Prediction of domain total using preliminary test

Abstract

In the paper the problem of prediction of the total of a variable under study in a population domain is considered. Two predictors are used in practice. One of them is the sub-sample mean (drawn from the domain) multiplied by the size of the domain. The other one is the average of the variable from the full sample (drawn from a whole population) multiplied by the size of the domain. Both predictors are unbiased if the expected values of variables in the domain and in the population are the same. The mean square error of the second predictor (based on the full sample) is not larger than the mean square error of the first one (based on the subsample). Hence, we should use the second predictor provided the domain mean is equal to the population mean. So, the hypothesis that both means are the same can be formulated. If the hypothesis is true we use the second predictor. Otherwise we use the first one. In the paper the unbiasedness of the defined preliminary test predictor is considered as well as the derivation of variance of prediction error. Moreover, some other test-predictors of a domain total are proposed, too. Let us note that a preliminary test predictor can be named, more simply, a test-predictor. The test predictor is constructed similarly to the well known preliminary test estimator. Their properties and applications are considered by Saleh (2006) and earlier e.g. by Bankroft (Ann Math Stat 15(2):190–204, 1944), Bankroft and Han (Int Stat Rev 45:117–127, 1977), Bock et al. (J Am Stat Assoc 68(341):109–116, 1973), Giles and Giles (J Econ Surv 7(2):146–197, 1993), Giles et al. (J Am Stat Assoc 87(420):1153–1157, 1992), Judge and Bock (1978) and Sclove et al. (Ann Math Stat 43(5):1481–1490, 1972).