Partha Lahiri

Mixed Model Prediction and Small Area Estimation

Over the last three decades, mixed models have been frequently used in a wide range of small area applications. Such models offer great flexibilities in combining information from various sources, and thus are well suited for solving most small area estimation problems. The present article reviews major research developments in the classical inferential approach for linear and generalized linear mixed models that are relevant to different issues concerning small area estimation and related problems.

Parametric bootstrap approximation to the distribution of EBLUP and related prediction intervals in linear mixed models

Empirical best linear unbiased prediction (EBLUP) method uses a linear mixed model in combining information from different sources of information. This method is particularly useful in small area problems. The variability of an EBLUP is traditionally measured by the mean squared prediction error (MSPE), and interval estimates are generally constructed using estimates of the MSPE. Such methods have shortcomings like under-coverage or over-coverage, excessive length and lack of interpretability. We propose a parametric bootstrap approach to estimate the entire distribution of a suitably centered and scaled EBLUP. The bootstrap histogram is highly accurate, and differs from the true EBLUP distribution by only O(d 3 n -3/2 ), where d is the number of parameters and n the number of observations. This result is used to obtain highly accurate prediction intervals. Simulation results demonstrate the superiority of this method over existing techniques of constructing prediction intervals in linear mixed models.

Estimation of Finite Population Domain Means: A Model-Assisted Empirical Best Prediction Approach

In this article we introduce a general methodology for producing a model-assisted empirical best predictor (EBP) of a finite population domain mean using data from a complex survey. Our method improves on the commonly used design-consistent survey estimator by using a suitable mixed model. Such a model combines information from related sources, such as census and administrative data. Unlike a purely model-based EBP, the proposed model-assisted EBP converges in probability to the customary design-consistent estimator as the domain and sample sizes increase. The convergence in probability is shown to hold with respect to the sampling design, irrespective of the assumed mixed model, a property commonly known as design consistency. This property ensures robustness of the proposed predictor against possible model failures. In addition, the convergence in probability is shown to be valid with respect to the assumed mixed model (model consistency). A new mean squared prediction error (MSPE) estimator is proposed. Unlike earlier MSPE estimators, our MSPE estimator is second-order unbiased. Our simulation results demonstrate the robustness properties of our proposed model-assisted predictor and the usefulness of the second-order unbiased MSPE estimator.

A new adjusted maximum likelihood method for the Fay-Herriot small area model

In the context of the Fay-Herriot model, a mixed regression model routinely used to combine information from various sources in small area estimation, certain adjustments to a standard likelihood (e.g., profile, residual, etc.) have been recently proposed in order to produce strictly positive and consistent model variance estimators. These adjustments protect the resulting empirical best linear unbiased prediction (EBLUP) estimator of a small area mean from the possible over-shrinking to the regression estimator. However, in certain cases, the existing adjusted likelihood methods can lead to high biases in the estimation of both model variance and the associated shrinkage factors and can even produce a negative second-order unbiased mean square error (MSE) estimate of an EBLUP. In this paper, we propose a new adjustment factor that rectifies the above-mentioned problems associated with the existing adjusted likelihood methods. In particular, we show that our proposed adjusted residual maximum likelihood and profile maximum likelihood estimators of the model variance and the shrinkage factors enjoy the same higher-order asymptotic bias properties of the corresponding residual maximum likelihood and profile maximum likelihood estimators, respectively. We compare performances of the proposed method with the existing methods using Monte Carlo simulations.

A second-order efficient empirical Bayes confidence interval

We introduce a new adjusted residual maximum likelihood method (REML) in the context of producing an empirical Bayes (EB) confidence interval for a normal mean, a problem of great interest in different small area applications. Like other rival empirical Bayes confidence intervals such as the well-known parametric bootstrap empirical Bayes method, the proposed interval is second-order correct, that is, the proposed interval has a coverage error of order

On the design-consistency property of hierarchical Bayes estimators in finite population sampling

O(m^{-{3}/{2}})

Methods and results for small area estimation using smoking data from the 2008 National Health Interview Survey

. Moreover, the proposed interval is carefully constructed so that it always produces an interval shorter than the corresponding direct confidence interval, a property not analytically proved for other competing methods that have the same coverage error of order

Estimating variance of random effects to solve multiple problems simultaneously

O(m^{-{3}/{2}})

A New Approximation to the True Randomization-Based Design Effect

. The proposed method is not simulation-based and requires only a fraction of computing time needed for the corresponding parametric bootstrap empirical Bayes confidence interval. A Monte Carlo simulation study demonstrates the superiority of the proposed method over other competing methods.

A unified jackknife theory for empirical best prediction with M-estimation

We obtain a limit of a hierarchical Bayes estimator of a finite population mean when the sample size is large. The limit is in the sense of ordinary calculus, where the sample observations are treated as fixed quantities. Our result suggests a simple way to correct the hierarchical Bayes estimator to achieve design-consistency, a well-known property in the traditional randomization approach to finite population sampling.We also suggest three different measures of uncertainty of our proposed estimator.