Empirical likelihood for the difference of quantiles under censorship

In this paper, we use a smoothed empirical likelihood method to investigate the difference of quantiles under censorship. An empirical log-likelihood ratio is derived and its asymptotic distribution is shown to be chi-squared. Approximate confidence regions based on this method are constructed. Simulation studies are used to compare the empirical likelihood and the normal approximation method in terms of its coverage accuracy. It is found that the empirical likelihood method provides a much better performance. The research is supported by NSFC (10231030) and RFDP.     

Empirical Likelihood for Threshold Autoregressive Models

This article develops empirical likelihood for threshold autoregressive models. We propose general estimating equations based on moment constraint. Under some suitable conditions, we show the empirical likelihood estimators for parameter are asymptotically normally distributed, and the proposed log empirical likelihood ratio statistic asymptotically follows a standard chi-squared distribution.     

Asymptotic properties of the MAMSE adaptive likelihood weights

The weighted likelihood is a generalization of the likelihood designed to borrow strength from similar populations while making minimal assumptions. If the weights are properly chosen, the maximum weighted likelihood estimate may perform better than the maximum likelihood estimate (MLE). In a previous article, the minimum averaged mean squared error (MAMSE) weights are proposed and simulations show that they allow to outperform the MLE in many cases. In this paper, we study the asymptotic properties of the MAMSE weights. In particular, we prove that the MAMSE-weighted mixture of empirical distribution functions converges uniformly to the target distribution and that the maximum weighted likelihood estimate is strongly consistent. A short simulation illustrates the use of bootstrap in this context.     

Combined multiple testing by censored empirical likelihood

We propose a new procedure for combining multiple tests in samples of right-censored observations. The new method is based on multiple constrained censored empirical likelihood where the constraints are formulated as linear functionals of the cumulative hazard functions. We prove a version of Wilks’ theorem for the multiple constrained censored empirical likelihood ratio, which provides a simple reference distribution for the test statistic of our proposed method. A useful application of the proposed method is, for example, examining the survival experience of different populations by combining different weighted log-rank tests. Real data examples are given using the log-rank and Gehan-Wilcoxon tests. In a simulation study of two sample survival data, we compare the proposed method of combining tests to previously developed procedures. The results demonstrate that, in addition to its computational simplicity, the combined test performs comparably to, and in some situations more reliably than previously developed procedures. Statistical software is available in the R package ‘emplik’.     

Empirical Likelihood-based Inference for Stationary-ergodicity of the Generalized Random Coefficient Autoregressive Model

In this article, we consider the application of the empirical likelihood method to the generalized random coefficient autoregressive (GRCA) model. When the order of the model is 1, we derive an empirical likelihood ratio test statistic to test the stationary-ergodicity. Some simulation studies are also conducted to investigate the finite sample performances of the proposed test.     

Semiparametric tests for change-points with epidemic alternatives

The empirical likelihood ratio-based semiparametric tests of change-points with epidemic alternatives are constructed and are proved to have the same limiting null distributions as some well-known tests. The maximum empirical likelihood estimates of the change-points and the epidemic duration are shown to be consistent. Data-based model tests are also provided. The method is applied to a stock market price data and the Nile river data.     

Empirical likelihood inference for error density estimators in first-order autoregression models

In this paper we apply empirical likelihood method to the error density estimators in first-order autoregressive models under some mild conditions. The log-likelihood ratio statistic is shown to be asymptotically chi-squared distributed at a fixed point. In simulation, we show that the empirical likelihood produces confidence intervals having theoretical coverage accuracy which is better than normal approximation.     

Estimation and inference of combining quantile and least-square regressions with missing data

In this paper, we consider how to incorporate quantile information to improve estimator efficiency for regression model with missing covariates. We combine the quantile information with least-squares normal equations and construct an unbiased estimating equations (EEs). The lack of smoothness of the objective EEs is overcome by replacing them with smooth approximations. The maximum smoothed empirical likelihood (MSEL) estimators are established based on inverse probability weighted (IPW) smoothed EEs and their asymptotic properties are studied under some regular conditions. Moreover, we develop two novel testing procedures for the underlying model. The finite-sample performance of the proposed methodology is examined by simulation studies. A real example is used to illustrate our methods.     

The first-order random coefficient integer valued autoregressive process with the occasional level shift random noise based on dual empirical likelihood

Abstract

This paper investigates the first-order random coefficient integer valued autoregressive process with the occasional level shift random noise based on dual empirical likelihood. The limiting distribution of log empirical likelihood ratio statistic is constructed. Asymptotic convergence and confidence region results of empirical likelihood ratio are given. Hypothesis testing is considering, and maximum empirical likelihood estimation for parameter is acquired. Simulations are given to show that the maximum empirical likelihood estimation is more efficient than the conditional least squares estimation.     

Empirical likelihood‐based inferences for a low income proportion

Low income proportion is an important index in comparisons of poverty in countries around the world. The stability of a society depends heavily on this index. An accurate and reliable estimation of this index plays an important role for government's economic policies. In this paper, the authors study empirical likelihood‐based inferences for a low income proportion under the simple random sampling and stratified random sampling designs. It is shown that the limiting distributions of the empirical likelihood ratios for the low income proportion are the scaled chi‐square distributions. The authors propose various empirical likelihood‐based confidence intervals for the low income proportion. Extensive simulation studies are conducted to evaluate the relative performance of the normal approximation‐based interval, bootstrap‐based intervals, and the empirical likelihood‐based intervals. The proposed methods are also applied to analyzing a real economic survey income dataset. The Canadian Journal of Statistics 39: 1–16; 2011 ©2011 Statistical Society of Canada     

Empirical Likelihood for an Autoregressive Model with Explanatory Variables

In this article, we use the empirical likelihood method to construct the confidence region for parameters in autoregressive model with martingale difference error. It is shown that the empirical log-likelihood ratio at the true parameter converges to the standard chi-square distribution. The simulation results suggest that the empirical likelihood method outperforms the normal approximation based method in terms of coverage probability.     

Statistical Analysis Of Mixture Vector Autoregressive Models

In this paper, we reconsider the mixture vector autoregressive model, which was proposed in the literature for modelling non‐linear time series. We complete and extend the stationarity conditions, derive a matrix formula in closed form for the autocovariance function of the process and prove a result on stable vector autoregressive moving‐average representations of mixture vector autoregressive models. For these results, we apply techniques related to a Markovian representation of vector autoregressive moving‐average processes. Furthermore, we analyse maximum likelihood estimation of model parameters by using the expectation–maximization algorithm and propose a new iterative algorithm for getting the maximum likelihood estimates. Finally, we study the model selection problem and testing procedures. Several examples, simulation experiments and an empirical application based on monthly financial returns illustrate the proposed procedures.     

A two-state regime switching autoregressive model with an application to river flow analysis

We propose a regime switching autoregressive model and apply it to analyze daily water discharge series of River Tisza in Hungary. The dynamics is governed by two regimes, along which both the autoregressive coefficients and the innovation distributions are altering, moreover, the hidden regime indicator process is allowed to be non-Markovian. After examining stationarity and basic properties of the model, we turn to its estimation by Markov Chain Monte Carlo (MCMC) methods and propose two algorithms. The values of the latent process serve as auxiliary parameters in the first one, while the change points of the regimes do the same in the second one in a reversible jump MCMC setting. After comparing the mixing performance of the two methods, the model is fitted to the water discharge data. Simulations show that it reproduces the important features of the water discharge series such as the highly skewed marginal distribution and the asymmetric shape of the hydrograph.     

The Empirical Likelihood for First-Order Random Coefficient Integer-Valued Autoregressive Processes

This article studies the empirical likelihood method for the first-order random coefficient integer-valued autoregressive process. The limiting distribution of the log empirical likelihood ratio statistic is established. Confidence region for the parameter of interest and its coverage probabilities are given, and hypothesis testing is considered. The maximum empirical likelihood estimator for the parameter is derived and its asymptotic properties are established. The performances of the estimator are compared with the conditional least squares estimator via simulation.     

Empirical likelihood for a partially linear model with covariate data missing at random

In this paper, we consider the partial linear model with the covariables missing at random. Empirical likelihood ratios for the regression coefficients and the baseline function are investigated, 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. The finite sample behavior of the proposed method is evaluated with simulation and illustrated with an AIDS clinical trial dataset.     

Empirical Likelihood for First-order Autoregressive Error-in-variable of Models With Validation Data

In this article, we consider the empirical likelihood for the autoregressive error-in-explanatory variable models. With the help of validation, we first develop an empirical likelihood ratio test statistic for the parameters of interest, and prove that its asymptotic distribution is that of a weighted sum of independent standard χ²₁ random variables with unknown weights. Also, we propose an adjusted empirical likelihood and prove that its asymptotic distribution is a standard χ². Furthermore, an empirical likelihood-based confidence region is given. Simulation results indicate that the proposed method works well for practical situations.     

Asymptotics of the weighted least squares estimation for AR(1) processes with applications to confidence intervals

For the first-order autoregressive model, we establish the asymptotic theory of the weighted least squares estimations whether the underlying autoregressive process is stationary, unit root, near integrated or even explosive under a weaker moment condition of innovations. The asymptotic limit of this estimator is always normal. It is shown that the empirical log-likelihood ratio at the true parameter converges to the standard chi-square distribution. An empirical likelihood confidence interval is proposed for interval estimations of the autoregressive coefficient. The results improve the corresponding ones of Chan et al. (Econ Theory 28:705–717, 2012). Some simulations are conducted to illustrate the proposed method.     

Data-dependent probability matching priors for highest posterior density and equal-tailed two-sided regions based on empirical-type likelihoods

We consider a very general class of empirical-type likelihoods which includes the usual empirical likelihood and all its major variants proposed in the literature. It is known that none of these likelihoods admits a data-free probability matching prior for the highest posterior density region. We develop necessary higher order asymptotics to show that at least for the usual empirical likelihood this difficulty can be resolved if data-dependent priors are entertained. A related problem concerning the equal-tailed two-sided posterior credible region is also investigated. A simulation study is seen to lend support to the theoretical results.     

Two‐sample empirical likelihood ratio tests for medians in application to biomarker evaluations

The median is a commonly used parameter to characterize biomarker data. In particular, with two vastly different underlying distributions, comparing medians provides different information than comparing means; however, very few tests for medians are available. We propose a series of two‐sample median‐specific tests using empirical likelihood methodology and investigate their properties. We present the technical details of incorporating the relevant constraints into the empirical likelihood function for in‐depth median testing. An extensive Monte Carlo study shows that the proposed tests have excellent operating characteristics even under unfavourable occasions such as non‐exchangeability under the null hypothesis. We apply the proposed methods to analyze biomarker data from Western blot analysis to compare normal cells with bronchial epithelial cells from a case–control study. The Canadian Journal of Statistics 39: 671–689; 2011. © 2011 Statistical Society of Canada     

Approximate jackknife empirical likelihood method for estimating equations

It is known that the profile empirical likelihood method based on estimating equations is computationally intensive when the number of nuisance parameters is large. Recently, Li, Peng, & Qi (2011) proposed a jackknife empirical likelihood method for constructing confidence regions for the parameters of interest by estimating the nuisance parameters separately. However, when the estimators for the nuisance parameters have no explicit formula, the computation of the jackknife empirical likelihood method is still intensive. In this paper, an approximate jackknife empirical likelihood method is proposed to reduce the computation in the jackknife empirical likelihood method when the nuisance parameters cannot be estimated explicitly. A simulation study confirms the advantage of the new method. The Canadian Journal of Statistics 40: 110–123; 2012 © 2012 Statistical Society of Canada