The Circular Structural Model

Abstract

Summary This paper deals with consistent estimation of the structural parameters of the circular structural model. Identifiability is proved and the difficulties of dealing with the structural model in practice mentioned. A uniform distribution is then assumed for the angular variables and maximum likelihood estimation of the parameters examined. Also discussed are use of an approximation to the ratio of modified Bessel functions, asymptotic behaviour of the estimators as the ratio of the circle radius to the underlying structural variance parameter increases, and efficiency of other estimators of the parameters. The circular functional model, and the difficulties associated with it, is briefly mentioned.

Statistics on Special Manifolds

“…Mardia and Holmes (1980) and Anderson (1981) modeled the positions of stones in megalithic stone "circles". Cruz-Orive, Hoppeler, Mathieu, and Weibel (1985) applied directional statistics to stereological analysis of anisotropic structures (such as capillaries in skeletal muscle).…”

The Statistical Behaviour of Some Least Squares Estimators of the Centre and Radius of a Circle

“…It is the behaviour [No.2, of the estimators under this form of the model that we shall consider later. Anderson (1981) has examined the Gaussian structural model and given a detailed analysis of maximum likelihood estimation when 0i '" Uniform (0, 2n].…”

“…Now suppose that there is some disturbance to the location of the particle, the size co, and direction cPi of the disturbance being independent of each other, and all directions being equally likely. This is probably a reasonable model for circle fitting problems arising in microwave engineering (Kasa, 1976;Berman, 1983); it has also been used for fitting circles to stone rings (Anderson, 1981;Berman, 1983).…”

“…In these examples, types of least squares estimators have been used. In the statistical literature, on the other hand, maximum likelihood estimators have been considered as well (Chan, 1965;Anderson, 1981;Berman, 1983;Berman and Griffiths, 1985). The latter estimators can be established only on the basis of a statistical model for the data, whereas least squares estimators can be set up without any reference to one.…”

On the Ratio of Two Folded Normal Distributions

“…Then, a tail probability of the ratio of errors of mean estimators, which may be defined as = Pr X − X / Y − Y > 1− , can be written as = Pr W 1 /W 2 > 1− for X = Y = 0 2 1 = 2 X /n 1 , and 2 2 = 2 Y /n 2 . Other examples occur in the setting of the circular functional model and the econometric model considered by Anderson (1981) and Yatchew (1986), respectively. Under the circular model, the ratio of deviations of the Cartesian coordinates of the ith observation X i Y i from the center of circle is defined by X i − / Y i − , that can be written as W 1 /W 2 , for 1 = cos i 2 = sin i , and 2 1 = 2 2 .…”