Empirical Likelihood for Censored Linear Regression

In this paper we investigate the empirical likelihood method in a linear regression model when the observations are subject to random censoring. An empirical likelihood ratio for the slope parameter vector is defined and it is shown that its limiting distribution is a weighted sum of independent chi-square distributions. This reduces to the empirical likelihood to the linear regression model first studied by Owen (1991) if there is no censoring present. Some simulation studies are presented to compare the empirical likelihood method with the normal approximation based method proposed in Lai et al. (1995). It was found that the empirical likelihood method performs much better than the normal approximation method.     

Empirical Likelihood Confidence Intervals for Response Mean with Data Missing at Random

Abstract.  A kernel regression imputation method for missing response data is developed. A class of bias-corrected empirical log-likelihood ratios for the response mean is defined. It is shown that any member of our class of ratios is asymptotically chi-squared, and the corresponding empirical likelihood confidence interval for the response mean is constructed. Our ratios share some of the desired features of the existing methods: they are self-scale invariant and no plug-in estimators for the adjustment factor and asymptotic variance are needed; when estimating the non-parametric function in the model, undersmoothing to ensure root- n consistency of the estimator for the parameter is avoided. Since the range of bandwidths contains the optimal bandwidth for estimating the regression function, the existing data-driven algorithm is valid for selecting an optimal bandwidth. We also study the normal approximation-based method. A simulation study is undertaken to compare the empirical likelihood with the normal approximation method in terms of coverage accuracies and average lengths of confidence intervals.     

Empirical Likelihood Ratio Test for a Change-Point in Linear Regression Model

A nonparametric method based on the empirical likelihood is proposed to detect the change-point in the coefficient of linear regression models. The empirical likelihood ratio test statistic is proved to have the same asymptotic null distribution as that with classical parametric likelihood. Under some mild conditions, the maximum empirical likelihood change-point estimator is also shown to be consistent. The simulation results show the sensitivity and robustness of the proposed approach. The method is applied to some real datasets to illustrate the effectiveness.     

Empirical Likelihood Confidence Region for Parameters in Semi-linear Errors-in-Variables Models

Abstract.  This paper proposes a constrained empirical likelihood confidence region for a parameter in the semi-linear errors-in-variables model. The confidence region is constructed by combining the score function corresponding to the squared orthogonal distance with a constraint on the parameter, and it overcomes that the solution of limiting mean estimation equations is not unique. It is shown that the empirical log likelihood ratio at the true parameter converges to the standard chi-square distribution. Simulations show that the proposed confidence region has coverage probability which is closer to the nominal level, as well as narrower than those of normal approximation of generalized least squares estimator in most cases. A real data example is given.     

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.     

Empirical Likelihood Method for Censored Median Regression Models

Based on the semiparametric median regression analysis for the right-censored data developed by Ying et al. (1995 Ying , Z. , Jung , S. H. , Wei , L. J. ( 1995 ). Survival analysis with median regression models . J. Amer. Statist. Assoc. 90 : 178 – 184 .[Taylor & Francis Online], [Web of Science ®] , [Google Scholar]), an empirical likelihood based inferential procedure for the regression coefficients is proposed. The limiting distribution of the proposed log-empirical likelihood ratio test statistic follows a chi-squared distribution, which corresponds to the standard asymptotic results of the empirical likelihood method. The inference about the subsets of the entire regression coefficients vector is discussed. The proposed method is illustrated by some simulation studies.     

Non-parametric Hypothesis Testing and Confidence Intervals with Doubly Censored Data

The non-parametric maximum likelihood estimator (NPMLE) of the distribution function with doubly censored data can be computed using the self-consistent algorithm (Turnbull, 1974). We extend the self-consistent algorithm to include a constraint on the NPMLE. We then show how to construct confidence intervals and test hypotheses based on the NPMLE via the empirical likelihood ratio. Finally, we present some numerical comparisons of the performance of the above method with another method that makes use of the influence functions.     

Nonlinear Censored Regression Using Synthetic Data

Abstract.  The problem of estimating a nonlinear regression model, when the dependent variable is randomly censored, is considered. The parameter of the model is estimated by least squares using synthetic data. Consistency and asymptotic normality of the least squares estimators are derived. The proofs are based on a novel approach that uses i.i.d. representations of synthetic data through Kaplan–Meier integrals. The asymptotic results are supported by a small simulation study.     

Confidence Intervals on Regression Models with Censored Data

Stute (1993, Consistent estimation under random censorship when covariables are present. Journal of Multivariate Analysis 45, 89–103) proposed a new method to estimate regression models with a censored response variable using least squares and showed the consistency and asymptotic normality for his estimator. This article proposes a new bootstrap-based methodology that improves the performance of the asymptotic interval estimation for the small sample size case. Therefore, we compare the behavior of Stute's asymptotic confidence interval with that of several confidence intervals that are based on resampling bootstrap techniques. In order to build these confidence intervals, we propose a new bootstrap resampling method that has been adapted for the case of censored regression models. We use simulations to study the improvement the performance of the proposed bootstrap-based confidence intervals show when compared to the asymptotic proposal. Simulation results indicate that, for the new proposals, coverage percentages are closer to the nominal values and, in addition, intervals are narrower.     

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.     

Statistical Inference of Type-II Progressively Hybrid Censored Data with Weibull Lifetimes

In this article, we discuss the maximum likelihood estimators and approximate maximum likelihood estimators of the parameters of the Weibull distribution with two different progressively hybrid censoring schemes. We also present the associated expressions of the expected total test time and the expected effective sample size which will be useful for experimental planning purpose. Finally, the efficiency of the point estimation of the parameters based on the two progressive hybrid censoring schemes are compared and the merits of each censoring scheme are discussed.     

Empirical Likelihood for Nonparametric Models Under Linear Process Errors

In this paper, we study the construction of confidence intervals for a nonparametric regression function under linear process errors by using the blockwise technique. It is shown that the blockwise empirical likelihood (EL) ratio statistic is asymptotically distributed. The result is used to obtain EL based confidence intervals for the nonparametric regression function. The finite‐sample performance of the method is evaluated through a simulation study.     

Exact Non-Parametric Confidence, Prediction and Tolerance Intervals with Progressive Type-II Censoring

Abstract.  This article extends recent results [Scand. J. Statist. 28 (2001) 699] about exact non-parametric inferences based on order statistics with progressive type-II censoring. The extension lies in that non-parametric inferences are now covered where the dependence between involved order statistics cannot be circumvented. These inferences include: (a) tolerance intervals containing at least a specified proportion of the parent distribution, (b) prediction intervals containing at least a specified number of observations in a future sample, and (c) outer and/or inner confidence intervals for a quantile interval of the parent distribution. The inferences are valid for any parent distribution with continuous distribution function. The key result shows how the probability of an event involving k dependent order statistics that are observable/uncensored with progressive type-II censoring can be represented as a mixture with known weights of corresponding probabilities involving k dependent ordinary order statistics. Further applications/developments concerning exact Kolmogorov-type confidence regions are indicated.     

Inferences Concerning Exponential Distributions in the Presence of Randomly Right Censored Data with Missing Censored Values

Inferences concerning exponential distributions are considered from a sampling theory viewpoint when the data are randomly right censored and the censored values are missing. Both one-sample and m-sample (m 2) problems are considered. Likelihood functions are obtained for situations in which the censoring mechanism is informative which leads to natural and intuitively appealing estimators of the unknown proportions of censored observations. For testing hypotheses about the unknown parameters, three well-known test statistics, namely, likelihood ratio test, score test, and Wald-type test are considered.     

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.     

Simultaneous Confidence Bands for Linear Regression with Covariates Constrained in Intervals

Abstract. The focus of this article is on simultaneous confidence bands over a rectangular covariate region for a linear regression model with k>1 covariates, for which only conservative or approximate confidence bands are available in the statistical literature stretching back to Working & Hotelling (J. Amer. Statist. Assoc. 24 , 1929; 73–85). Formulas of simultaneous confidence levels of the hyperbolic and constant width bands are provided. These involve only a k‐dimensional integral; it is unlikely that the simultaneous confidence levels can be expressed as an integral of less than k‐dimension. These formulas allow the construction for the first time of exact hyperbolic and constant width confidence bands for at least a small k(>1) by using numerical quadrature. Comparison between the hyperbolic and constant width bands is then addressed under both the average width and minimum volume confidence set criteria. It is observed that the constant width band can be drastically less efficient than the hyperbolic band when k>1. Finally it is pointed out how the methods given in this article can be applied to more general regression models such as fixed‐effect or random‐effect generalized linear regression models.     

Empirical Likelihood Ratio for Linear Transformation Models with Doubly Censored Data

Double censoring arises when T represents an outcome variable that can only be accurately measured within a certain range, [L, U], where L and U are the left- and right-censoring variables, respectively. When L is always observed, we consider the empirical likelihood inference for linear transformation models, based on the martingale-type estimating equation proposed by Chen et al. (2002 Chen , K. , Jin , Z. , Ying , Z. ( 2002 ). Semiparametric analysis of transformation models with censored data . Biometrika 89 : 659 – 668 .[Crossref], [Web of Science ®] , [Google Scholar]). It is demonstrated that both the approach of Lu and Liang (2006 Lu , W. , Liang , Y. ( 2006 ). Empirical likelihood inference for linear transformation models . Journal of Multivariate Analysis 97 : 1586 – 1599 .[Crossref], [Web of Science ®] , [Google Scholar]) and that of Yu et al. (2011 Yu , W. , Sun , Y. , Zheng , M. ( 2011 ). Empirical likelihood method for linear transformation models . Annals of the Institute of Statistical Mathematics 63 : 331 – 346 .[Crossref], [Web of Science ®] , [Google Scholar]) can be extended to doubly censored data. Simulation studies are conducted to investigate the performance of the empirical likelihood ratio methods.     

Methods of Statistical Inference for Median Regression Models with Doubly Censored Data

Recently, least absolute deviations (LAD) estimator for median regression models with doubly censored data was proposed and the asymptotic normality of the estimator was established. However, it is invalid to make inference on the regression parameter vectors, because the asymptotic covariance matrices are difficult to estimate reliably since they involve conditional densities of error terms. In this article, three methods, which are based on bootstrap, random weighting, and empirical likelihood, respectively, and do not require density estimation, are proposed for making inference for the doubly censored median regression models. Simulations are also done to assess the performance of the proposed methods.     

Marginal Likelihood Estimation for Proportional Odds Models with Right Censored Data

One majoraspect in medical research is to relate the survival times ofpatients with the relevant covariates or explanatory variables.The proportional hazards model has been used extensively in thepast decades with the assumption that the covariate effects actmultiplicatively on the hazard function, independent of time.If the patients become more homogeneous over time, say the treatmenteffects decrease with time or fade out eventually, then a proportionalodds model may be more appropriate. In the proportional oddsmodel, the odds ratio between patients can be expressed as afunction of their corresponding covariate vectors, in which,the hazard ratio between individuals converges to unity in thelong run. In this paper, we consider the estimation of the regressionparameter for a semiparametric proportional odds model at whichthe baseline odds function is an arbitrary, non-decreasing functionbut is left unspecified. Instead of using the exact survivaltimes, only the rank order information among patients is used.A Monte Carlo method is used to approximate the marginal likelihoodfunction of the rank invariant transformation of the survivaltimes which preserves the information about the regression parameter.The method can be applied to other transformation models withcensored data such as the proportional hazards model, the generalizedprobit model or others. The proposed method is applied to theVeteran's Administration lung cancer trial data.     

Confidence Intervals for Nonparametric Regression Functions with Missing Data

Suppose that we have a nonparametric regression model Y = m(X) + ε with X ∈ R^p, where X is a random design variable and is observed completely, and Y is the response variable and some Y-values are missing at random. Based on the “complete” data sets for Y after nonaprametric regression imputation and inverse probability weighted imputation, two estimators of the regression function m(x₀) for fixed x₀ ∈ R^p are proposed. Asymptotic normality of two estimators is established, which is used to construct normal approximation-based confidence intervals for m(x₀). We also construct an empirical likelihood (EL) statistic for m(x₀) with limiting distribution of χ²₁, which is used to construct an EL confidence interval for m(x₀).