Optimum design-based ratio estimators of the distribution function. (English) Zbl 1514.62766
Summary: The ratio method is commonly used to the estimation of means and totals. This method was extended to the problem of estimating the distribution function. An alternative ratio estimator of the distribution function is defined. A result that compares the variances of the aforementioned ratio estimators is used to define optimum design-based ratio estimators of the distribution function. Different empirical results indicate that the optimum ratio estimators can be more efficient than alternative ratio estimators. In addition, we show by simulations that alternative ratio estimators can have large biases, whereas biases of the optimum ratio estimators are negligible in this situation.
MSC:
| 62-XX | Statistics |
References:
| [1] | R.L. Chambers and R. Dunstan, Estimating distribution functions from survey data, Biometrika 73 (1986), pp. 597-604. · Zbl 0614.62005 |
| [2] | A.H. Dorfman, Inference on distribution functions and quantiles, in Handbook of Statitistics 29B Sample Surveys: Inference and Analysis, D. Pffefermann and C.R. Rao, eds. Elsevier, B.V., North-Holland, 2009, pp. 371-395. · doi:10.1016/S0169-7161(09)00236-3 |
| [3] | Eurostat, Low-wage employees in EU countries, in Statistics in Focus: Population and Social Conditions. Office for Official Publications, European Communities, Luxembourg, 2000, pp. 1-12. |
| [4] | J. Haughton and S.R. Khandker, Handbook on Poverty and Inequality, The World Bank, Washington, DC, 2009. |
| [5] | D.G. Horvitz and D.J. Thompson, A generalization of sampling without replacement from a finite universe, J. Am. Stat. Assoc. 47 (1952), pp. 663-685. doi: 10.1080/01621459.1952.10483446 · Zbl 0047.38301 |
| [6] | A.Y.C. Kuk, Estimation of distribution functions and medians under sampling with unequal probabilities, Biometrika 75 (1988), pp. 97-103. doi: 10.1093/biomet/75.1.97 · Zbl 0632.62009 · doi:10.1093/biomet/75.1.97 |
| [7] | A.Y.C. Kuk and T.K. Mak, A functional approach to estimating finite population distribution functions, Theory Meth. 23 (1994), pp. 883-896. doi: 10.1080/03610929408831293 · Zbl 0825.62108 |
| [8] | N.T. Longford, M.G. Pittau, R. Zelli, and R. Massari, Poverty and inequality in European regions, J. Appl. Stat. 39 (2012), pp. 1557-1576. doi: 10.1080/02664763.2012.661705 · Zbl 1250.91087 |
| [9] | M.N. Murthy, Sampling Theory and Method, Statistical Publishing Society, Calcutta, 1967. · Zbl 0183.20602 |
| [10] | J.N.K. Rao, J.G. Kovar, and H.J. Mantel, On estimating distribution function and quantiles from survey data using auxiliary information, Biometrika 77 (1990), pp. 365-375. doi: 10.1093/biomet/77.2.365 · Zbl 0716.62013 · doi:10.1093/biomet/77.2.365 |
| [11] | M. Rueda, S. Martínez, H. Martínez, and A. Arcos, Estimation of the distribution function with calibration methods, J. Stat. Plan. Infer. 137 (2007), pp. 435-448. doi: 10.1016/j.jspi.2005.12.011 · Zbl 1103.62010 · doi:10.1016/j.jspi.2005.12.011 |
| [12] | B. Saalvedra, B. Nolan, and T. Smmeding, (eds.), The Oxford Handbook of Economic Inequality, Oxford University Press, Oxford, 2009. |
| [13] | C.E. Särndal, B. Swensson, and J. Wretman, Model Assisted Survey Sampling, Springer Verlag Inc, New York, 1992. · Zbl 0742.62008 · doi:10.1007/978-1-4612-4378-6 |
| [14] | P.L.D. Silva and C.J. Skinner, Estimating distribution functions with auxiliary information using poststratification, J. Off. Stat. 11 (1995), pp. 277-294. |
| [15] | S. Singh, A.H. Joarder, and D.S. Tracy, Median estimation using double sampling, Aust. NZ J. Stat. 43 (2001), pp. 33-46. doi: 10.1111/1467-842X.00153 · Zbl 1180.62022 · doi:10.1111/1467-842X.00153 |
| [16] | S. Singh, J.M. Kim, H. Jang, and S. Horn, Truncated Midzuno-Sen sampling schemes for estimating distribution functions, Commun. Stat. Simulat. Comput. 40 (2011), pp. 1096-1110. doi: 10.1080/03610918.2011.563006 · Zbl 1219.62019 |
| [17] | R. Singh and M. Kumar, A note on transformations on auxiliary variable in survey sampling, Model Assisted Stat. Appl. 6 (2011), pp. 17-19. |
| [18] | H.P. Singh, S. Singh, and M. Kozak, A family of estimators of finite-population distribution functions using auxiliary information, Acta Appl. Math. 104 (2008), pp. 115-130. doi: 10.1007/s10440-008-9243-1 · Zbl 1168.62011 · doi:10.1007/s10440-008-9243-1 |
| [19] | S. Singh, H.P. Singh, and L.N. Upadhyaya, Chain ratio and regression type estimators for median estimation in survey sampling, Stat. Pap. 48 (2007), pp. 23-46. doi: 10.1007/s00362-006-0314-y · Zbl 1132.62304 · doi:10.1007/s00362-006-0314-y |
| [20] | S. Wang and A.H. Dorfman, A new estimator for the finite population distribution function, Biometrika 83 (1996), pp. 639-652. doi: 10.1093/biomet/83.3.639 · Zbl 0865.62008 · doi:10.1093/biomet/83.3.639 |
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