Abstract
This paper proposes a new model-based approach to estimate small areas that extends the Fay–Herriot methodology. The new model is additive, with a random term to characterize the inter-area variability and a nonparametric mean function specification, defined using the information on an auxiliary variable. The most significant advantage of the proposal is that it avoids the model misspecification problem. The monotonicity is the only assumption about the functional form of the relationship between the variable of interest and the auxiliary one. Estimators for the area means are derived combining “Order Restricted Inference” and standard mixed model approaches. A large simulation experiment shows how the new approach outperforms the Fay–Herriot methodology in many scenarios. Besides, the new method is applied to the Australian farms data.
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References
Chambers R, Tzavidis N (2006) M-quantile models for small area estimation. Biometrika 93(2):255–268
Chatterjee S, Lahiri P, Li H (2008) Parametric bootstrap approximation to the distribution of EBLUP and related prediction intervals in linear mixed models. Ann Stat 36(3):1221–1245
David B (2003) Choosing a method for poverty mapping. Food and Agriculture Organization of the United Nations, Rome
Efron B, Morris CN (1975) Data analysis using Stein’s estimator and its generalizations. J Am Stat Assoc 70:311–319
Fay RE, Herriot RA (1979) Estimates of income for small places: an application of James–Stein procedures to census data. J Am Stat Assoc 74:341–353
González-Manteiga W, Lombardía MJ, Molina I, Morales D, Santamaría L (2008) Bootstrap mean squared error of a small-area EBLUP. J Stat Comput Simul 78(5):443–462
Hall P, Maiti T (2006) On parametric bootstrap methods for small area prediction. J R Stat Soc, Ser B 68(2):221–238
Huang ET, Bell W (2006) Using the t-distribution in small area estimation: an application to SAIPE state poverty models. SAIPE publications. US Census Bureau
Mammen E, Yu K (2007) Additive isotonic regression. Lecture notes monograph series, vol 55, pp 179–195
Opsomer J, Claeskens G, Ranalli M, Kauermann G, Breidt FJ (2008) Non-parametric small area estimation using penalized spline regression. J R Stat Soc B 70(1):265–286
Pan G, Khattree R (1999) On estimation and testing arising from order restricted balanced mixed models. J Stat Plan Inference 77:281–292
Rao JNK (2003) Small area estimation. Wiley, New York
Robertson T, Wright FT, Dykstra RL (1988) Order restricted statistical inference. Wiley, New York
Rueda C, Menéndez JA (2005) A restricted model approach to improve the precision of estimators. Stat Transit J 7(3):697
Silvapulle MJ, Sen PK (2005) Constrained statistical inference. Wiley, New York
Ugarte MD, Goicoa T, Militino AF, Durbán M (2009a) Spline smoothing in small area trend estimation and forecasting. Comput Stat Data Anal 53:3616–3629
Ugarte MD, Militino AF, Goicoa T (2009b) Benchmarked estimators in small areas using linear mixed models with restrictions. Test 18(2):342–364
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Rueda, C., Menéndez, J.A. & Gómez, F. Small area estimators based on restricted mixed models. TEST 19, 558–579 (2010). https://doi.org/10.1007/s11749-010-0186-2
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DOI: https://doi.org/10.1007/s11749-010-0186-2
Keywords
- James–Stein estimator
- Small areas
- Order restricted inference
- Mixed models
- EBLUP estimator
- Fay–Herriot model