Abstract
The problem of estimating a polynomial model with a classical error in the input factor is under consideration in the functional case. The nonparametric method recently introduced for estimating structural dependences does not use any additional information, but it is very effortconsuming computationally and needs samples of large size.We propose some easier methods. The first approach is based on a preliminary estimation of the Berkson error variance under assumption of its normal distribution by the maximum likelihood method for a piecewise linearmodel. This estimate of variance is used for recovering the parameters of a polynomial by the methods of general and adjusted least squares. In case the error variance deviates from normal distribution, an adaptive method is developed that is based on the generalized lambda distribution. These approaches were applied for solving the problem of knowledge level evaluation.
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References
W. A. Fuller, Measurement ErrorModels (JohnWiley & Sons, New York, 1987).
R. J. Carroll, D. Ruppert, and L. A. Stefanski, Measurement Error in Nonlinear Models (Chapman and Hall, London, 1995).
A. Yu. Timofeeva and O. E. Avrunev, “Locally Weighted Smoothing of Structural Relationships for the Student Progress Analysis,” Dokl. Akad. Nauk Vyssh. Shkoly Ross. Feder. No. 1 (22), 135–146 (2014).
O. V. Pol’din, “Predicting Success in College on the Basis of the Results of Unified National Exam,” Prikl. Ekonometrika 21 (1), 87–106 (2011).
T. E. Khavenson and A. A. Solov’eva, “Studying the Relation between the Unified State Exam Points and Higher Education Performance,” Voprosy Obrazovaniya No. 1, 176–199 (2014) [Educational Studies (Moscow) No. 1, 176–199 (2014)].
J. L. Kobrin, B. F Patterson, E. J. Shaw, K. D. Mattern, and S. M. Barbuti, Validity of the SAT for Predicting First-Year College Grade Point Average, College Board Research Rep. No. 2008–5 (The College Board, New York, 2008).
A. Timofeeva, “Orthogonal Regression for Nonparametric Estimation of Errors-in-Variables Models,” J. Math. Comput. Phys. Quantum Engrg. 8 (8), 470–474 (2014).
J. P. Marini, K. D. Mattern, and E. J. Shaw, Examining the Linearity of the PSAT/NMSQTR-FYGPA Relationship, College Board Research Report No. 2011–7 (The College Board, New York, 2011).
P. J. Rousseeuw and K. Van Driessen, Computing LTS Regression for Large Data Sets (Univ. of Antwerpen, Antwerpen, 1999).
V. S. Timofeev and E. A. Khailenko, “Adaptive Estimation of Parameters of RegressionModelswithUsing the Generalized Lambda Distribution,” Dokl. Akad. Nauk Vyssh. Shkoly Ross. Feder. No. 2 (15), 25–36 (2010).
M. Cendall and A. Stuart, Statistical Conclusions and Relations (Nauka, Moscow, 1973) [in Russian].
S. VanHuffel and J. Vandewalle, The Total Least Squares Problem: Computational Aspects and Analysis (SIAM, 1991).
A. A. Greshilov, V. A. Stakun, and A. A. Stakun, Mathematical Methods of Forecasting (Radio i Svyaz’, Moscow, 1997) [in Russian].
V. I. Denisov, A. Yu. Timofeeva, E. A. Khailenko, and O. I. Buzmakova, “Robust Estimation of Nonlinear Structural Dependences,” Sibirsk. Zh. Industr. Mat. 16 (4), 47–60 (2013).
C.-L. Cheng and H. Schneeweiss, “Polynomial Regression with Errors in the Variables,” J. Royal Statist. Soc. Ser. B, 60, 189–199 (1998).
S. M. Schennach, “Estimation of NonlinearModels withMeasurement Error,” Econometrica 72 (1), 33–75 (2004).
J. Hausman, W. K. Newey, and J. L. Powell, “Nonlinear Errors in Variables: Estimation of Some Engel Curves,” J. Econometrics 65 (1), 205–233 (1995).
A. Yu. Timofeeva, “On Endogeneity of Consumer Expenditures in the Estimation of Household Demand System,” Prikl. Ekonometrika 37 (1), 87–106 (2015).
S. M. Schennach and Y. Hu, “Nonparametric Identification and Semiparametric Estimation of Classical Measurement Error Models without Side Information,” J. American Statist. Assoc. 108 (501), 177–186 (2013).
A. Yu. Timofeeva and O. I. Buzmakova, “Semiparametric Estimation of Dependences between Stochastic Variables,” Nauchn. Vestnik Novosib. Gos. Tekhn. Univ. 49 (4), 29–37 (2012).
L. Wang, “Estimation of Nonlinear Models with Berkson Measurement Errors,” Annals of Statist. 32 (6), 2559–2579 (2004).
L. Huwang and Y. H. S. Huang, “On Errors-in-Variables in Polynomial Regression—Berkson Case,” Statistica Sinica 10, 923–936 (2000).
Z. A. Karian and E. J. Dudewicz, Fitting Statistical Distributions: The Generalized Lambda Distribution and Generalized BootstrapMethods (CRC Press LLC, New York, 2000).
A. Lakhany and H. Mausser, “Estimation of the Parameters of theGeneralized Lambda Distribution,” ALGO Res. Quart. 3 (3), 27–58 (2000).
M. V. Stasyshin, O. E. Avrunev, E. V. Afonina, and K. N. Lyakh, “An Information System of University: Experience of Creating and Current State,” Otkrytoe i Distants. Obrazovanie No. 2 (46), 9–15 (2012).
P. J. Rousseeuw and C. Croux, “Alternatives to the Median Absolute Deviation,” J. Amer. Statist. Assoc. 88 (424), 1273–1283 (1993).
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Original Russian Text © V.I. Denisov, A.Yu. Timofeeva, E.A. Khailenko, 2016, published in Sibirskii Zhurnal Industrial’noi Matematiki, 2016, Vol. XIX, No. 3, pp. 15–27.
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Denisov, V.I., Timofeeva, A.Y. & Khailenko, E.A. Estimating parameters of polynomial models with errors in variables and no additional information. J. Appl. Ind. Math. 10, 322–332 (2016). https://doi.org/10.1134/S1990478916030029
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DOI: https://doi.org/10.1134/S1990478916030029