Empirical Likelihood On Density Ratio Models And Small Area Estimation

Empirical likelihood(EL)is a nonparametric statistical method.Since it was formally proposed by Owen(1988,1990),it has been extensively studied by many researchers,see Owen(2001)and the extensive references therein.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;EL is Bartlett correctable,and seamlessly incorporate auxiliary information,and so on.Now it has been more and more widely used because of its flexibility and efficiency.With respect to the density ratio model(DRM),EL under DRM has been demon-strated to be a flexible and useful platform for semiparametric inferences.Based on DRM-based EL inferences,it is well known that "dual likelihood" function has the same maximum value and point as the profile EL,thus the parameter estimation is often based on the so-called dual EL since it has a closed form and is easy to calculate.Although it shares many properties with the DRM-based EL,the dual EL is not a real likelihood,let alone the real likelihood based on the data.Unlike the dual EL,the standard EL can easily incorporate auxiliary information defined through additional estimating equations.Naturally we may wonder whether inferences based on dual EL are always the same as those based on the standard EL under DRM.If yes,we shall recommend the former in?stead of the latter.Otherwise,when do they perform similarly and when does the former lose remarkable efficiency?One contribution of this thesis is to point that it is not always true.Take the estimation problem of a general parameter in Chapter 2 for example,We make a careful comparison of the dual EL and the standard EL estimations based on DRM.Our theoretical comparison indicates that their point estimators are still identi-cal,however our numerical comparison shows that the corresponding interval estimators may have different performances,especially when the underlying populations are severely skewed.A real example is analyzed for illustration purpose.Furthermore,auxiliary population information is often available in finite population inference problems,and the empirical likelihood(EL)approach has been demonstrated to be flexible and useful for such problems.Chapter 3 considers EL when interest centers on inference for the mean of the baseline distribution under two-sample density ratio models.An interesting problem is whether inferences based on EL using auxiliary information are really much better than those based on dual EL under DRM.If yes,we shall combine such auxiliary information into the likelihood as much as possible.Otherwise,when do they perform similarly and when does the former increase remarkable efficiency?The answer to this problem makes up a part of this thesis,see Chapter 3.In addition,semiparametric and nonparametric models and methods of small area estimation(SAE)have received increasing attention.In Chapter 4,we introduce em-pirical likelihood with application to small area estimation.SAE originated in sampling survey.The term small area usually denotes any subpopulation for which direct estimates of adequate precision cannot be produced.The sampling design of a typical national sur-vey may result in few or even no sampling units in many sub-areas or small areas,the problem of SAE thus occurs.The heart of SAE problem is to borrow strength and pro-duce reliable small area estimators.Among small area models,one of the most popular methods is the use of nested error regression model(NER)to derive empirical best linear unbiased predictors(EBLUP)for the estimation of small area mean.A problem with the traditional EBLUP under the NER model is that its optimality depends largely on the normality assumptions of models.To overcome this deficiency,another contribution of this thesis is to propose a transformed NER model with an invertible transformation,and employ the maximum likelihood method to obtain the parameter estimators over-all of the transformed NER model.Motivated by Duan(1983)'s smearing estimator and Chen et al.(2008)'s adjusted empirical likelihood method.,we propose two small area mean estimators depending on whether all the population covariates or only the population co-variate means are available in addition to sample covariates.We conduct two design-based simulation studies to investigate their finite-sample performance.The simulation results indicate that compared with existing methods such as EBLUP,the proposed estimators are nearly the same reliable when the NER model is valid and become more reliable in general when the NER model is violated.In particular,our method does benefit from incorporating auxiliary covariate information.In applications,to protect against possible model breakdowns and for consistency in publication,the aggregation of those model-based small area estimates is often constrained to agree with an appropriate direct estimate for an aggregate of the areas.The process of adjusting estimates to correct this problem is known as calibration,or benchmarking.The last contribution of this thesis is to further propose a parametric bootstrap procedure to estimate the mean squared error of the benchmarked small area estimator under the transformed NER model.With a design-based simulation study based on real data from the Survey of Labour and Income Dynamics provided by Statistics Canada(2014),the proposed estimator is compared with existing methods such as the empirical best linear unbiased prediction and other competitive benchmarked estimators,and found to have better performances for small sample sizes.