Theses and Dissertations from UMD
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Item GENERALIZED OBSERVED BEST PREDICTION WITH EMPIRICAL BAYES PARAMETRIC BOOTSTRAP MODEL BUILDING(2020) WALDRON, WILLIAM; Lahiri, Partha; Mathematics; Digital Repository at the University of Maryland; University of Maryland (College Park, Md.)The observed best predictor (OBP) has been recently offered as a more robust alternative to the remarkable empirical best linear unbiased predictor (EBLUP). Although the latter has become a pervasive tool among applied statisticians, there are critical reasons why the OBP should almost always be used in conjunction with the EBLUP. In particular, mathematical models are often oversimplified or misspecified, lacking key predictors within the available set of data. For more complex models such as time-series applications, model robustness becomes even more imperative. We will provide some results related to the OBP theory and introduce a generalized, or weighted version of the OBP for different loss functions. This will first be defined on the Fay-Herriot model and then extended to the General Linear Mixed model. Finally, we will apply the best predictive estimator (BPE) to both parameter coefficients and variance parameters within the Fay-Herriot and cross-sectional time series models. Model building strategies abound, and have continued to evolve. These are instrumental for applied statisticians and analysts passing judgement on whether statistical models are suitable for drawing conclusions or producing official estimates. A number of methodologies and approaches have been developed to consider this critical question of model selection and diagnostics. We endeavor to view this problem from the perspective of empirical Bayes (EB) - in a similar fashion as the EBLUP. As such, we define and develop an EB parametric bootstrap approach not only to estimate mean squared error, but also for finding the best model from a set of candidates (e.g., variable selection). This could be done for general criteria by considering leave-one-out predictive distributions. Once a viable model is selected, we can continue the model-building process by performing appropriate validation. Thus, the method is not only versatile, but has some computational advantages over other model building strategies.Item Small area estimation: an empirical best linear unbiased prediction approach(2007-09-17) Li, Huilin; Lahiri, Partha; Mathematical Statistics; Digital Repository at the University of Maryland; University of Maryland (College Park, Md.)In a large scale survey, we are usually concerned with estimation of some characteristics of interest for a large area (e.g., a country). But we are frequently interested in estimating similar characteristics for a subpopulation using the same survey data. The direct survey estimator which utilizes data only from the small area of interest has been found to be highly unreliable due to small sample size. Model-based methods have been used in small area estimation in order to combine information available from the survey data and various administrative and census data. We study the empirical best linear unbiased prediction (EBLUP) and its inferences under the general Fay-Herriot small area model. Considering that the currently used variance estimation methods could produce zero estimates, we propose the adjusted density method (ADM) following Morris' comments. This new method always produces positive estimates. Morris only suggested such adjustment to the restricted maximum likelihood. Asymptotic theory of ADM is unknown. We prove the consistency for the ADM estimator. We also propose an alternate consistent ADM estimator by adjusting the maximum likelihood. By comparing these two ADM estimators both in theory and simulation, we find that the ADM estimator using maximum likelihood is better than the one using the restricted likelihood in terms of bias. We provide a concrete proof for the positiveness and consistency of both ADM estimators. We also propose EBLUP estimator of $\theta_i$ where we use two ADM estimators of $A$. The associated second-order unbiased Taylor linearization MSE estimators are also proposed. In addition, a new parametric bootstrap prediction interval method using ADM estimator is proposed. The positiveness of ADM estimators is emphasized in the construction of the prediction interval. We also show that the coverage probability of this new method is accurate up to $O(m^{-3/2})$. Extensive Monte Carlo simulations are conducted. A data analysis for the SAIPE data set is also presented. The positiveness of ADM estimators plays a vital role here since for this data set the method-of-moments, REML, ML and FH methods could be all zero. We observe that ADM methods produce EBLUP's which generally put more weights to the direct survey estimates than the corresponding EBLUP's that use the other methods of variance component estimation.