Perturbation and scaled Cook's distance

Cook's distance [Technometrics 19 (1977) 15-18] is one of the most important diagnostic tools for detecting influential individual or subsets of observations in linear regression for cross-sectional data. However, for many complex data structures (e.g., longitudinal data), no rigorous approach has been developed to address a fundamental issue: deleting subsets with different numbers of observations introduces different degrees of perturbation to the current model fitted to the data, and the magnitude of Cook's distance is associated with the degree of the perturbation. The aim of this paper is to address this issue in general parametric models with complex data structures. We propose a new quantity for measuring the degree of the perturbation introduced by deleting a subset. We use stochastic ordering to quantify the stochastic relationship between the degree of the perturbation and the magnitude of Cook's distance. We develop several scaled Cook's distances to resolve the comparison of Cook's distance for different subset deletions. Theoretical and numerical examples are examined to highlight the broad spectrum of applications of these scaled Cook's distances in a formal influence analysis.Comment: Published in at http://dx.doi.org/10.1214/12-AOS978 the Annals of Statistics (http://www.imstat.org/aos/) by the Institute of Mathematical Statistics (http://www.imstat.org

Perturbation selection and influence measures in local influence analysis

Cook's [J. Roy. Statist. Soc. Ser. B 48 (1986) 133--169] local influence approach based on normal curvature is an important diagnostic tool for assessing local influence of minor perturbations to a statistical model. However, no rigorous approach has been developed to address two fundamental issues: the selection of an appropriate perturbation and the development of influence measures for objective functions at a point with a nonzero first derivative. The aim of this paper is to develop a differential--geometrical framework of a perturbation model (called the perturbation manifold) and utilize associated metric tensor and affine curvatures to resolve these issues. We will show that the metric tensor of the perturbation manifold provides important information about selecting an appropriate perturbation of a model. Moreover, we will introduce new influence measures that are applicable to objective functions at any point. Examples including linear regression models and linear mixed models are examined to demonstrate the effectiveness of using new influence measures for the identification of influential observations.Comment: Published in at http://dx.doi.org/10.1214/009053607000000343 the Annals of Statistics (http://www.imstat.org/aos/) by the Institute of Mathematical Statistics (http://www.imstat.org

Import Demand for Dairy Products in Cote d'Ivoire

We estimate an LA/AIDS model of demand for imported dairy products for Cote d'Ivoire. We employ a unique set of Ivorian customs data, spanning seven dairy products observed monthly from January 1996 to December 2005. Demand for milk powder is found to be inelastic, as substitutes for milk powder in the domestic processing industry are scarce. Demand for fluid milk, yogurt, and cream are found to be elastic, as these domestic products produced from imported powder may substitute for the imports. With the exception of condensed milk, dairy products are found to be necessities.International Relations/Trade,