Abstract
In a multivariate stratified sampling more than one characteristic are defined on every unit of the population. An optimum allocation which is optimum for one characteristic will generally be far from optimum for others. A compromise criterion is needed to work out a usable allocation which is optimum, in some sense, for all the characteristics. When auxiliary information is also available the precision of the estimates of the parameters can be increased by using it. Furthermore, if the travel cost within the strata to approach the units selected in the sample is significant the cost function remains no more linear. In this paper an attempt has been made to obtain a compromise allocation based on minimization of individual coefficients of variation of the estimates of various characteristics, using auxiliary information and a nonlinear cost function with fixed budget. A new compromise criterion is suggested. The problem is formulated as a multiobjective all integer nonlinear programming problem. A solution procedure is also developed using goal programming technique.
Similar content being viewed by others
References
Ahsan MJ (1975–1976) A procedure for the problem of optimum allocation in multivariate stratified random sampling stratified random sampling. Aligarh Bull Math 5–6: 37–42
Ahsan MJ (1978) Allocation problem in multivariate stratified random sampling. J Indian Stat Assoc 16: 1–5
Ahsan MJ, Khan SU (1977) Optimum allocation in multivariate stratified random sampling using prior information. J Indian Stat Assoc 15: 57–67
Ahsan MJ, Khan SU (1982) Optimum allocation in multivariate stratified random sampling with overhead cost. Metrika 29: 71–78
Ansari AH, Najmussehar , Ahsan MJ (2009) On multiple response stratified random sampling design. J Stat Sci Kolkata, India 1(1): 45–54
Aoyama H (1963) Stratified random sampling with optimum allocation for multivariate population. Ann Inst Stat Math 14: 251–258
Beardwood J, Halton JH, Hammersley JM (1959) The shortest path through many points. Math Proc Camb Philos Soc 55: 299–327
Bethel J (1985) An optimum allocation algorithm for multivariate surveys. Proc Surv Res Methods Sect, Am Stat Assoc 209–212
Bethel J (1989) Sample allocation in multivariate surveys. Surv Methodol 15: 47–57
Chatterjee S (1967) A note on optimum allocation. Scand Actuar J 50: 40–44
Chatterjee S (1968) Multivariate stratified surveys. J Am Stat Assoc 63: 530–534
Chromy JR (1987) Design optimization with multiple objectives. Proc Surv Res Methods Sect, Am Stat Assoc 194–199
Cochran WG (1977) Sampling techniques. Wiley, New York
Dalenius T (1957) Sampling in Sweden: contributions to the methods and theories of sample survey practice. Almqvist and Wiksell, Stockholm
Dayal S (1985) Allocation of sample using values of auxiliary Characteristic. J Stat Plan Inference 11(3): 321–328
Dıaz-García JA, Cortez LU (2006) Optimum allocation in multivariate stratified sampling: multi-objective programming. Comunicación Técnica No. I-06-07/28-03-2006 (PE/CIMAT)
Díaz-García JA, Cortez LU (2008) Multi-objective optimisation for optimum allocation in multivariate stratified sampling. Surv Methodol 34(2): 215–222
Ericson WA (1965) Optimum stratified sampling using prior information. J Am Stat Assoc 60(311): 750–771
Folks JL, Antle CE (1965) Optimum allocation of sampling units to strata when there are R responses of interest. J Am Stat Assoc 60: 225–233
Ghosh SP (1958) A note on stratified random sampling with multiple characters. Calcutta Stat Assoc Bull 8: 81–89
Jahan N, Ahsan MJ (1995) Optimum allocation using separable programming. Dhaka Univ J Sci 43(1): 157–164
Jahan N, Khan MGM, Ahsan MJ (1994) A generalized compromise allocation. J Indian Stat Assoc 32: 95–101
Jahan N, Khan MGM, Ahsan MJ (2001) Optimum compromise allocation using dynamic programming. Dhaka Univ J Sci 49(2): 197–202
Khan MGM, Ahsan MJ, Jahan N (1997) Compromise allocation in multivariate stratified sampling: an integer solution. Nav Res Logist 44: 69–79
Khan MGM, Khan EA, Ahsan MJ (2003) An optimal multivariate stratified sampling design using dynamic programming. Aust N Z J Stat 45(1): 107–113
Khan MGM, Khan EA, Ahsan MJ (2008) Optimum allocation in multivariate stratified sampling in presence of non-response. J Ind Soc Agric Stat 62(1): 42–48
Khan MGM, Maiti T, Ahsan MJ (2010) An optimal multivariate stratified sampling design using auxiliary information: an integer solution using goal programming approach. J Off Stat 26(4): 695–708
Kokan AR, Khan SU (1967) Optimum allocation in multivariate surveys: an analytical solution. J R Stat Soc B 29(1): 115–125
Kozak M (2006a) On sample allocation in multivariate surveys. Commun Stat Simul Comput 35: 901–910
Kozak M (2006b) Multivariate sample allocation: application of random search method. Stat Transit 7(4): 889–900
Kreienbrock L (1993) Generalized measures of dispersion to solve the allocation problem in multivariate stratified random sampling. Commun Stat Theory Methods 22(1): 219–239
LINGO-User’s Guide (2001) LINGO-User’s Guide. LINDO SYSTEM INC., USA
Neyman J (1934) On the two different aspects of the representative method: the method of stratified sampling and the method of purposive selection. J R Stat Soc 97(4): 558–625
Rahim MA (1995) Use of distance function to optimize sample allocation in multivariate surveys: a new perspective. Proc Surv Res Methods Sect, Am Stat Assoc 365–369
Raiffa H, Schlaifer R (1961) Applied statistical decision theory. Graduate School of Business Adminitration Harvard University, Boston
Schittkowski K (1985–1986) NLPQL: a FORTRAN subroutine solving constrained nonlinear programming problems. Ann Oper Res 5: 485–500
Schniederjans MJ (1995) Goal programming: methodology and applications. Kluwer, Dordrecht
Semiz M (2004) Determination of compromise integer strata sample sizes using goal programming. Hacet J Math Stat 33: 91–96
Singh S (2003) Advanced sampling theory with applications: how Michael ‘selected’ Amy. Kluwer, Dordrecht
Stuart A (1954) A simple presentation of optimum sampling results. J R Stat Soc B16(2): 239–241
Sukhatme PV, Sukhatme BV, Sukhatme S, Asok C (1984) Sampling Theory of Surveys with Applications. Iowa State University Press, Iowa, USA and Indian Society of Agricultural Statistics, New Delhi, India
Tschuprow AA (1923) On the mathematical expectation of the moments of frequency distributions in the case of correlated observations. Metron 2: 461–493
Yates F (1960) Sampling methods for censuses and surveys. Charles Griffin and Co Ltd., London
Author information
Authors and Affiliations
Corresponding author
Rights and permissions
About this article
Cite this article
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). https://doi.org/10.1007/s00186-012-0380-y
Received:
Published:
Issue Date:
DOI: https://doi.org/10.1007/s00186-012-0380-y