OPTIMUM STRATIFICATION WITH AUXILIARY INFORMATION USING MATHEMATICAL PROGRAMMING
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Date
2018
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Sher-e-Kashmir University of Agricultural Sciences and Technology of Jammu
Abstract
The method of choosing the best boundaries that makes strata internally homogeneous as far as possible is known as optimum stratification. To achieve this, the strata should be constructed in such a way that the strata variances for the characteristic under study be as small as possible. If the frequency distribution of the study variable is known, the stratification points could be obtained by cutting the range of the distribution at suitable points. However, if the frequency distribution of the study variable is unknown, it may be approximated from the past experience or some prior knowledge obtained at a recent study. In the present investigation entitled optimum stratification with auxiliary information using mathematical programming”, theories have been developed for optimum stratification, when one character of the population is under study, using two auxiliary variables as stratification variables. Minimal equations giving optimum strata boundaries (OSB) have been obtained for stratified random sampling under different methods of allocations, by minimizing the variance of the sample estimates. For all these cases, Cum 3 Di x, z rules (i=1,2,3,4) ,where the functionDi x, ztakes different forms for different allocations, have been developed. Under the classical optimization technique the empirical studies on uniform, right triangular and exponential distribution have been made for the gain in efficiency that could be achieved through the proposed rules. It has been observed that proposed technique under classical optimization approach performed better than the techniques using single auxiliary variable proposed by Ekman (1959a), Singh (1971) and other workers. Methods have also been developed for obtaining OSB using mathematical programming technique for different allocations. The problem of determining OSB are formulated as nonlinear programming problem (NLPP),which turn out to be multistage decision problem and are solved using dynamic programming approach. The basic advantage of mathematical programming over the classical optimization is that it can determine OSB efficiently, when the density function of the population is approximately known from the previous studies. Iterative methods required initial solutions and there is no guarantee that they will converge and give the global minimum variance in the absence of a suitably chosen initial solution while as the proposed methods do not require any initial approximation solution. More importantly, the proposed techniques have a wide scope of application compared to other methods. In practice, thye complete data set of the study variable is unknown, which diminishes the uses of many satisfaction techniques. In such a situation the proposed techniques can be used as it requires only the values of parameters of the population, which can easily be available from the past studies. The proposed methods using the simulated data are presented to illustrate the applications and the computational details using R and LINGO softwares. In case of general variance, the results are presented together with the results of Cum √f method of Dalenius and Hodges (1959), the geometric method by Gunning and Horgan (2004) and the generalized method of Lavallee-Hidiroglou (1988) Khan et al. (2008) for computational analysis. It has been observed that proposed method leads to substantial gains in the precision of the estimates. Besides, the proposed method under equal allocation also showed more efficient results than the method proposed by singh (1977). Under proportional allocation, by empirical study it is found that the method proposed by Thomson (1973) and Khan et al. (2005) are inferior to the proposed method. The proposed technique, under Neyman allocation, has been found to be more efficient than the methods given by Singh and Sukhatme (1969), Khan et al. (2008), Khan et al. (2014), and Nand and Khan (2008). Through the empirical investigations it is to be concluded that the method proposed by Fonolahi and Khan (2014) is having greater variance than the proposed method. However, it is to be noted that the variance obtained by the classical optimization technique (Cum √Di(x,z)) of the present investigation for all the allocations. Thus, it is to be concluded that the mathematical programming technique leads to substantial increase in the precision as compared to classical optimization technique.
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