The problem of fitting a circle to a set of data points on a plane arises in many areas, and types of least squares estimators are often used to estimate the center and radius of the circle. In this paper, we show that, for a general circular model, these estimators are inconsistent, whereas the maximum likelihood estimators may be nonexistent, inconsistent, or difficult to compute. Consistent and asymptotically normal estimators are then obtained from unbiased modified least squares estimating equations, and their small-sample superiority relative to the least squares estimators is supported by numerical results. Further attention is paid to a special case of the model, also a generalization of Berman's model to account for angular measurement errors, for which a two-step estimator is suggested and its small-sample performance is examined. Keywords: Asymptotic normality, circle fitting, consistency, estimating equation, incidental parameter, inconsistency, least squares, maximum likelihood.
 

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