Singular asymptotic normality of an estimator in the conic section fitting problem. I

The conic section fitting problem is considered. True points are assumed to lie on a conic section. The points are observed with additive errors, which are independent and have bivariate normal distribution $N(0, \sigma ^2 I)$ with unknown $\sigma ^2$. We study asymptotic properties of the estimator of conic section parameters introduced by Kukush, Markovsky, and Van Huffel in Computational Statistics and Data Analysis 47 (2004), 123–147. Sufficient conditions for singular asymptotic normality of the estimator are provided. The asymptotic covariance matrix is singular and has defect 1 because the unit sphere in Euclidean space is taken as a parameter space.