On calibrated weights in stratified sampling

Abstract

In this paper, we propose a calibration estimator of population mean in stratified sampling using the known mean and variance information from multi-auxiliary variables. The problem of determining the optimum calibrated weights is formulated as an optimisation problem and is solved using the Lagrange multiplier technique. A numerical example with real data is presented to illustrate the computational details of the proposed estimator. A comparison study is also carried out using real and simulated data to evaluate the performance and the usefulness of the proposed estimator. The study reveals that the proposed estimator with multi-auxiliary information is the most efficient estimator of the population mean when compared to other estimators as it provides least estimated variance and highest gain in relative efficiency (RE).

A Goal Programming Approach for Multivariate Calibration Weights Estimation in Stratified Random Sampling

“…are the sample of j th auxiliary variable and population means in h th stratum respectively. Further, Rao [8] proposed MCE for the population mean in SRS using different constraints related to two auxiliary variables. They also suggested minimizing (3.1) as their objective function subject to the same three constraints used in [7] but with respect to multi-auxiliary variables.…”

“…To assess the performance of different calibration estimators compared to the unbiased estimator y st , the measure of Relative Efficiency (RE) is computed as follows [8]:…”

“…It is the estimated variance of the unbiased estimator y st , v(Ŷ) is the estimated variance of the calibrated estimators, that can be computed through substituting by the calibrated weights instead of the design weights in (5.2) [8].…”

“…The performance of the different methods for calibration estimators under comparison are evaluated by the Relative Root Mean Squared Error "RRMSE" that can be defined by [11]. denotes the calibration estimators; specifically, α = Singh [3], Tracy [4], Rao [5], Rao [8] & the proposed GP estimator. Table (6) summarizes the simulation results for the three cases.…”

“…Table 6 shows that in case 1, where equal weights are given for both of the variance and the bias of the calibrated estimator, the suggested multivariate calibration estimator is better than other existing calibration estimators. The simulated data's results showed that its RRMSE is lower than the RRMSE of [3], [4], [5] and [8] calibration estimators. Moreover, the proposed calibrated estimator in case 2 has a lower RRMSE than [8] calibration estimator but higher than the other estimators.…”