Geometric programming approach in multivariate stratified sample surveys in case of non-response


Abstract


This paper provides an attempt to utilize the geometric programming approach in multivariate stratified sample surveys in case of non-response. The problem has been solved in two phases. In first phase the multivariate stratified sample surveys in case of non-response has been formulated as geometric programming problem (GPP) and the solution is obtained. The obtained solution is the dual solution of the formulated GPP. In second phase with the help of dual solutions of formulated GPP and primal-dual relationship theorem the optimum allocation of sample sizes of respondents and non respondents are obtained. A numerical example is given to illustrate the procedure.


DOI Code: 10.1285/i20705948v8n1p28

Keywords: geometric programming, convex programming, non-response, optimum allocation, multivariate stratified sampling.

References


.Ahmed, J. and. Bonham Charles D. (1987). Application of Geometric Programming to Optimum Allocation Problems in Multivariate Double Sampling. Appl.Maths. and Comm.21(2), 157-169.

. Ahsan, M.J. (1975–76).A procedure for the problem of optimum allocation in multivariate stratified random sampling, Aligarh Bull. Math. 5–6, 37–42.

..Ahsan, M.J. and Khan, S.U. (1977).Optimum allocation in multivariate stratified random sampling using prior information, J. Indian Statist. Assoc. 15,57–67.

.Ahsan, M.J. (1978).Allocation problem in multivariate stratified random sampling, J. Indian Statist. Assoc. 16, 1–5.

.Ali, I., Raghav, Y. S. and Bari, A.(2013).Compromise allocation in multivariate stratified surveys with stochastic quadratic cost function, Journal of Statistical computation and Simulation, 83:5, 960-974.

.Aoyama, H. (1963).Stratified random sampling with optimum allocation for multivariate populations, Ann. Inst. Statist. Math. 14 , 251–258.

.Chatterjee, S. (1967).A note on optimum allocation, Scand. Actuarial J. 50 pp. 40–44.

.Chatterjee, S. (1968).Multivariate stratified surveys, J. Amer. Statist. Assoc. 63 pp. 530–534.

.Chromy, J.R.(1987).Design optimization with multiple objectives, Proceedings of the Survey Research Methods section, American Statistical Association, pp. 194–199.

.Cochran, W.G. (1977).Sampling Techniques. 3rd ed. New York: Wiley and Sons.

.Dalenius, T. (1957).Sampling in Sweden, Contributions to the Methods and Theories of Sample Survey Practice, Almqvist and Wicksell, Stockholm.

.Davis, M., Rudolf, E.S. (1987). Geometric programming for optimal allocation of integrated samples in quality control. Comm.Stat.Theo.Meth., 16 (11), 3235-3254.

.Duffin, R.J., Peterson, E.L., Zener, C. (1967).Geometric programming: Theory & applications. New York: John Wiley & Sons.

.Dupačová, J. (2010). Stochastic Geometric Programming with an application.Kybernetika, Vol. 46, No. 3, 374—386.

.Fatima, U. and Ahsan, M. J. (2011). Non-response in stratified sampling: A mathematical programming approach.The South Pacific Journal of Natural and Applied Sciences, No. 29, 40-42.

.Folks, J.L. and Antle, C.E. (1965).Optimum allocation of sampling units to the strata when there are r responses of interest, J. Amer. Statist. Assoc. 60, 225–233.

.Geary, R.C. (1949). Sampling methods applied to Irish agricultural statistics. Technical series, Central Statistical office, Dublin.

.Ghosh, S.P. (1958).A note on stratified random sampling with multiple characters, Calcutta Statist. Assoc. Bull. 8, 81–89.

.Islam, S., and Roy, T.K., (2005). Modified geometric programming problem and its applications, Journal of Appl. Maths. and Comp., Korea, 17(1-2), 121-144.

.J. Neyman, (1934).On the two different aspects of the representative method: The method of stratified sampling and the method of purposive selection, J. Roy. Statist. Soc. 97, 558–625.

.Jahan, N. Khan, M.G.M. and Ahsan, M.J. (1994).A generalized compromise allocation, J. Indian Statist. Assoc. 32, 95–101.

.Jahan, N. and Ahsan, M.J. (1995). Optimum allocation using separable programming, Dhaka Univ. J. Sci. 43(1), 157–164.

.Jahan,N., Khan, M.G.M. and Ahsan, M.J. (2001).Optimum compromise allocation using dynamic programming, Dhaka Univ. J. Sci., 49 (2), 197–202.

.Khan, M.G.M., Ahsan, M.J. and Jahan, N. (1997).Compromise allocation in multivariate stratified sampling: An integer solution, Naval Res. Logist., 44,69–79.

.Khan, M.G.M., Khan, E.A. and Ahsan, M.J. (2003).An optimal multivariate stratified sampling design using dynamic programming, Austral. New Zealand J. Statist. 45(1), 107–113.

.Khan, M.G.M., Khan, E.A., Ahsan, M.J. (2008). Optimum allocation in multivariate stratified sampling in presence of non-response. J. Indian Soc. Agr. Stat. 62(1), 42–48.

.Khan, M.G.M., Khan, E.A., and Ahsan, M.J. (2008).Optimum allocation in multivariate stratified sampling in presence of non-response, J. Ind. Soc. Agric. Statist. 62(1), 42–48.

.Khan, M.G.M., Maiti, T. and Ahsan, M.J. (2010).An optimal multivariate stratified sampling design using auxiliary information: An integer solution using goal programming approach, J. Official Statist. 26(4), 695–708.

.Khan,M.F. Ali, I., Raghav ,Y. S. and Bari, (2012), A. Allocation in Multivariate Stratified Surveys with Non-Linear Random Cost Function,American Journal of Operations Research, 2, 100-105.

.Khare, B.B. (1987). Allocation in stratified sampling in presence of nonresponse. Metron45(1–2), 213–221.

.Khowaja, S., Ghufran, S. and Ahsan, M.J. (2011).Estimation of population means in multivariate stratified random sampling, Commun. Statist. Simul. Comput. 40(5),710–718(9).

.Kokan, A.R. and Khan, S.U. (1967).Optimum allocation in multivariate surveys: An analytical solution, J. Roy. Statist. Soc. Ser. B 29, 115–125.

.LINGO User’s Guide: Published by Lindo Systems Inc., 1415 North Dayton Street, hicago, Illinois-60622, USA (2001).

.M.H. Hansen, Hurwitz, W.N. (1946). The problem of non-response in sample surveys. J. Am. Stat. Assoc. 41, 517–529.

.Maqbool, S., Mir, A. H. and Mir, S. A. (2011). Geometric Programming Approach to Optimum Allocation in Multivariate Two-Stage Sampling Design. Electronic Journal of Applied Statistical Analysis,4(1), 71 – 82.

.Ojha, A.K and Biswal, K.K. (2010). Posynomial Geometric Programming Problems with Multiple Parameters. Jour. of comp., 7(2).

.Ojha, A.K. and Das, A.K. (2010). Multi-Objective Geometric Programming Problem being cost coefficients as continuous function with weighted mean. Jour. of comp.2(2), 2151-9617.

.Raghav, Y.S., Ali, I., and Bari, A., (2012): Multi-objective Nonlinear Programming Problem Approach in Multivariate Stratified Sample Surveys in case of Non-Response, Journal of Statistical Computation and Simulation (Taylor & Francis Doi:10.1080/00949655.2012.692370).

.Rao, S.S. (1979). Optimization Theory and Applications. Wiley Eastern Limited.

.Särndal, C.E., Lundström, S. (2005), Estimation in Surveys with Nonresponse. Wiley, New York.

.Shafiullah, Ali, I. and Bari, A. (2013). Geometric Programming Approach in Three – Stage Sampling Design, International Journal of Scientific & Engineering Research (France),4(6), 2229-5518.

.Shaojian, Qu, Kecun, Z., Fusheng, W. (2008). A global optimization using linear relaxation for generalized geometric programming. Europ. Jour.ofOper. Res., 190, 345-356.

.Shiang-Tai Liu. (2008).Posynomial geometric programming with interval exponents and coefficients. Europ. Jour. of Oper. Res., 186, 17-27.

.Singh, S. (2003).Advanced Sampling Theory with Application, Vol. II, Kluwer Academic Publishers, Dordrecht, The Netherlands.

.Sukhatme, P.V., Sukhatme, B.V., Sukhatme, S., Asok, C. (1984). Sampling Theory of Surveys with Applications. Iowa State University Press, Iowa, U.S.A. and Indian Society of Agricultural Statistics, New Delhi, India.

.Varshney, R. Najmussehar andAhsan, M. J. (2012).Estimation of more than one parameters in stratified sampling with fixed budget. Math Meth Oper. Res. 75:185–197.

.Yates, F. (1960).Sampling Methods for Censuses and Surveys, 2nd ed., Charles Griffin, London.


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