A note on regression estimation with unknown population size 5. Conclusions

The regression estimator can be quite efficient if the auxiliary data that it uses are well correlated with the variable of interest. Furthermore, it requires that population totals corresponding to the auxiliary variables are available. In this article, we investigated the behavior of the regression estimator ( Y ^ SREG ) MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiFu0Je9sqqrpepC0xbbL8F4rqqrpipeea0xe9Lqpe0x e9q8qqvqFr0dXdbrVc=b0P0xb9peuD0xXddrpe0=1qpeea0=yrVue9 Fve9Fve8meaabaqaciaacaGaaeqabaWaaeaaeaaakeaadaqadaqaai qadMfagaqcamaaBaaaleaacaqGtbGaaeOuaiaabweacaqGhbaabeaa aOGaayjkaiaawMcaaaaa@3E4A@  proposed by Singh and Raghunath (2011). This estimator uses estimated population count as a control total and the known population totals for the auxiliary variables. We compared it to the Generalized Regression estimator ( Y ^ GREG ) , MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiFu0Je9sqqrpepC0xbbL8F4rqqrpipeea0xe9Lqpe0x e9q8qqvqFr0dXdbrVc=b0P0xb9peuD0xXddrpe0=1qpeea0=yrVue9 Fve9Fve8meaabaqaciaacaGaaeqabaWaaeaaeaaakeaadaqadaqaai qadMfagaqcamaaBaaaleaacaqGhbGaaeOuaiaabweacaqGhbaabeaa aOGaayjkaiaawMcaaiaacYcaaaa@3EEE@  its optimal analogue ( Y ^ OPT ) , MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiFu0Je9sqqrpepC0xbbL8F4rqqrpipeea0xe9Lqpe0x e9q8qqvqFr0dXdbrVc=b0P0xb9peuD0xXddrpe0=1qpeea0=yrVue9 Fve9Fve8meaabaqaciaacaGaaeqabaWaaeaaeaaakeaadaqadaqaai qadMfagaqcamaaBaaaleaacaqGpbGaaeiuaiaabsfaaeqaaaGccaGL OaGaayzkaaGaaiilaaaa@3E39@  and to an alternative estimator ( Y ^ KREG ) MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiFu0Je9sqqrpepC0xbbL8F4rqqrpipeea0xe9Lqpe0x e9q8qqvqFr0dXdbrVc=b0P0xb9peuD0xXddrpe0=1qpeea0=yrVue9 Fve9Fve8meaabaqaciaacaGaaeqabaWaaeaaeaaakeaadaqadaqaai qadMfagaqcamaaBaaaleaacaqGlbGaaeOuaiaabweacaqGhbaabeaa aOGaayjkaiaawMcaaaaa@3E42@  that uses the first-order inclusion probabilities and auxiliary data for which the population totals are known. As the optimal regression estimator requires the computation of second-order inclusion probabilities, we also included a pseudo-optimal estimator ( Y ^ POPT ) MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiFu0Je9sqqrpepC0xbbL8F4rqqrpipeea0xe9Lqpe0x e9q8qqvqFr0dXdbrVc=b0P0xb9peuD0xXddrpe0=1qpeea0=yrVue9 Fve9Fve8meaabaqaciaacaGaaeqabaWaaeaaeaaakeaadaqadaqaai qadMfagaqcamaaBaaaleaacaqGqbGaae4taiaabcfacaqGubaabeaa aOGaayjkaiaawMcaaaaa@3E5C@  that does not require them. We investigated the properties of these estimators in terms of bias and efficiency via a simulation that included various sampling designs, and different values of the intercept in the model for a generated artificial population. We compared the results when the population size was known and unknown.

When the population size is known, the most efficient estimator is the optimal estimator Y ^ OPT MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiFu0Je9sqqrpepC0xbbL8F4rqqrpipeea0xe9Lqpe0x e9q8qqvqFr0dXdbrVc=b0P0xb9peuD0xXddrpe0=1qpeea0=yrVue9 Fve9Fve8meaabaqaciaacaGaaeqabaWaaeaaeaaakeaaceWGzbGbaK aadaWgaaWcbaGaci4taiaaccfacaGGubaabeaaaaa@3BFB@ . However, since this estimator can be unstable, the pseudo-optimal estimator Y ^ POPT MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiFu0Je9sqqrpepC0xbbL8F4rqqrpipeea0xe9Lqpe0x e9q8qqvqFr0dXdbrVc=b0P0xb9peuD0xXddrpe0=1qpeea0=yrVue9 Fve9Fve8meaabaqaciaacaGaaeqabaWaaeaaeaaakeaaceWGzbGbaK aadaWgaaWcbaGaaeiuaiaab+eacaqGqbGaaeivaaqabaaaaa@3CC9@ is a good alternative to it. This is in line with Rao (1994) who favoured the optimal estimator Y ^ POPT MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiFu0Je9sqqrpepC0xbbL8F4rqqrpipeea0xe9Lqpe0x e9q8qqvqFr0dXdbrVc=b0P0xb9peuD0xXddrpe0=1qpeea0=yrVue9 Fve9Fve8meaabaqaciaacaGaaeqabaWaaeaaeaaakeaaceWGzbGbaK aadaWgaaWcbaGaaeiuaiaab+eacaqGqbGaaeivaaqabaaaaa@3CC9@ over the Generalized Regression estimator Y ^ GREG . MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiFu0Je9sqqrpepC0xbbL8F4rqqrpipeea0xe9Lqpe0x e9q8qqvqFr0dXdbrVc=b0P0xb9peuD0xXddrpe0=1qpeea0=yrVue9 Fve9Fve8meaabaqaciaacaGaaeqabaWaaeaaeaaakeaaceWGzbGbaK aadaWgaaWcbaGaae4raiaabkfacaqGfbGaae4raaqabaGccaGGUaaa aa@3D67@ The Singh and Raghunath (2011) proposition to use Y ^ SREG MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiFu0Je9sqqrpepC0xbbL8F4rqqrpipeea0xe9Lqpe0x e9q8qqvqFr0dXdbrVc=b0P0xb9peuD0xXddrpe0=1qpeea0=yrVue9 Fve9Fve8meaabaqaciaacaGaaeqabaWaaeaaeaaakeaaceWGzbGbaK aadaWgaaWcbaGaae4uaiaabkfacaqGfbGaae4raaqabaaaaa@3CB7@ is not viable, as it can be quite inefficient. When the population size is not known, the alternative regression estimator Y ^ KREG MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiFu0Je9sqqrpepC0xbbL8F4rqqrpipeea0xe9Lqpe0x e9q8qqvqFr0dXdbrVc=b0P0xb9peuD0xXddrpe0=1qpeea0=yrVue9 Fve9Fve8meaabaqaciaacaGaaeqabaWaaeaaeaaakeaaceWGzbGbaK aadaWgaaWcbaGaae4saiaabkfacaqGfbGaae4raaqabaaaaa@3CAF@ is the best one to use.

Acknowledgements

The authors kindly acknowledge suggestions for improved readability provided by the Associate Editor and the referees.

References

Cassel, C.M., Särndal, C.-E. and Wretman, J.H. (1976). Some results on generalized difference estimators and generalized regression estimation for finite populations. Biometrika, 63, 615-620.

Fuller, W.A. (2009). Sampling Statistics. New York: John Wiley & Sons, Inc.

Horvitz, D.G., and Thompson, D.J. (1952). A generalization of sampling without replacement from a finite universe. Journal of the American Statistical Association, 47, 663-685.

Isaki, C.T., and Fuller, W.A. (1982). Survey design under the regression superpopulation model. Journal of the American Statistical Association, 77, 89-96.

Midzuno, H. (1952). On the sampling system with probability proportional to sum of size. Annals of the Institute of Statistical Mathematics, 3, 99-107.

Montanari, G.E. (1987). Post-sampling efficient QR-prediction in large-scale surveys. International Statistical Review, 55, 191-202.

Montanari, G.E. (1998). On regression estimation of finite population means. Survey Methodology, 24, 1, 69-77.

Rao, J.N.K. (1994). Estimating totals and distribution functions using auxiliary data information at the estimation stage. Journal of Official Statistics, 10(2), 153‑165.

Rosner, B. (2006). Fundamentals of Biostatistics. Sixth edition, Duxbury Press.

Sampford, M.R. (1967). On sampling without replacement with unequal probabilities of section. Biometrika, 54, 499-513.

Särndal, C.-E., Swensson, B. and Wretman, J. (1992). Model Assisted Survey Sampling. New York: Springer-Verlag.

Singh, S., and Raghunath, A. (2011). On calibration of design weights. METRON International Journal of Statistics, vol. LXIX, 2, 185-205.

Report a problem on this page

Is something not working? Is there information outdated? Can't find what you're looking for?

Please contact us and let us know how we can help you.

Privacy notice

Date modified: