Empirical Log-likelihood Ratio Research Articles

Monitoring of high-dimensional and high-frequency data streams: A nonparametric approach

ABSTRACT High-dimensional and high-frequency data streams (HHDS) have become widely available with the prevalence of data-acquisition technologies. Because HHDS have the characteristics of high dimensionality and temporal correlation, traditional statistical process control (SPC) techniques cannot be used directly to address the problem of monitoring HHDS. Most extant monitoring methods commonly assume that the process observations are independent or can be described by some parametric models and face the challenge posed by high-dimensional data. Little research has been conducted on the robust monitoring of HHDS that can simultaneously accommodate high dimensions and high frequencies. Therefore, in this paper, we propose a novel nonparametric approach for monitoring HHDS, based on data decorrelation, dimension reduction, and SPC. Specifically, process observations are first sequentially decorrelated and standardized using Cholesky decomposition, and then the k -nearest neighbors classification algorithm is applied to transform uncorrelated high-dimensional data into one-dimensional data. Finally, the traditional cumulative-sum procedure is used for online monitoring, based on an empirical log-likelihood ratio test. Numerical studies have shown that our monitoring scheme is sufficiently reliable and efficient for the online monitoring of HHDS. An illustrative example about the Dow Jones 30 industrial stock prices is presented to further validate the proposed approach.

Decorrelated empirical likelihood for generalized linear models with high-dimensional longitudinal data

This paper focuses on decorrelated empirical likelihood-based inference for longitudinal data with ultrahigh-dimensional covariates. The primary issues we aim to address involve parameter estimation and hypothesis testing for a low-dimensional parameter of interest. Under the framework of the generalized linear model, we initially consider the within-subject correlation by linearizing the precision matrix with certain known matrices, which retains optimality even if the working correlated structure is misspecified. Coupled with the decorrelated matrix, we then eliminate the influence of nuisance parameters on the estimation procedure. The proposed approach not only yields more efficient estimators compared to generalized decorrelated estimating equations but also shares the same asymptotic variance as quadratic decorrelated inference function based methods. Furthermore, we define the decorrelated empirical log-likelihood ratio test statistic to assess the significance of regression coefficients. Finally, to evaluate the performance of the proposed procedure, we conduct simulation studies and apply it to a real data example.

Empirical likelihood and estimation in a partially linear varying coefficient model with right censored data

In this paper, we study the empirical likelihood and estimation of parameters of interest in a partially linear varying coefficient model with right censored data. Two cases are considered: censoring is independent of the covariates, and censoring depends on the covariates. The bias-corrected empirical log-likelihood ratios for the regression parameters are presented, Wilks' theorem is proved, and thus the confidence regions of the regression parameters are constructed. The estimators of parametric and nonparametric components are constructed, their asymptotic distributions are obtained, and the consistent estimators of the asymptotic variances are also given. Furthermore, the partial profile empirical log-likelihood ratios for each component of the regression parameters and the coefficient functions are constructed, and show they are asymptotically chi-squared. The obtained results can be directly used to construct the confidence regions/intervals for the regression parameters and the pointwise confidence intervals for the coefficient functions. Our approach is to directly calibrate the empirical log-likelihood ratio so that the resulting ratio is asymptotically chi-squared, undersmoothing of the coefficient functions is avoided, and existing data-driven methods can be effectively used to select the optimal bandwidth. Simulation studies compare the empirical likelihood method with one based on normal approximation and perform real data analysis.

Statistical Inference for Partially Linear Varying Coefficient Spatial Autoregressive Panel Data Model

This paper studies the estimation and inference of a partially linear varying coefficient spatial autoregressive panel data model with fixed effects. By means of the basis function approximations and the instrumental variable methods, we propose a two-stage least squares estimation procedure to estimate the unknown parametric and nonparametric components, and meanwhile study the asymptotic properties of the proposed estimators. Together with an empirical log-likelihood ratio function for the regression parameters, which follows an asymptotic chi-square distribution under some regularity conditions, we can further construct accurate confidence regions for the unknown parameters. Simulation studies show that the finite sample performance of the proposed methods are satisfactory in a wide range of settings. Lastly, when applied to the public capital data, our proposed model can also better reflect the changing characteristics of the US economy compared to the parametric panel data models.

Robust estimation and bias-corrected empirical likelihood in generalized linear models with right censored data

In this paper, we study the robust estimation and empirical likelihood for the regression parameter in generalized linear models with right censored data. A robust estimating equation is proposed to estimate the regression parameter, and the resulting estimator has consistent and asymptotic normality. A bias-corrected empirical log-likelihood ratio statistic of the regression parameter is constructed, and it is shown that the statistic converges weakly to a standard distribution. The result can be directly used to construct the confidence region of regression parameter. We use the bias correction method to directly calibrate the empirical log-likelihood ratio, which does not need to be multiplied by an adjustment factor. We also propose a method for selecting the tuning parameters in the loss function. Simulation studies show that the estimator of the regression parameter is robust and the bias-corrected empirical likelihood is better than the normal approximation method. An example of a real dataset from Alzheimer's disease studies shows that the proposed method can be applied in practical problems.

Empirical Likelihood for a First-Order Generalized Random Coefficient Integer-Valued Autoregressive Process.

In this paper, the authors consider the empirical likelihood method for a first-order generalized random coefficient integer-valued autoregressive process. The authors establish the log empirical likelihood ratio statistic and obtain its limiting distribution. Furthermore, the authors investigate the point estimation, confidence regions and hypothesis testing for the parameters of interest. The performance of empirical likelihood method is illustrated by a simulation study and a real data example.

Statistical Inference for Partially Linear Varying Coefficient Quantile Models with Missing Responses

The construction of confidence intervals is investigated for the partially linear varying coefficient quantile model with missing random responses. Combined with quantile regression, an imputation-based empirical likelihood method is proposed to construct confidence intervals for parametric and varying coefficient components. Then, it is proved that the proposed empirical log-likelihood ratios are asymptotically Chi-square in theory. Finally, the symmetry confidence intervals of the parametric components and the point-by-point confidence intervals of the varying coefficient components are constructed in the simulation studies to demonstrate further that the proposed method yields smaller confidence intervals and higher coverage probabilities.

Empirical likelihood in single-index partially functional linear model with missing observations

In this paper, we focus on the empirical likelihood in the single-index partially functional linear model. When the response variables or/and part of the covariates are missing at random, we construct the empirical likelihood ratio of the parameter in the model based on B-spline approximation for the link and slope functions, and define maximum empirical likelihood (MEL) estimator of the parameter. Under suitable assumptions, the asymptotic distributions of the proposed empirical log-likelihood ratio and MEL estimator are established. At the same time, based on penalized empirical likelihood (PEL) approach, we define the PEL estimator of the parameter and investigate variable selection of the model. A simulation study is done to evaluate the finite sample performance for the proposed methods.

Empirical likelihood analysis for accelerated failure time model using length-biased data

Length-biased data arise when the probability of observing a subject in a sample is proportional to its corresponding value. The statistical tools that should be applied to analyse such sets of data differ from those used for random samples. Ignoring this issue in an investigation results in a biased inference. In this paper, we propose an empirical likelihood-based method to draw inference on covariate effects in an accelerated failure time model while the observations are subject to length-bias. The asymptotic distribution of the empirical log-likelihood ratio statistic is derived to be a weighted sum of independent chi-square distributions. Hence, we then extend the results by an exploration into the adjusted empirical likelihood. We derive an asymptotic standard chi-square distribution for the adjusted log-likelihood statistic. The limiting distributions are applied to obtain confidence regions for the regression parameters. A simulation study is carried out to evaluate and compare the performance of the proposed procedures with an existing method based on the normal approximation approach. Finally, the procedures are illustrated by modelling the regression parameter and estimating confidence intervals for a set of real data on widths of shrubs.

Empirical likelihood without population covariance matrix

In this paper, we show that the log empirical likelihood ratio statistic for the population mean converges in distribution to achi-squared distribution when the population is in the generalized domain of attraction of multivariate normal law.

Estimation for a hybrid model of functional and linear measurement errors regression with missing response

In this paper, we consider least squares estimation and empirical likelihood inference for partial functional linear models when the covariates in the non-functional linear component are measured with additive error and the responses are missing at random. Asymptotic properties of the proposed estimators for the parametric and nonparametric components are established. A class of empirical log-likelihood ratio functions of the parametric component and response mean are developed, and the corresponding maximum empirical likelihood estimators are constructed. We also prove that the empirical log-likelihood ratio functions are asymptotic standard chi-squared distribution. The results can be used to construct confidence intervals for the parametric component and response mean. The asymptotic distributions of the corresponding maximum empirical likelihood estimators are also established. Meanwhile, a simulation study is conducted to demonstrate the finite sample performance of the proposed procedure. A real data analysis is also used to illustrate our methods.

Wilks’ theorem for semiparametric regressions with weakly dependent data

The empirical likelihood inference is extended to a class of semiparametric models for stationary, weakly dependent series. A partially linear single-index regression is used for the conditional mean of the series given its past, and the present and past values of a vector of covariates. A parametric model for the conditional variance of the series is added to capture further nonlinear effects. We propose suitable moment equations which characterize the mean and variance model. We derive an empirical log-likelihood ratio which includes nonparametric estimators of several functions, and we show that this ratio behaves asymptotically as if the functions were given.

Novel empirical likelihood inference for the mean difference with right-censored data.

This paper focuses on comparing two means and finding a confidence interval for the difference of two means with right-censored data using the empirical likelihood method combined with the independent and identically distributed random functions representation. In the literature, some early researchers proposed empirical link-based confidence intervals for the mean difference based on right-censored data using the synthetic data approach. However, their empirical log-likelihood ratio statistic has a scaled chi-squared distribution. To avoid the estimation of the scale parameter in constructing confidence intervals, we propose an empirical likelihood method based on the independent and identically distributed representation of Kaplan-Meier weights involved in the empirical likelihood ratio. We obtain the standard chi-squared distribution. We also apply the adjusted empirical likelihood to improve coverage accuracy for small samples. In addition, we investigate a new empirical likelihood method, the mean empirical likelihood, within the framework of our study. The performances of all the empirical likelihood methods are compared via extensive simulations. The proposed empirical likelihood-based confidence interval has better coverage accuracy than those from existing methods. Finally, our findings are illustrated with a real data set.

Statistical Inference for the Heteroscedastic Partially Linear Varying-Coefficient Errors-in-Variables Model with Missing Censoring Indicators

In this paper, we focus on heteroscedastic partially linear varying-coefficient errors-in-variables models under right-censored data with censoring indicators missing at random. Based on regression calibration, imputation, and inverse probability weighted methods, we define a class of modified profile least square estimators of the parameter and local linear estimators of the coefficient function, which are applied to constructing estimators of the error variance function. In order to improve the estimation accuracy and take into account the heteroscedastic error, reweighted estimators of the parameter and coefficient function are developed. At the same time, we apply the empirical likelihood method to construct confidence regions and maximum empirical likelihood estimators of the parameter. Under appropriate assumptions, the asymptotic normality of the proposed estimators is studied. The strong uniform convergence rate for the estimators of the error variance function is considered. Also, the asymptotic chi-squared distribution of the empirical log-likelihood ratio statistics is proved. A simulation study is conducted to evaluate the finite sample performance of the proposed estimators. Meanwhile, one real data example is provided to illustrate our methods.

Jackknife empirical likelihood confidence intervals for assessing heterogeneity in meta-analysis of rare binary event data

In meta-analysis, the heterogeneity of effect sizes across component studies is typically described by a variance parameter in a random-effects (Re) model. In the literature, methods for constructing confidence intervals (CIs) for the parameter often assume that study-level effect sizes are normally distributed. However, this assumption might be violated in practice, especially in meta-analysis of rare binary events. We propose to use jackknife empirical likelihood (JEL), a nonparametric approach that uses jackknife pseudo-values, to construct CIs for the heterogeneity parameter. To compute jackknife pseudo-values, we employ a moment-based estimator and consider two commonly used weighing schemes (i.e., equal and inverse variance weights). We prove that with each scheme, the resulting log empirical likelihood ratio follows a chi-square distribution asymptotically. We further examine the performance of the proposed JEL methods and compare them with existing CIs through simulation studies and data examples that focus on data of rare binary events. Our numerical results suggest that the JEL method with equal weights compares favorably to alternatives, especially when (observed) effect sizes are non-normal and the number of component studies is large. Thus, it is worth serious consideration in statistical inference.

Weighted empirical likelihood inferences for a class of varying coefficient ARCH-M models

In this paper, we consider the empirical likelihood inferences for a class of varying coefficient ARCH-M models, which is an extended version of parametric ARCH-M models. By constructing a weighted auxiliary random vector, we propose a weighted empirical likelihood method for estimating the functional-coefficients. Under some regularity conditions, the constructed empirical log-likelihood ratio is shown to be asymptotically , and then the pointwise confidence interval for functional-coefficient is constructed. Some simulation studies are carried out to compare finite sample performances of the proposed empirical likelihood estimation method with some existing estimation methods under various model settings. A real data analysis is also undertaken to illustrate practical implementation and performance of the proposed estimation procedure.

Empirical likelihood for varying coefficient partially nonlinear model with missing responses

In this paper, we consider the statistical inferences for varying coefficient partially nonlinear model with missing responses. Firstly, we employ the profile nonlinear least squares estimation based on the weighted imputation method to estimate the unknown parameter and the nonparametric function, meanwhile the asymptotic normality of the resulting estimators is proved. Secondly, we consider empirical likelihood inferences based on the weighted imputation method for the unknown parameter and nonparametric function, and propose an empirical log-likelihood ratio function for the unknown parameter vector in the nonlinear function and a residual-adjusted empirical log-likelihood ratio function for the nonparametric component, meanwhile construct relevant confidence regions. Thirdly, the response mean estimation is also studied. In addition, simulation studies are conducted to examine the finite sample performance of our methods, and the empirical likelihood approach based on the weighted imputation method (IEL) is further applied to a real data example.

Estimation and testing of multivariate random coefficient autoregressive model based on empirical likelihood

This paper studies the empirical likelihood method for a multivariate first-order random coefficient autoregressive (RCAR(1)) model. Based on the modified least square score equation, empirical likelihood method is used to test the randomness of coefficients and estimate the unknown parameters. First, the limiting distribution of the log empirical likelihood ratio statistic is established. Second, the maximum empirical likelihood estimator is derived and its asymptotic properties are established. Finally, the performances of the estimator are compared with the modified least square estimator via simulation. Furthermore, simulation result shows that the test has the correct size and very good power for all cases considered.

Empirical likelihood for mean difference between two samples with missing data

In this paper, we consider the empirical likelihood confidence intervals for mean difference under the assumption of missing at random (MAR). We prove that the empirical log-likelihood ratio of mean difference converges to a standard chi-squared distribution. We conduct simulation study to compare the proposed empirical likelihood with the normal approximation method in terms of the coverage accuracies and average lengths of confidence intervals. By Monte Carlo simulations, we verify that the proposed method performs well.

Penalized empirical likelihood for high-dimensional partially linear errors-in-function model with martingale difference errors

ABSTRACT For the high-dimensional partially linear errors-in-function model with martingale difference errors, we, in this paper, propose an empirical log-likelihood ratio function for the regression parameter. The empirical log-likelihood ratio is proven to be asymptotically chi-squared. In addition, we propose penalized empirical likelihood (PEL) for the parameter. By using an appropriate penalty function, we show that PEL has the oracle property. Also, we obtain the PEL ratio is defined and its limiting distribution is asymptotically chi-squared under the null hypothesis. Moreover, simulation studies and a real data example are undertaken to assess the finite sample performance of our proposed method.