Adjusted Empirical Likelihood Method for Comparison of Treatment Effects in Linear Model Setting

Empirical likelihood is a nonparametric method of statistical inference which was introduced by Owen. It allows the data analyst to use it without making distribution assumptions. Empirical likelihood method has been widely used not only for nonparametric models but also for semi-parametric models, with the effectiveness of the likelihood approach and good power properties. However, when the sample size is small or the dimension is high, the method is poorly calibrated, producing tests that generally have a higher type I error. In addition, it suffers from a limiting convex hull constraint. Many statisticians have proposed methods to address the performance. We explore the method proposed by Chen which makes an adjustment on empirical likelihood method. This thesis derives an adjusted empirical likelihood-based method for comparing two treatment effects in a linear model setting. We use the adjusted empirical likelihood-based method to make inference for the difference by comparing the parameters in two linear models. Our method is free of the assumptions of normally distributed and homogeneous errors, and equal sample size. In addition, the adjusted empirical likelihood method is Bartlett correctable. We apply the Bartlett correction procedure to further improve the coverage of our proposed method. Simulation experimental are used to illustrate that our method outperforms the published ones and also empirical likelihood-based method. This method can be extended into multiple treatment effects comparison

Adjusted empirical likelihood with high-order precision

Empirical likelihood is a popular nonparametric or semi-parametric statistical method with many nice statistical properties. Yet when the sample size is small, or the dimension of the accompanying estimating function is high, the application of the empirical likelihood method can be hindered by low precision of the chi-square approximation and by nonexistence of solutions to the estimating equations. In this paper, we show that the adjusted empirical likelihood is effective at addressing both problems. With a specific level of adjustment, the adjusted empirical likelihood achieves the high-order precision of the Bartlett correction, in addition to the advantage of a guaranteed solution to the estimating equations. Simulation results indicate that the confidence regions constructed by the adjusted empirical likelihood have coverage probabilities comparable to or substantially more accurate than the original empirical likelihood enhanced by the Bartlett correction.Comment: Published in at http://dx.doi.org/10.1214/09-AOS750 the Annals of Statistics (http://www.imstat.org/aos/) by the Institute of Mathematical Statistics (http://www.imstat.org

Second-order accurate confidence regions based on members of the generalised power divergence family

Recently, a technique based on pseudo-observations has been proposed to tackle the so called convex hull problem for the empirical likelihood statistic. The resulting adjusted empirical likelihood also achieves the highorder precision of the Bartlett correction. Nevertheless, the technique induces an upper bound on the resulting statistic that may lead, in certain circumstances, to worthless confidence regions equal to the whole parameter space. In this paper we show that suitable pseudo-observations can be deployed to make each element of the generalised power divergence family Bartlett-correctable and released from the convex hull problem. Our approach is conceived to achieve this goal by means of two distinct sets of pseudo-observations with dfferent tasks. An important effect of our formulation is to provide a solution that permits to overcome the problem of the upper bound. The proposal, whose effectiveness is confirmed by simulation results, gives back attractiveness to a broad class of statistics that potentially contains good alternatives to the empirical likelihood

Second-order Refinement of Empirical Likelihood for Testing Overidentifying Restrictions

This paper studies second-order properties of the empirical likelihood overidentifying restriction test to check the validity of moment condition models. We show that the empirical likelihood test is Bartlett correctable and suggest second-order refinement methods for the test based on the empirical Bartlett correction and adjusted empirical likelihood. Our second-order analysis supplements the one in Chen and Cui (2007) who considered parameter hypothesis testing for overidentified models. In simulation studies we find that the empirical Bartlett correction and adjusted empirical likelihood assisted by bootstrapping provide reasonable improvements for the properties of the null rejection probabilities.Empirical likelihood, GMM, Overidentification test, Bartlett correction, Higher order analysis

Adjusted Empirical Likelihood for Long-memory Time Series Models

Empirical likelihood method has been applied to short-memory time series models by Monti (1997) through the Whittle's estimation method. Yau (2012) extended this idea to long-memory time series models. Asymptotic distributions of the empirical likelihood ratio statistic for short and long-memory time series have been derived to construct confidence regions for the corresponding model parameters. However, computing profile empirical likelihood function involving constrained maximization does not always have a solution which leads to several drawbacks. In this paper, we propose an adjusted empirical likelihood procedure to modify the one proposed by Yau (2012) for autoregressive fractionally integrated moving average (ARFIMA) model. It guarantees the existence of a solution to the required maximization problem as well as maintains same asymptotic properties obtained by Yau (2012). Simulations have been carried out to illustrate that the adjusted empirical likelihood method for different long-time series models provides better confidence regions and coverage probabilities than the unadjusted ones, especially for small sample sizes

Bartlett-type Adjustments for Empirical Discrepancy Test Statistics

This paper derives two Bartlett-type adjustments that can be used to obtain higher-order improvements to the distribution of the class of empirical discrepancy test statistics recently introduced by Corcoran (1998) as a generalisation of Owen's (1988)empirical likelihood. The corrections are illustrated in the context of the so-called Cressie-Read goodness-of-fit statistic Baggerly, and their effectiveness in finite samples is evaluated using simulations.asymptotic expansions; Bartlett and Bartlett-type corrections; empirical likelihood; nonparametric likelihood inference

Empirical likelihood for high frequency data

This paper introduces empirical likelihood methods for interval estimation and hypothesis testing on volatility measures in some high frequency data environments. We propose a modified empirical likelihood statistic that is asymptotically pivotal under infill asymptotics, where the number of high frequency observations in a fixed time interval increases to infinity. The proposed statistic is extended to be robust to the presence of jumps and microstructure noise. We also provide an empirical likelihood-based test to detect the presence of jumps. Furthermore, we study higher-order properties of a general family of nonparametric likelihood statistics and show that a particular statistic admits a Bartlett correction: a higher-order refinement to achieve better coverage or size properties. Simulation and a real data example illustrate the usefulness of our approach