Experience Likelihood Of High-end Nature Of Some Research

Empirical likelihood(EL) is a nonparametric statistical method used to construct confidence regions for multivariate means,and more generally,for parameters defined by estimating equations.Since it was proposed by Owen(1988,1990),it has been extensively studied by many researchers,see Owen(2001) and the extensive references therein.Now it has been more and more widely used because of its flexibility and efficiency.The EL method has many nice properties:for example,it does not assume the data come from a parametric distribution;there is no need to estimate variance;the shape of confidence regions is automatically determined by data;EL based confidence regions are range preserving and transformation invariant,and so on.Even so,it still has at least two problems in practice.One is that,the coverage accuracy of EL based confidence regions is often not satisfactory.In theory the coverage error is of order O(n^-1),where n is the sample size,however simulation indicates that an EL based confidence region with nominal level 90%may have coverage probability as low as 83%.Second,in computing a profile EL function,the required numerical problem may have no solution.In this case the profile EL function has no definition and the EL method does not work.This thesis makes contributions to the empirical likelihood in solving both the two problems.With respect to the first problem,it is found that Bartlett correction can improve the coverage accuracy of EL based confidence regions if the EL is Bartlett correctable. Through Bartlett correction,the coverage error of EL based confidence regions is sharply reduced from O(n^-1) to O(n^-2).EL for many models have been shown to be Bartlett correctable.DiCiccio,Hall and Romano(1991) disclosed that the EL owns such a property when the parameter can be expressed as a smooth function of the population mean of the data.Unfortunately,the formula of the Bartlett correction factor given in this paper is incorrect for vector-valued data although it is right for scalar data. One contribution of this thesis is to point out this error and give the right formula. EL based confidence regions for linear regression coefficients and quantiles also admit Bartlett correction(Chen 1993,1994;Chen and Hall 1993).Jing(1995) extended the EL method to the two sample mean problem for scalar observations and pointed out that it is still Bartlett correctable.However,Jing(1995) gave an incorrect Bartlett correction factor.Liu,Zou and Zhang(2008) considered this problem for multivariate observations by the EL method and presented the right Bartlett correction factor,which makes up a part of this thesis.See Chapter 5.In general estimating equation set-up,Lazar and Mykland(1999) disclosed that an EL defined by two estimating functions with a nuisance parameters need not be Bartlett correctable.While if the nuisance parameter is profiled out given the value of the parameter of interest,the EL in the framework of general estimating equations is still Bartlett correctable(Cui and Chen 2006,2007). Generally speaking,the EL is Bartlett correctable for commonly-used models,which means that the precision of the EL method in this situation can be improved by Bartlett correction.To solve the second problem,the convention is to define the profile EL function to be 0 when it has no definition.However,there are at least two limitations with this strategy.One is that it is hard to determine at what parameter value the profile EL function has no definition.The other is that the plausibility of the parameter values where the likelihood is set to 0 is in doubt.Chen,Variyath,and Abranham(2008) introduced an adjusted empirical likelihood(AEL),which completely solves this problem,is easy to apply and has many other benefits.Another contribution of this thesis is to show that the AEL,with the adjustment level being half the Bartlett correction factor of the usual empirical likelihood(AEL with Bartlett correction for short),not only guarantees the existence of the profile the AEL function,but also achieves the same precision of the EL with Bartlett correction.Although in theory,Bartlett correction can reduce the coverage errors of the EL based or AEL based confidence regions from O(n^-1) to O(n^-2),simulation indicates that it does not perform so satisfactorily as expected,especially for small or moderate sample sizes,which we found may arise from the severe underestimation of Bartlett correction factor by its usual moment estimate.A new estimate of Bartlett correction factor is proposed and shown by simulations to less-biased than the usual moment esti- mate.As expected,all the relevant results are improved with this new estimate.Typically,confidence intervals such as the EL,Bartlett corrected EL and AEL based confidence intervals are two-sided ranges when the parameter of interest is univariate. In some circumstances,it can make more sense to express the confidence interval in only one direction—to either the lower or upper confidence limit.As has been shown in the literature,the two-sided confidence intervals based on the usual EL or AEL have coverage error of order O(n^-1) and each endpoint of these confidence intervals has coverage error of order O(n^-1/2).Although the AEL with Bartlett correction or the Bartlett-corrected EL have much smaller coverage error(O(n^-2)) of two-sided confidence intervals,their resulting one-sided confidence intervals still have coverage error of order O(n^-1/2).The last contribution of this thesis is to propose an AEL with two pseudo observations,the two-sided confidence intervals based on which have coverage error of order O(n^-2) and each endpoint of these confidence intervals has coverage error of order O(n^-3/2).