Circular functional relationships

This paper develops techniques for fitting a circular functional relationship. The model is formulated assuming the centre of the circle is known and taken to be the origin. This incurs some loss of generality, but considerably simplifies the analysis. Methods of estimating the parameters of the relationship, assuming that the co-ordinates have uncorrelated errors with equal variance, are developed and compared by way of simulations. Consistent estimators are obtained from unbiased estimating equations and by maximising the marginal likelihood. Approximate estimators are obtained by approximating the estimating equations and by maximising a modified profile likelihood function. The ideas and methodology are then applied to the relationship with the more realistic assumption of an unknown centre, with an example highlighting the differences between estimation techniques.