Statistical analysis of curve fitting methods in errors-in-variables models

Abstract

Abstract. Regression models in which all variables are subject to errors are known as errors-in-variables (EIV) models. The respective parameter estimates have many unusual properties: their exact distributions are very hard to determine, and their absolute moments are often infinite (so that their mean and variance do not exist). In our paper, Error analysis for circle fitting algorithms, Electr. J. Stat. 3 (2009), 886-911, we developed an unconventional statistical analysis that allowed us to effectively assess EIV parameter estimates and design new methods with superior characteristics. In this paper we validate our approach in a series of numerical tests. We also prove that in the case of fitting circles, the estimates of the parameters are absolutely continuous (have densities).

Further statistical analysis of circle fitting

“…Therefore, the mean square error of HyperLS fit is smaller than that of hyper fit, which is smaller than geometric fit, and so one. This validates our conclusions in [1,2].…”

“…Our main goal in [1,2] was to compare the most popular circle fits (geometric fit and other various algebraic fits such as Kåsa's fit, Pratt's fit, and Taubin's fit). We characterized the accuracy of estimators based on (Total) Mean Squared Error (MSE):…”

“…On other hand, we noticed in some situations that the geometric fit still practically performs better than hyper and HyperLS. Of course, our conclusions in [1,2] are solely based on first-order error terms of the variances and the firstorder of the bias for each algebraic fits, while the second-order term of the variance dropped. Therefore, we believe that our findings in [1] do not reveal the whole picture about the performance of circle fits.…”

“…This means that if we restricted our error analysis to the first leading term, then the geometric fit would be an unbiased and efficient estimator of θ in the sense that the variance of its linear approximation achieves its minimal possible bound σ 2 Ṽmin in the small-noise limit; this was proven by Fuller (Theorem 3.2.1 in [16]) for the geometric fit and independently by Chernov and Lesort [11]. In fact, the same property holds for all algebraic fits.…”

A new perspective in functional EIV linear model: Part I

“…Moreover, the estimatesα 1 andβ 1 do not have finite moments (Anderson, 1976). As a result, they have infinite variances and infinite mean squared errors, though such an erratic behavior is barely seen in practice (Anderson, 1976;Anderson and Sawa, 1982;Chernov, 2010;Al-Sharadqah and Chernov, 2011). Another difficulty in studying the simple linear regression in the functional EIV model stems from dealing with large sample problem n → ∞.…”