On mean squared prediction error estimation in small area estimation problems Shijie Chen, RTI International P. Lahiri, University of Maryland at College Park Shijie Chen, STRD, Research Triangle Institute, 1615 M ST N.W. Suite 740, Washington, DC, 20036, USA (schen@rti.org) Key Words: Mixed Linear Models, Small Area Es- timation, Mean Squared Prediction Error, Empirical Best Linear Unbiased Predictor, Jackknife. Abstract: In this paper, we consider a Taylor-series approxima- tion to the weighted jackknife mean squared predic- tion error (MSPE) of an empirical best linear un- biased predictor (EBLUP). Like the Taylor series method, this approximation provides a closed-form expression and saves computation. We compare var- ious MSPE estimators using a Monte Carlo simula- tion study. 1. Introduction For effective planning of health, social and other ser- vices, and for apportioning government funds, there is a growing demand to produce reliable estimates for smaller geographic areas and sub-populations, called small areas, for which adequate samples are not available. The usual design-based small area es- timators are unreliable since they are based on a very few observations that are available from the area. An empirical best linear prediction (EBLUP) approach has been found suitable in many small area estimation problems. The method essentially uses an appropriate mixed linear model which captures various salient features of the sampling design and combines information from censuses or administra- tive records in conjunction with the survey data. For a review of small area estimation, see Rao (2003). The estimation of MSPE of EBLUP is a challeng- ing problem. The naive MSPE estimator, i.e., the MSPE of the BLUP with estimated model parame- ters, usually underestimates the true MSPE. There are two reasons for this underestimation problem. First, it fails to incorporate the extra variabilities incurred due to the estimation of various model pa- rameters and the order of this underestimation is O(m −1 ), where m is the number of the small areas. The research of S. Chen was supported in part by Profes- sional Development Award from the RTI International, RTP NC. Secondly, the naive MSPE estimator even underesti- mates the true MSPE of the BLUP, the order of un- derestimation being O(m −1 ). Several attempts have been made in the literature to account for these two sources of underestimation and to produce MSPE estimators that are correct up to the order O(m −1 ). These are called second-order unbiased MSPE esti- mators. Jiang, Lahiri and Wan (2002) proposed a jackknife method to estimate the MSPE of an empirical best predictor for a general situation. Bell (2001) pointed out that the Jiang-Lahiri-Wan jackknife MSPE esti- mator could take negative values in certain circum- stances. However, Chen and Lahiri (2002, 2003) found that this is not a severe problem in their sim- ulation studies and can be easily rectified by con- sidering an alternative bias correction formula. For the well-known Fay-Herriot model, Chen and Lahiri (2003) provided an approximation to the jackknife MSPE estimator using a Taylor series approxima- tion. Like the Prasad-Rao formula, this provides a closed-form formula. In this paper, we follow up on Chen and Lahiri (2003) and obtain the Taylor series approximation to the jackknife MSPE formula for a general case. In section 2, we define the BLUP and EBLUP of a general mixed effect. We provide a Taylor series approximation to the jackknife method in section 3. In section 4, the method is illustrated using the simple but important Fay-Herriot model (see Fay and Herriot 1979). To demonstrate the efficiency of our proposed method, results from a Monte carlo simulation study are reported in section 5. 2. The BLUP and EBLUP Consider the following general normal mixed linear model in small area estimation considered in Prasad and Rao (1990) and Datta and Lahiri (2000) : y i = X i β + Z i v i + e i ,i =1, ..., m, (1) where X i (n i × p) and Z i (n i × b i ) are known ma- trices, v i and e i are independently distributed with ASA Section on Survey Research Methods 2852