Statistica Sinica 14(2004), 547-570 KURTOSIS AND CURVATURE MEASURES FOR NONLINEAR REGRESSION MODELS L. M. Haines, T. E. O’Brien and G. P. Y. Clarke University of KwaZulu-Natal, Loyola University Chicago and Agriculture Western Australia Abstract: An expression for the second-order approximation to the kurtosis asso- ciated with the least squares estimate of an individual parameter in a nonlinear regression model is derived, and connections between this and various other mea- sures of curvature are made. Furthermore a means of predicting the reliability of the commonly-used Wald confidence intervals for individual model parameters, based on measures of skewness and kurtosis, is developed. Numerous examples illustrating the theoretical results are provided. Key words and phrases: Bias, confidence intervals, parameter-effects curvature, relative overlap, skewness. 1. Introduction There has been considerable interest over the past twenty years in devel- oping measures of curvature for nonlinear regression models which in some way quantify the deviation of the model from linearity. Specifically, in a landmark paper in 1980, Bates and Watts built on the seminal work of Beale (1960) and in- troduced relative intrinsic and parameter-effects curvatures which provide global measures of the nonlinearity of the model. However these measures are not al- ways helpful when the individual parameters in the model are of interest (see e.g., Cook and Witmer (1985)) and as a consequence a number of researchers have developed measures of curvature which are specifically associated with the individual parameters. In particular Ratkowsky (1983) suggested examining the skewness and kurtosis of the least squares estimates of the parameters by means of simulation and Hougaard (1985) reinforced this idea by deriving a formula for the second-order approximation to skewness. Further Cook and Goldberg (1986) and Hamilton (1986) extended the ideas of Bates and Watts (1980) to accommo- date individual parameters, while Clarke (1987) introduced a marginal curvature measure derived from the second-order approximation to the profile likelihood. The list seems rather daunting and the curvature measures diverse and uncon- nected. In fact Clarke (1987), and more recently Kang and Rawlings (1998), have