A comparative study for two newly developed estimators for the slope in the functional EIV linear model
Author:
A. A. Al-Sharadqah
Journal:
Theor. Probability and Math. Statist. 97 (2018), 211-236
MSC (2010):
Primary 68T10, 68K45, 68K40, 62P30
DOI:
https://doi.org/10.1090/tpms/1058
Published electronically:
February 21, 2019
MathSciNet review:
3746009
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Additional Information
Abstract: Two estimators were recently developed for the slope of a line in the functional EIV model. Both are unbiased, up to order $\sigma ^4$, where $\sigma$ is the error standard deviation. One estimator was constructed as a function of the maximum likelihood estimator (MLE). Therefore, it was called the Adjusted MLE (AMLE). The second estimator was constructed using a completely different approach. Although both the estimators are unbiased, up to the order $\sigma ^4$, the latter estimator is much more accurate than the AMLE. We study these two estimators more rigorously here, and we show why one estimator outperforms the other one.
References
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References
- A. Al-Sharadqah, A new perspective in functional EIV linear model: Part I, Comm. Statist. Theory Methods 47 (2017), no. 14, 7039–7062. MR 3640166
- A. Al-Sharadqah and N. Chernov, Statistical analysis of curve fitting methods in errors-in-variables models, Theory Probab. Math. Statist. 84 (2011), 4–17. MR 2857412
- A. Al-Sharadqah, N. Chernov, and Q. Huang, Errors-in-variables regression and the problem of moments, Brazilian Journal of Probability and Statistics 84 (2013), 401–415. MR 3105036
- Y. Amemiya and W. A. Fuller, Estimation for the nonlinear functional relationship, Annals Statist. 16 (1988), 147–160. MR 924862
- T. W. Anderson, Estimation of linear functional relationships: Approximate distributions and connections with simultaneous equations in econometrics, J. R. Statist. Soc. B 38 (1976), 1–36. MR 0411025
- T. W. Anderson and T. Sawa, Distributions of estimates of coefficients of a single equation in a simultaneous system and their asymptotic expansions, Econometrica 41 (1973), 683–714. MR 0438637
- T. W. Anderson and T. Sawa, Exact and approximate distributions of the maximum likelihood estimator of a slope coefficient, J. R. Statist. Soc. B 44 (1982), 52–62. MR 655374
- C.-L. Cheng and J. W. Van Ness, Statistical Regression with Measurement Error, Arnold, London, 1999. MR 1719513
- C. L. Cheng and A. Kukush, Non-existence of the first moment of the adjusted least squares estimator in multivariate errors-in-variables model, Metrika 64 (2006), 41–46. MR 2242556
- N. Chernov, Circular and Linear Regression: Fitting Circles and Lines by Least Squares, CRC Monographs on Statistics & Applied Probability, vol. 117, Chapman & Hall, 2010. MR 2723019
- N. Chernov, Fitting circles to scattered data: parameter estimates have no moments, Metrika 73 (2011), 373–384. MR 2785031
- N. Chernov and C. Lesort, Statistical efficiency of curve fitting algorithms, Comp. Stat. Data Anal. 47 (2004), 713–728. MR 2101548
- L. J. Gleser, Functional, structural and ultrastructural errors-in-variables models, Proc. Bus. Econ. Statist. Sect. Am. Statist. Assoc., 1983, pp. 57–66.
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- K. Kanatani, Statistical Optimization for Geometric Computation: Theory and Practice, Elsevier, Amsterdam, 1996. MR 1392697
- K. Kanatani, Cramer–Rao lower bounds for curve fitting, Graph. Mod. Image Process. 60 (1998), 93–99.
- K. Kanatani, For geometric inference from images, what kind of statistical model is necessary?, Syst. Comp. Japan 35 (2004), 1–9.
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- J. R. Magnus and H. Neudecker, The commutation matrix: some properties and applications, Annals Statist. 7 (1979), 381–394. MR 520247
- K. M. Wolter and W. A. Fuller, Estimation of nonlinear errors-in-variables models, Annals Statist. 10 (1982), 539–548. MR 653528
- E. Zelniker and V. Clarkson, A statistical analysis of the Delogne–Kåsa method for fitting circles, Digital Signal Proc. 16 (2006), 498–522.
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Additional Information
A. A. Al-Sharadqah
Affiliation:
Department of Mathematics and Interdisciplinary Research Institute for the Sciences, California State University-Northridge, Northridge, California 91330-8313
Email:
ali.alsharadqah@csun.edu
Keywords:
Simple linear regression,
errors-in-variables models,
small-noise model,
maximum likelihood estimator,
bias correction,
mean squared errors
Received by editor(s):
May 3, 2017
Published electronically:
February 21, 2019
Article copyright:
© Copyright 2019
American Mathematical Society
