An optimum multivariate stratified double sampling design in presence of non-response

In stratified sampling when strata weights are unknown double sampling technique may be used to estimate them. At first a large simple random sample from the population without considering the stratification is drawn and sampled units belonging to each stratum are recorded to estimate the unknown strata weights. A stratified random sample is then obtained comprising of simple random subsamples out of the previously selected units of the strata. If the problem of non-response is there, then these subsamples may be divided into classes of respondents and non-respondents. A second subsample is then drawn out of non-respondents and an attempt is made to obtain the information. This procedure is called Double Sampling for Stratification (DSS). Okafor (Aligarh J Statist 14:13–23, 1994) derived DSS estimators based on the subsampling of non-respondents. Najmussehar and Bari (Aligarh J Statist 22:27–41, 2002) discussed an optimum double sampling design by formulating the problem as a mathematical programming problem and used the dynamic programming technique to solve it. In the present paper a multivariate stratified population is considered with unknown strata weights and an optimum sampling design is proposed in the presence of non-response to estimate the unknown population means using DSS strategy. The problem turns out to be a multiobjective integer nonlinear programming problem. A solution procedure is developed using Goal Programming technique. A numerical example is presented to illustrate the computational details.