum likelihood.     

Converting observed likelihood to levels of significance for transformation models

Inference for a scalar parameter in the pressence of nuisance parameters requires high dimensional integrations of the joint density of the pivotal quantities. Recent development in asymptotic methods provides accurate approximations for significance levels and thus confidence intervals for a scalar component parameter. In this paper, a simple, efficient and accurate numerical procedure is first developed for the location model and is then extended to the location-scale model and the linear regression model. This numerical procedure only requires a fine tabulation of the parameter and the observed log likelihood function, which can be either the full, marginal or conditional observed log likelihood function, as input and output is the corresponding significance function. Numerical results showed that this approximation is not only simple but also very accurate. It outperformed the usual approximations such as the signed likelihood ratio statistic, the maximum likelihood estimate and the score statistic.     

Application of Quasi-Least Squares to Analyse Replicated Autoregressive Time Series Regression Models

Time series regression models have been widely studied in the literature by several authors. However, statistical analysis of replicated time series regression models has received little attention. In this paper, we study the application of the quasi-least squares method to estimate the parameters in a replicated time series model with errors that follow an autoregressive process of order p. We also discuss two other established methods for estimating the parameters: maximum likelihood assuming normality and the Yule-Walker method. When the number of repeated measurements is bounded and the number of replications n goes to infinity, the regression and the autocorrelation parameters are consistent and asymptotically normal for all three methods of estimation. Basically, the three methods estimate the regression parameter efficiently and differ in how they estimate the autocorrelation. When p=2, for normal data we use simulations to show that the quasi-least squares estimate of the autocorrelation is undoubtedly better than the Yule-Walker estimate. And the former estimate is as good as the maximum likelihood estimate almost over the entire parameter space.     

空间回归模型估计中的最小二乘法

空间回归模型由于引入了空间地理信息而使得其参数估计变得复杂,因为主要采用最大似然法,致使一般人认为在空间回归模型参数估计中不存在最小二乘法。通过分析空间回归模型的参数估计技术,研究发现,最小二乘法和最大似然法分别用于估计空间回归模型的不同的参数,只有将两者结合起来才能快速有效地完成全部的参数估计。数理论证结果表明,空间回归模型参数最小二乘估计量是最佳线性无偏估计量。空间回归模型的回归参数可以在估计量为正态性的条件下而实施显著性检验,而空间效应参数则不可以用此方法进行检验。     

Maximum likelihood computation for fitting semiparametric mixture models

Three general algorithms that use different strategies are proposed for computing the maximum likelihood estimate of a semiparametric mixture model. They seek to maximize the likelihood function by, respectively, alternating the parameters, profiling the likelihood and modifying the support set. All three algorithms make a direct use of the recently proposed fast and stable constrained Newton method for computing the nonparametric maximum likelihood of a mixing distribution and employ additionally an optimization algorithm for unconstrained problems. The performance of the algorithms is numerically investigated and compared for solving the Neyman-Scott problem, overcoming overdispersion in logistic regression models and fitting two-level mixed effects logistic regression models. Satisfactory results have been obtained.     

The four-parameter Burr XII distribution: Properties,regression model,and applications

This paper introduces a new four-parameter lifetime model called the Weibull Burr XII distribution. The new model has the advantage of being capable of modeling various shapes of aging and failure criteria. We derive some of its structural properties including ordinary and incomplete moments, quantile and generating functions, probability weighted moments, and order statistics. The new density function can be expressed as a linear mixture of Burr XII densities. We propose a log-linear regression model using a new distribution so-called the log-Weibull Burr XII distribution. The maximum likelihood method is used to estimate the model parameters. Simulation results to assess the performance of the maximum likelihood estimation are discussed. We prove empirically the importance and flexibility of the new model in modeling various types of data.     

Empirical Likelihood Local Polynomial Regression Analysis of Clustered Data

Abstract. In this article, a naive empirical likelihood ratio is constructed for a non‐parametric regression model with clustered data, by combining the empirical likelihood method and local polynomial fitting. The maximum empirical likelihood estimates for the regression functions and their derivatives are obtained. The asymptotic distributions for the proposed ratio and estimators are established. A bias‐corrected empirical likelihood approach to inference for the parameters of interest is developed, and the residual‐adjusted empirical log‐likelihood ratio is shown to be asymptotically chi‐squared. These results can be used to construct a class of approximate pointwise confidence intervals and simultaneous bands for the regression functions and their derivatives. Owing to our bias correction for the empirical likelihood ratio, the accuracy of the obtained confidence region is not only improved, but also a data‐driven algorithm can be used for selecting an optimal bandwidth to estimate the regression functions and their derivatives. A simulation study is conducted to compare the empirical likelihood method with the normal approximation‐based method in terms of coverage accuracies and average widths of the confidence intervals/bands. An application of this method is illustrated using a real data set.     

ASYMPTOTIC DISTRIBUTIONS OF SEMIPARAMETRIC MAXIMUM LIKELIHOOD ESTIMATORS WITH ESTIMATING EQUATIONS FOR GROUP-CENSORED DATA

Semiparametric maximum likelihood estimation with estimating equations (SMLE) is more flexible than traditional methods; it has fewer restrictions on distributions and regression models. The required information about distribution and regression structures is incorporated in estimating equations of the SMLE to improve the estimation quality of non‐parametric methods. The likelihood of SMLE for censored data involves complicated implicit functions without closed‐form expressions, and the first derivatives of the log‐profile‐likelihood cannot be expressed as summations of independent and identically distributed random variables; it is challenging to derive asymptotic properties of the SMLE for censored data. For group‐censored data, the paper shows that all the implicit functions are well defined and obtains the asymptotic distributions of the SMLE for model parameters and lifetime distributions. With several examples the paper compares the SMLE, the regular non‐parametric likelihood estimation method and the parametric MLEs in terms of their asymptotic efficiencies, and illustrates application of SMLE. Various asymptotic distributions of the likelihood ratio statistics are derived for testing the adequacy of estimating equations and a partial set of parameters equal to some known values.     

Empirical Likelihood Based Synthetic Data Method for Censored Regression Analysis

In this article, we propose a new empirical likelihood method for linear regression analysis with a right censored response variable. The method is based on the synthetic data approach for censored linear regression analysis. A log-empirical likelihood ratio test statistic for the entire regression coefficients vector is developed and we show that it converges to a standard chi-squared distribution. The proposed method can also be used to make inferences about linear combinations of the regression coefficients. Moreover, the proposed empirical likelihood ratio provides a way to combine different normal equations derived from various synthetic response variables. Maximizing this empirical likelihood ratio yields a maximum empirical likelihood estimator which is asymptotically equivalent to the solution of the estimating equation that are optimal linear combination of the original normal equations. It improves the estimation efficiency. The method is illustrated by some Monte Carlo simulation studies as well as a real example.     

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.     

A New Estimator for Cox Proportional Hazard Regression Model in Presence of Collinearity

We propose a new approach to estimate the parameters of the Cox proportional hazards model in the presence of collinearity. Generally, a maximum partial likelihood estimator is used to estimate parameters for the Cox proportional hazards model. However, the maximum partial likelihood estimators can be seriously affected by the presence of collinearity since the parameter estimates result in large variances.

In this study, we develop a Liu-type estimator for Cox proportional hazards model parameters and compare it with a ridge regression estimator based on the scalar mean squared error (MSE). Finally, we evaluate its performance through a simulation study.     

Estimation of regression parameters in a semiparametric transformation model

A semiparametric approach to model skewed/heteroscedastic regression data is discussed. We work with a semiparametric transform-both-sides regression model, which contains a parametric regression function and a nonparametric transformation. This model is adequate when the relationship between the median response and the explanatory variable has been specified by a theoretical result or a previous empirical study. The transform-both-sides model with a parametric transformation has been studied extensively and applied successfully to a number data sets. Allowing a nonparametric transformation function increases the flexibility of the model. In this article, we estimate the nonparametric transformation function by the conditional kernel density approach developed by Wang and Ruppert (1995), and then use a pseudo-maximum likelihood estimator to estimate the regression parameters. This estimate of the regression parameters has not been studied previously. In this article, the asymptotic distribution of this pseudo-MLE is derived. We also show that when σ, the standard deviation of the error, goes to zero (small σ asymptotics), this estimator is adaptive. Adaptive means that the regression parameters are estimated as precisely as when the transformation is known exactly. A similar result holds in the parametric approaches of Carroll and Ruppert (1984) and Ruppert and Aldershof (1989). Simulated and real examples are provided to illustrate the performance of the proposed estimator for finite sample size.     

Empirical Likelihood Inference of the Partial Linear Isotonic Errors-in-variables Regression Models with Missing Data

This article is concerned with statistical inference of the partial linear isotonic regression model missing response and measurement errors in covariates. We proposed an empirical likelihood ratio test statistics and show that it has a limiting weighted chi-square distribution. An adjusted empirical likelihood ratio statistic, which is shown to have a limiting standard central chi-square distribution, is then proposed further. A maximum empirical likelihood estimator is also developed. A simulation study is conducted to examine the finite-sample property of proposed procedure.     

The Burr XII System of densities: properties,regression model and applications

We introduce a new class of distributions called the Burr XII system of densities with two extra positive parameters. We provide a comprehensive treatment of some of its mathematical properties. We estimate the model parameters by maximum likelihood. We assess the performance of the maximum likelihood estimators in terms of biases and mean squared errors by means of a simulation study. We also introduce a new family of regression models based on this system of densities. The usefulness of the proposed models is illustrated by means of three real data sets.