Curvature and Convergence

Fisher’s method of scoring is probably the most important general algorithm in statistics. This paper picks out those aspects of curvature, normal and statistical, which are relevant to its convergence properties. For any particular data set the convergence of the algorithm near the maximum likelihood estimate depends on the eigenvalues of the convergence matrix, the derivative of the iteration function. In the least squares case, these eigenvalues can be interpretted as normal curvatures of one-dimensional curves on the response surface. Statistical curvature is shown to provide a before-the-data estimate of the squared sizes of the components of the convergence matrix. Bates and Watts (1980)’s intrinsic curvature is shown to correspond to the expected size of the convergence matrix in particular directions. A theme of the paper is that there is a close relationship between the convergence properties of the method of scoring, and the statistical properties of the model being fitted. This relationship is in large part due to mutual dependence on curvature. Citation: Smyth, G.K. (1987). Curvature and convergence. Proceedings of the Statistical Computing Section. American Statistical Association, Virginia, 278–283.