Influence diagnostics in Birnbaum-Saunders nonlinear regression models

We consider the issue of assessing influence of observations in the class of Birnbaum-Saunders nonlinear regression models, which is useful in lifetime data analysis. Our results generalize those in Galea et al. [2004, Influence diagnostics in log-Birnbaum-Saunders regression models. Journal of Applied Statistics, 31, 1049-1064] which are confined to Birnbaum-Saunders linear regression models. Some influence methods, such as the local influence, total local influence of an individual and generalized leverage are discussed. Additionally, the normal curvatures of local influence are derived under various perturbation schemes.

💡 Research Summary

The paper addresses the problem of assessing the influence of individual observations in Birnbaum‑Saunders (BS) nonlinear regression models, a class of models that are particularly useful for analyzing lifetime data with asymmetric, positive‑valued responses. While earlier work by Galea et al. (2004) developed influence diagnostics for log‑BS linear regression, the present study extends those methods to the more general nonlinear setting.

The authors first define a BS nonlinear regression model in which the response (y_i) follows a BS distribution with shape parameter (\alpha) and a mean function (\mu_i(\boldsymbol\beta)) that is a nonlinear function of a parameter vector (\boldsymbol\beta). The log‑likelihood is written explicitly, and the score vector and observed information matrix are derived using matrix‑calculus techniques that accommodate the nonlinear mean structure.

Building on Cook’s (1986) local influence framework, the paper introduces a perturbed log‑likelihood (\ell(\boldsymbol\theta|\boldsymbol\omega)) where (\boldsymbol\omega) represents a vector of small perturbations. Three perturbation schemes are examined: (1) case‑weight perturbation, (2) response‑value perturbation, and (3) design‑matrix perturbation. For each scheme the authors compute the second‑order derivative matrix (\mathbf{H}) of the perturbed log‑likelihood with respect to the model parameters and the perturbation vector, and they obtain the normal curvature \