A Mathematical Programming Approach in Optimum Stratification Under Neyman Allocation for Two Stratifying Variables

Abstract

The current study discusses the solution for obtaining stratification points under Neyman allocation having one study variable and two auxiliary variables. Using dynamic programming approach non-linear programming problem has been solved. The proposed technique has gained in precision rather than using only one auxiliary variable. Numerical illustration has been given in which each of the auxiliary variable is supposed to follow different distribution. Through the empirical study, the proposed method has been compared with the Ravindra and Sukhatme (1969) and Khan et al.(2005) methods with the conclusion of having its more relative efficiency.

Heuristic algorithm for univariate stratification problem

“…that the use of (i), (ii), and (iii) guaranteed the fulfillment of ( 19) and (20). The satisfaction of constraint ( 19) is given by the definition of step (iii).…”

“…In [17,18], an approach based on dynamic programming for the two-variable stratification problem uses proportional and Neyman allocations. Finally, in [20], the bivariate stratification problem is also solved via dynamic programming, evaluating different probability distribution functions.…”

“…The satisfaction of constraint ( 19) is given by the definition of step (iii). In the case of the restrictions in (20), by the definition of (i) and by the recurrence relation in (ii), it is observed that each 𝑤 ℎ (ℎ = 1, . .…”

Optimum Stratification Using Dynamic Programming with a Mixture of Ratio and Product Estimators under Super Population Model

“…Danish and Rizvi [8] proposed a method for obtaining stratifcation points using two highly related variables. Two stratifcation variables have been used by Danish et al [13], Danish and Rizvi [9], and Danish and Rizvi [14]. Abo-El Hassan et al [15] proposed goal programming for obtaining the stratifcation points.…”

“…Abo-El Hassan et al [15] proposed goal programming for obtaining the stratifcation points. Tere has been a dramatic increase in studies regarding obtaining strata boundaries, with some of them being the most recent work of Brito et al [16], Reddy and Khan [17], and Danish et al [18]. Alshqaq et al [19] discussed the linear approximation of the multivariate stratifed sampling problem with examples.…”