American Journal of Theoretical and Applied Statistics
Volume 4, Issue 5, September 2015, Pages: 396-403
Received: Aug. 17, 2015;
Accepted: Aug. 29, 2015;
Published: Sep. 9, 2015
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Gladys Gakenia Njoroge, Department of Physical Sciences, Chuka University, Chuka, Kenya
This study sought to estimate finite population total using spline functions. The emerging patterns from spline smoother were compared with those that were obtained from the model-based, the model-assisted and the non-parametric estimators. To measure the performance of each estimator, three aspects were considered: the average bias, the efficiency by use of the average mean square error and the robustness using the rate of change of efficiency. We used six populations: four natural and two simulated. The findings showed that the model-based estimator works very well in terms of efficiency while the model-assisted is almost unbiased when the model is linear and homoscedastic. However, the estimators break down when the underlying model assumptions are violated. The Kernel Estimator (Nadaraya-Watson) is found to be the most robust of the five estimators considered. Between the two spline functions that we considered, the periodic spline was found to perform better. The spline functions were found to provide good results whether or not the design points were uniformly spaced. We also found out that, under certain conditions, a smoothing spline estimator and a Kernel estimator are equivalent. The study recommends that both the ratio estimator and the local polynomial estimator should be used within the confines of a linear homoscedastic model. The Nadaraya-Watson and the periodic spline estimators, both of which are non-parametric, are highly robust. The Nadaraya-Watson however is even more robust than the periodic spline.
Gladys Gakenia Njoroge,
Estimation of Population Total Using Spline Functions, American Journal of Theoretical and Applied Statistics.
Vol. 4, No. 5,
2015, pp. 396-403.
Aerts, M., Augustyns, I. and Janssen, P., “Smoothing Sparse Multinomial Data Using Local Polynomial Fitting,” Journal of Nonparametric Statistics, 8, 127-147, 1997.
Breidt, F. J. and Opsomer, J. D., “Local Polynomial Regression Estimators in Survey Sampling,” Annals of Statistics, 28, 1026-1053, 2000.
Cardot, H., “Local Roughness Penalties for Regression Splines,”Computational Statistics, 17, 89-102, 2002.
Fuller, W.A, Sampling Statistics, Wiley, Hoboken, 2009.
Harms, T. and Duchesne, P., “On Kernel Non- Parametric Regression Designed for Complex Survey”, Metrika, 72 (1), 111-138, July2010.
Kauermann, G., Krivobokova, T. and Fahrmeir, L., “Some Asymptotic Results on Generalized Penalized Spline Smoothing,” J. R. Statistic. Soc.Series B, 71, 487-503, 2009.
Lu, J. and Yan, Z., “A Class of Ratio Estimators of a Finite Population Mean Using Two Auxiliary Variables,”PLoS ONE 9(2): e89538.doi:10.1371/journal.pone.0089538, 2014.
Nadaraya, E. A., “On Estimating Regression,” Jour. Theory Probab. Appl.9 (1), 141-142, 1964.
Royall, R. M., “Likelihood Functions in Finite Population Sampling Theory,”Biometrika, 63, 605-614, 1976.
Sarda, P. and Vieu, P., Kernel Regression in Smoothing and Regression: Approaches Computation and Application, Ed M. G. Schimek, Wiley Series in Probability and Statistics, 2000, 43-70.
Schoenberg, I. J., “Spline Functions and the Problem of Graduation,”Proc. Nat. Acad. Sci. U.S.A., 52, 947-950, 1946.
Schumaker, L. L., Spline Functions: Computational Methods, SIAM, Philadelphia, 2015.
Silverman, B. W., “Spline Smoothing: The Equivalent Variable Kernel Method,” The Annals of Statistics, 12(3), 898-916, 1984.
Wahba, G., “Smoothing Noisy Data with Spline Functions,”Numerische Mathematik, 24, 383-393, 1975.
Watson, G. S., “Smooth Regression Analysis,”Sankhya Ser. A.,26, 359-372, 1664
Whittaker, E., “On a New Method of Graduation,”Proc. Edinburgh Math. Soc., 41, 63-75, 1923.