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An Optimal Chance Constraint Multivariate Stratified Sampling Design Using Auxiliary Information

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Journal of Mathematical Modelling and Algorithms in Operations Research

Abstract

When we are dealing with multivariate problem then we need an allocation which is optimal for all the characteristics in some sense because the individual optimum allocations usually differ widely unless the characteristics are highly correlated. So an allocation called “Compromise allocation” is to be worked out suggested by Cochran. When auxiliary information is also available, it is customary to use it to increase the precision of the estimates. Moreover, for practical implementation of an allocation, we need integer values of the sample sizes. In the present paper the problem is to determine the integer optimum compromise allocation when the population means of various characteristics are of interest and auxiliary information is available for the separate and combined ratio and regression estimates. This paper considers the optimum compromise allocation in multivariate stratified sampling with non-linear objective function and probabilistic non-linear cost constraint. The probabilistic non-linear cost constraint is converted into equivalent deterministic one by using Chance Constrained programming. The formulated multi-objective nonlinear programming problem is solved by Fuzzy Goal programming approach and Chebyshev approximation. Numerical illustration is also given to show the practical utility of the approaches.

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References

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

    Google Scholar 

  2. Ahsan, M.J., Khan, S.U.: Optimum allocation in multivariate stratified random sampling with overhead cost. Metrika 29, 71–78 (1982)

    Article  MathSciNet  MATH  Google Scholar 

  3. Ahsan, M.J., Najmussehar, Khan, M.G.M.: Mixed allocation in stratified sampling. Aligarh J. Stat. 25, 87–97 (2005)

    MathSciNet  Google Scholar 

  4. Ali, I, Raghav, Y.S., Bari, A.: Compromise allocation in multivariate stratified surveys with stochastic quadratic cost function. J. Stat. Comp. Simul. 83(5), 960–974 (2011). doi:10.1080/00949655.2011.643890

    Google Scholar 

  5. Ansari, A.H., Najmussehar, Ahsan, M.J.: On multiple responses stratified random sampling design. J. Stat. Sci. 1(1), 45–54 (2009)

    Google Scholar 

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

    Article  MathSciNet  Google Scholar 

  7. Bethel, J.: Sample allocation in multivariate surveys. Surv. Methodol. 15, 47–57 (1989)

    Google Scholar 

  8. Chatterjee, S.: A study of optimum allocation in multivariate stratified surveys. Skandinavisk Actuarietidskrift 55, 73–80 (1972)

    Google Scholar 

  9. Chromy, J.: Design optimization with multiple objectives. In: Proceedings of the American Statistical Association, Section on Survey Methods Research, pp. 194–199 (1987)

  10. Dalenius, T.: The multivariate sampling problem. Skandinavisk Aktuarietidskrift 36, 92–102 (1953)

    MathSciNet  Google Scholar 

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

    MATH  Google Scholar 

  12. Dayal, S.: Allocation of sample using values of auxiliary characteristic. J. Stat. Plan. Inference 11, 321–328 (1985)

    Article  MathSciNet  MATH  Google Scholar 

  13. Diaz-Garcia, J.A., Ulloa Cortez, L.: Optimum Allocation in Multivariate Stratified Sampling: Multi-Objective Programming. Comunicación Técnica No. I-06-07/28-03-206 (PE/CIMAT). Centro de Investigación en Matemáticas, A.C., Guanajuato, México (2006)

  14. Diaz-Garcia, J.A., Ulloa Cortez, L.: Multi-objective optimization for optimum allocation in multivariate stratified sampling. Surv. Methodol. 34(2), 215–222 (2008)

    Google Scholar 

  15. Folks, J.L., Antle, C.E.: Optimum allocation of sampling units to the strata when there are R responses of sampling units of interest. J. Am. Stat. Assoc. 60, 225–233 (1965)

    MathSciNet  MATH  Google Scholar 

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

    Google Scholar 

  17. Gren, J.: Some methods of sample allocation in multivariate stratified sampling. Przeglad Statystyczny 11, 361–369 (1964) (in Polish)

    Google Scholar 

  18. Gren, J.: Some application of non-linear programming in sampling methods. Przeglad Statystyczny 13, 203–217 (1966) (in Polish)

    Google Scholar 

  19. Gupta, N., Shafiullah, Iftekhar, S., Bari, A.: Fuzzy goal programming approach to solve non-linear bi-level programming problem in stratified double sampling design in presence of non-response. Int. J. Sci. Eng. Res. 3(10), 1–9 (2012). October—2012 ISSN 2229–5518

  20. Hartley, H.O.: Multiple purpose optimum allocation in stratified sampling. In: Proceedings of the American Statistical Association, Social Statistics Section, pp. 258–261. American Statistical Association, Alexandria (1965)

    Google Scholar 

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

    Google Scholar 

  22. Khan, M.G.M., Ahsan, M.J.: A note on optimum allocation in multivariate stratified sampling. S. Pac. J. Nat. Sci. 21, 91–95 (2003)

    Google Scholar 

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

    Article  MATH  Google Scholar 

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

    Google Scholar 

  25. Khan, M.F., Ali, I., Ahmad, Q.S.: Chebyshev approximate solution to allocation problem in multiple objective surveys with random costs. 1, 247–251 (2011). doi:10.4236/ajcm.2011.14029

  26. Khan, M.F., Ali, I, Raghav, Y.S., Bari, A.: Allocation in multivariate stratified surveys with non- linear random cost function. Am. J. Oper. Res. (USA) 2(1), 122–125 (2012)

    Article  Google Scholar 

  27. Kokan, A.R., Khan, S.U.: Optimum allocation in multivariate surveys: an analytical solution. J. R. Stat. Soc. Ser. B 29, 115–125 (1967)

    MathSciNet  MATH  Google Scholar 

  28. Kozak, M.: On sample allocation in multivariate surveys. Commun. Stat. Simul. Comp. 35 901–910 (2006)

    Article  MathSciNet  MATH  Google Scholar 

  29. Kreienbrock, L.: Generalized measures of dispersion to solve the allocation problem in multivariate stratified random sampling. Commun. Stat. Theor. Methods 22(1), 219–239 (1993)

    Article  MATH  Google Scholar 

  30. Neyman, J.: On the two different aspects of the representative methods: the method stratified sampling and the method of purposive selection. J. R. Stat. Soc. 97, 558–606 (1934)

    Google Scholar 

  31. Varshney, R., Najmussehar, Ahsan, M.J.: Estimation of more than one parameters in stratified sampling with fixed budget. Math. Meth. Oper. Res. 75, 185–197 (2012). doi:10.1007/s00186-012-0380-y

    Article  MathSciNet  MATH  Google Scholar 

  32. Yates, F.: Sampling methods for census and surveys (Second Edition). Charles Griffin and Co. Ltd., London (1960)

    Google Scholar 

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Authors and Affiliations

  1. Department of Statistics & Operations Research, Aligarh Muslim University, Aligarh, India

    Neha Gupta, Irfan Ali & Abdul Bari

Authors
  1. Neha Gupta
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  2. Irfan Ali
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  3. Abdul Bari
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Correspondence to Neha Gupta or Irfan Ali.

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Gupta, N., Ali, I. & Bari, A. An Optimal Chance Constraint Multivariate Stratified Sampling Design Using Auxiliary Information. J Math Model Algor 13, 341–352 (2014). https://doi.org/10.1007/s10852-013-9237-5

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  • DOI: https://doi.org/10.1007/s10852-013-9237-5

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