Empirical likelihood-based inference in nonlinear regression models with missing responses at random

This paper investigates the estimations of regression parameters and response mean in nonlinear regression models in the presence of missing response variables that are missing with missingness probabilities depending on covariates. We propose four empirical likelihood (EL)-based estimators for the regression parameters and the response mean. The resulting estimators are shown to be consistent and asymptotically normal under some general assumptions. To construct the confidence regions for the regression parameters as well as the response mean, we develop four EL ratio statistics, which are proven to have the χ² distribution asymptotically. Simulation studies and an artificial data set are used to illustrate the proposed methodologies. Empirical results show that the EL method behaves better than the normal approximation method and that the coverage probabilities and average lengths depend on the selection probability function.     

On the asymptotic distributions of mean,autocovariance, autocorrelation,crossgovariancb and impulse response estimators of a stationary multidimensional random field

Asymptotic properties of mean, autocovariance, autocorrelation, crosscovariance and impulse response estimators of a stationary M-dimensionai (M-D) random field are studied. It is shown that only unbiased-type estimators of autocovariances, autocorrelations, crosscovariances and impulse responses have the asymptotic distributions when M≧ 2. Moreover, the asymptotic distributions of mean, autocovariance, autocorrelation, crosscovariance and impulse response estimators are presented.     

Weighted composite quantile regression method via empirical likelihood for non linear models

In this paper, we investigate empirical likelihood (EL) inferences via weighted composite quantile regression for non linear models. Under regularity conditions, we establish that the proposed empirical log-likelihood ratio is asymptotically chi-squared, and then the confidence intervals for the regression coefficients are constructed. The proposed method avoids estimating the unknown error density function involved in the asymptotic covariance matrix of the estimators. Simulations suggest that the proposed EL procedure is more efficient and robust, and a real data analysis is used to illustrate the performance.     

Efficient mean estimation in log-normal linear models

Log-normal linear models are widely used in applications, and many times it is of interest to predict the response variable or to estimate the mean of the response variable at the original scale for a new set of covariate values. In this paper we consider the problem of efficient estimation of the conditional mean of the response variable at the original scale for log-normal linear models. Several existing estimators are reviewed first, including the maximum likelihood (ML) estimator, the restricted ML (REML) estimator, the uniformly minimum variance unbiased (UMVU) estimator, and a bias-corrected REML estimator. We then propose two estimators that minimize the asymptotic mean squared error and the asymptotic bias, respectively. A parametric bootstrap procedure is also described to obtain confidence intervals for the proposed estimators. Both the new estimators and the bootstrap procedure are very easy to implement. Comparisons of the estimators using simulation studies suggest that our estimators perform better than the existing ones, and the bootstrap procedure yields confidence intervals with good coverage properties. A real application of estimating the mean sediment discharge is used to illustrate the methodology.     

Penalized empirical likelihood for quantile regression with missing covariates and auxiliary information

Based on the inverse probability weight method, we, in this article, construct the empirical likelihood (EL) and penalized empirical likelihood (PEL) ratios of the parameter in the linear quantile regression model when the covariates are missing at random, in the presence and absence of auxiliary information, respectively. It is proved that the EL ratio admits a limiting Chi-square distribution. At the same time, the asymptotic normality of the maximum EL and PEL estimators of the parameter is established. Also, the variable selection of the model in the presence and absence of auxiliary information, respectively, is discussed. Simulation study and a real data analysis are done to evaluate the performance of the proposed methods.     

Generalized empirical likelihood inference in partially linear model for longitudinal data with missing response variables and error-prone covariates

In this article, we consider statistical inference for longitudinal partial linear models when the response variable is sometimes missing with missingness probability depending on the covariate that is measured with error. A generalized empirical likelihood (GEL) method is proposed by combining correction attenuation and quadratic inference functions. The method that takes into consideration the correlation within groups is used to estimate the regression coefficients. Furthermore, residual-adjusted empirical likelihood (EL) is employed for estimating the baseline function so that undersmoothing is avoided. The empirical log-likelihood ratios are proven to be asymptotically Chi-squared, and the corresponding confidence regions for the parameters of interest are then constructed. Compared with methods based on NAs, the GEL does not require consistent estimators for the asymptotic variance and bias. The numerical study is conducted to compare the performance of the EL and the normal approximation-based method, and a real example is analysed.     

Absolute error criteria for bandwidth selection in density estimation from censored data

In Kernel density estimation, a criticism of bandwidth selection techniques which minimize squared error expressions is that they perform poorly when estimating tails of probability density functions. Techniques minimizing absolute error expressions are thought to result in more uniform performance and be potentially superior. An asympotic mean absolute error expression for nonparametric kernel density estimators from right-censored data is developed here. This expression is used to obtain local and global bandwidths that are optimal in the sense that they minimize asymptotic mean absolute error and integrated asymptotic mean absolute error, respectively. These estimators are illustrated fro eight data sets from known distributions. Computer simulation results are discussed, comparing the estimation methods with squared-error-based bandwidth selection for right-censored data.     

Relative error prediction via kernel regression smoothers

In this article, we introduce and study local constant and local linear nonparametric regression estimators when it is appropriate to assess performance in terms of mean squared relative error of prediction. We give asymptotic results for both boundary and non-boundary cases. These are special cases of more general asymptotic results that we provide concerning the estimation of the ratio of conditional expectations of two functions of the response variable. We also provide a good bandwidth selection method for the estimators. Examples of application, limited simulation results and discussion of related problems and approaches are also given.     

Empirical likelihood for generalized linear models with missing responses

The paper uses the empirical likelihood method to study the construction of confidence intervals and regions for regression coefficients and response mean in generalized linear models with missing response. By using the inverse selection probability weighted imputation technique, the proposed empirical likelihood ratios are asymptotically chi-squared. Our approach is to directly calibrate the empirical likelihood ratio, which is called as a bias-correction method. Also, a class of estimators for the parameters of interest is constructed, and the asymptotic distributions of the proposed estimators are obtained. A simulation study indicates that the proposed methods are comparable in terms of coverage probabilities and average lengths/areas of confidence intervals/regions. An example of a real data set is used for illustrating our methods.     

Shrinkage estimation in lognormal regression model for censored data

We introduce in this paper, the shrinkage estimation method in the lognormal regression model for censored data involving many predictors, some of which may not have any influence on the response of interest. We develop the asymptotic properties of the shrinkage estimators (SEs) using the notion of asymptotic distributional biases and risks. We show that if the shrinkage dimension exceeds two, the asymptotic risk of the SEs is strictly less than the corresponding classical estimators. Furthermore, we study the penalty (LASSO and adaptive LASSO) estimation methods and compare their relative performance with the SEs. A simulation study for various combinations of the inactive predictors and censoring percentages shows that the SEs perform better than the penalty estimators in certain parts of the parameter space, especially when there are many inactive predictors in the model. It also shows that the shrinkage and penalty estimators outperform the classical estimators. A real-life data example using Worcester heart attack study is used to illustrate the performance of the suggested estimators.     

Influence functions applied to the estimation of mean rain rate

In this paper we illustrate the usefulness of influence functions for studying properties of various statistical estimators of mean rain rate using space-borne radar data. In Martin (1999), estimators using censoring, minimum chi-square, and least squares are compared in terms of asymptotic variance. Here, we use influence functions to consider robustness properties of the same estimators. We also obtain formulas for the asymptotic variance of the estimators using influence functions, and thus show that they may also be used for studying relative efficiency. The least squares estimator, although less efficient, is shown to be more robust in the sense that it has the smallest gross-error sensitivity. In some cases, influence functions associated with the estimators reveal counterintuitive behaviour. For example, observations that are less than the mean rain rate may increase the estimated mean. The additional information gleaned from influence functions may be used to understand better and improve the estimation procedures themselves.     

Empirical likelihood for nonlinear models with missing responses

In this paper, a nonlinear model with response variables missing at random is studied. In order to improve the coverage accuracy for model parameters, the empirical likelihood (EL) ratio method is considered. On the complete data, the EL statistic for the parameters and its approximation have a χ² asymptotic distribution. When the responses are reconstituted using a semi-parametric method, the empirical log-likelihood on the response variables associated with the imputed data is also asymptotically χ². The Wilks theorem for EL on the parameters, based on reconstituted data, is also satisfied. These results can be used to construct the confidence region for the model parameters and the response variables. It is shown via Monte Carlo simulations that the EL methods outperform the normal approximation-based method in terms of coverage probability for the unknown parameter, including on the reconstituted data. The advantages of the proposed method are exemplified on real data.     

Combining least-squares and quantile regressions

Least-squares and quantile regressions are method of moments techniques that are typically used in isolation. A leading example where efficiency may be gained by combining least-squares and quantile regressions is one where some information on the error quantiles is available but the error distribution cannot be fully specified. This estimation problem may be cast in terms of solving an over-determined estimating equation (EE) system for which the generalized method of moments (GMM) and empirical likelihood (EL) are approaches of recognized importance. The major difficulty with implementing these techniques here is that the EEs associated with the quantiles are non-differentiable. In this paper, we develop a kernel-based smoothing technique for non-smooth EEs, and derive the asymptotic properties of the GMM and maximum smoothed EL (MSEL) estimators based on the smoothed EEs. Via a simulation study, we investigate the finite sample properties of the GMM and MSEL estimators that combine least-squares and quantile moment relationships. Applications to real datasets are also considered.     

Testing and Merging Information for Effect Size Estimation

A large-sample test for testing the equality of two effect sizes is presented. The null and non-null distributions of the proposed test statistic are derived. Further, the problem of estimating the effect size is considered when it is a priori suspected that two effect sizes may be close to each other. The combined data from all the samples leads to more efficient estimator of the effect size. We propose a basis for optimally combining estimation problems when there is uncertainty concerning the appropriate statistical model-estimator to use in representing the sampling process. The objective here is to produce natural adaptive estimators with some good statistical properties. In the context of two bivariate statistical models, the expressions for the asymptotic mean squared error of the proposed estimators are derived and compared with the parallel expressions for the benchmark estimators. We demonstrate that the suggested preliminary test estimator has superior asymptotic mean squared error performance relative to the benchmark and pooled estimators. A simulation study and application of the methodology to real data are presented.     

Testing symmetry based on empirical likelihood

In this paper, we propose a general kth correlation coefficient between the density function and distribution function of a continuous variable as a measure of symmetry and asymmetry. We first propose a root-n moment-based estimator of the kth correlation coefficient and present its asymptotic results. Next, we consider statistical inference of the kth correlation coefficient by using the empirical likelihood (EL) method. The EL statistic is shown to be asymptotically a standard chi-squared distribution. Last, we propose a residual-based estimator of the kth correlation coefficient for a parametric regression model to test whether the density function of the true model error is symmetric or not. We present the asymptotic results of the residual-based kth correlation coefficient estimator and also construct its EL-based confidence intervals. Simulation studies are conducted to examine the performance of the proposed estimators, and we also use our proposed estimators to analyze the air quality dataset.     

HAZARD RATE ESTIMATION FOR CENSORED DATA BY WAVELET METHODS

ABSTRACT

We study the estimation of a hazard rate function based on censored data by non-linear wavelet method. We provide an asymptotic formula for the mean integrated squared error (MISE) of nonlinear wavelet-based hazard rate estimators under randomly censored data. We show this MISE formula, when the underlying hazard rate function and censoring distribution function are only piecewise smooth, has the same expansion as analogous kernel estimators, a feature not available for the kernel estimators. In addition, we establish an asymptotic normality of the nonlinear wavelet estimator.     

Presmoothed Estimation with Left-Truncated and Right-Censored Data

We propose a new method to estimate the cumulative hazard function and the corresponding distribution function of survival times under randomly left-truncated and right-censored observations (LTRC). The new estimators are based on presmoothing ideas, the estimation of the conditional expectation m of the censoring indicator. An almost sure representation for both estimators is established, from which a strong consistency rate and asymptotic normality are derived. It is shown that the presmoothed modification leads to a gain in terms of asymptotic mean squared error. This efficiency with respect to the classical estimators is also shown in a simulation study. Finally, an application to a real data set is provided.     

A location-invariant non-positive moment-type estimator of the extreme value index

This paper investigates a class of location invariant non-positive moment-type estimators of extreme value index, which is highly flexible due to the tuning parameter involved. Its asymptotic expansions and its optimal sample fraction in terms of minimal asymptotic mean square error are derived. A small scale Monte Carlo simulation turns out that the new estimators, with a suitable choice of the tuning parameter driven by the data itself, perform well compared to the known ones. Finally, the proposed estimators with a bootstrap optimal sample fraction are applied to an environmental data set.     

On the mean squared error of nonparametric quantile estimators under random right-censorship

For randomly right-censored data, new asymptotic expressions for the mean squared errors of the product-limit quantile estimator and a kernel-type quantile estimator are presented in this paper. From these results a comparison of the two quantile estimators with respect to their mean squared errors is given.     

Parametric and nonparametric confidence intervals for estimating the difference of means of two skewed populations

In this paper, we have reviewed and proposed several interval estimators for estimating the difference of means of two skewed populations. Estimators include the ordinary-t, two versions proposed by Welch [17] and Satterthwaite [15], three versions proposed by Zhou and Dinh [18], Johnson [9], Hall [8], empirical likelihood (EL), bootstrap version of EL, median t proposed by Baklizi and Kibria [2] and bootstrap version of median t. A Monte Carlo simulation study has been conducted to compare the performance of the proposed interval estimators. Some real life health related data have been considered to illustrate the application of the paper. Based on our findings, some possible good interval estimators for estimating the mean difference of two populations have been recommended for the researchers.