The problem of fitting two concentric objects (circles and ellipses) to noisy data plays a vital role in many fields. In this paper, statistical methodologies are deployed by additionally assuming that the angular differences between successively measured data points are known, which is known as Berman’s model. The heteroscedasticity between covariates is also assumed here, and hence, several estimators for the problem of fitting concentric circles and ellipses are developed and their statistical properties are established. Unlike concentric circles, the problem of fitting concentric ellipses turns out to be nonlinear even with Berman’s assumption, and as such, iterative algorithms are implemented. Extensive numerical experiments were conducted to validate our results.

将两个同心对象（圆和椭圆）拟合到噪声数据中的问题在许多领域都起着至关重要的作用。在本文中，通过额外假设连续测量的数据点之间的角度差异已知来部署统计方法，这被称为伯曼模型。这里还假设了协变量之间的异方差性，因此，开发了几个用于拟合同心圆和椭圆的问题的估计器，并建立了它们的统计特性。与同心圆不同，即使采用 Berman 的假设，拟合同心椭圆的问题也是非线性的，因此，实现了迭代算法。进行了广泛的数值实验来验证我们的结果。