Area specific confidence intervals for a small area mean under the Fay-Herriot model

Abstract

Small area estimates have received much attention from both private and public sectors due to the growing demand for effective planning of health services, apportioning of government funds and policy and decision making. Surveys are generally designed to give representative estimates at national or district level, but estimates of variables of interest are often also needed at lower levels. These cannot be reliably obtained from the survey data as the sample sizes at these levels are too small. Census data are often available, but only give limited information with respect to the variables of interest. This problem is addressed by using small area estimation techniques, which combine the estimates from the survey and census data sets. The main purpose of this paper is obtaining confidence intervals based on the empirical best linear unbiased predictor (EBLUP) estimates. One of the criticism of the mean squared error (MSE) estimators is that it is not area-specific since it does not involve the direct estimator in its expression. However, most of the confidence intervals in the literature are constructed based on those MSEs. In this paper, we propose area specific confidence intervals for small area parameters under the Fay-Herriot model using area specific MSEs. We extend these confidence intervals to the difference between two small area means. The effectiveness of the proposed methods are also investigated via simulation studies and compared with the Cox (1975) and Prasad and Prasad and Rao (1990) methods. Our simulation results show that the proposed methods have higher coverage probabilities. Those methods are applied to the percentage of food expenditure measures in Ethiopia using the 2010/11 Household Consumption Expenditure (HCE) survey and the 2007 census data sets.

Comparison of mean squared error estimators under the Fay-Herriot model: application to poverty and percentage of food expenditure data

“…These results are shown separately for each of the sampling variances ψ i , distinguishing among the cases = M L , = REM L and A = F H . The FH, ML and REML correspond to the FH, ML and REML estimators of the MSE of the EBLUP, respectively [24]. Table 1 shows the RB and RR values of the MSE estimators that were obtained through the simulation study for m = 15 without covariates.…”

“…This g 3i (A) is not area-specific since it does not incorporate the direct survey estimate y i within it. Thus, [19] and [22] proposed alternative area-specific g 3i (A, y i ) ′ s under the FH model. The [19] g 3i (A, y i ) is given as…”

“…Note that MSE_Rao, MSE_JY and MSE_JY1 are the MSE estimators based on (2.12), (2.13) and (2.14) respectively under the three different methods of estimation. However, MSE_REML, MSE_ML and MSE_FH are the MSE estimators based on (2.11) under the REML, ML and FH methods of estimation [24]. The MSEs we want to compare are given as follows.…”

“…The area specific corrected confidence intervals for a small area mean θ i ; i = 1, · · · , m based onθ EB i are proposed by [22] as follows:…”

“…Another study by [14] considered a range of small areas m = 15 and 45 to study the asymptotic behaviour of their proposed confidence intervals. Further, [22] considered a sample of small areas such as m = 15, 20, 30, 40, 50, 60, 80 and 100 for comparing their proposed area specific confidence intervals under the FH model. Similarly [23] considered a sample of small areas such as m = 10, 15, 30, 45 and 60 in comparing their proposed confidence intervals.…”