Page 94 - Contributed Paper Session (CPS) - Volume 8

CPS2204 T. von R. et al.
(e.g. Beckman et al., 1987; Lawrance, 1988; Thomas and Cook, 1990; Wu and
Luo, 1993; St. Laurent and Cook, 1993; Shi, 1997; Poon and Poon, 1999; Zhu
and Lee, 2001; Zhu et al., 2007). The existing literature on influence analysis in
nonlinear regression is not as extensive as for linear regression. One reason
for this can be that there do not generally exist closed form estimators for the
parameters in the nonlinear regression model.
Detection of influential observations on the fit of the nonlinear regression
model is discussed by Cook and Weisberg (1982) and St. Laurent and Cook
(1993). Cook and Weisberg (1982) developed a nonlinear version of Cook’s
distance, and St. Laurent and Cook (1993) proposed an approach for assessing
the influence of the observations on the fitted values and on the estimate of
the variance in a nonlinear regression model. Detection of multiple influential
observations is in general a more difficult task due to masking and swamping
effects (Atkinson, 1985; Rousseeuw & Leory, 1987; Chatterjee & Hadi, 1988;
Lawrence, 1995). Masking occurs when an observation is not identified as
influential unless another observation is deleted first. Swamping occurs when
”good” observations are incorrectly identified as influential because of the
presence in data of another observation. The available approaches to dealing
with the problem of masking effects include the use of multiple 1 case
deletions (Chatterjee and Hadi, 1988; Hadi and Simonoff, ; Lawrance, 1995) or
stepwise procedures (Belsley et al., 1980; Bruce and Martin, 1989; Shi and
Huang, 2011).
The aim of this article is to develop influence measures to assess the
influence of a single observation as well as multiple influential observations on
the parameter estimates in nonlinear regression models. The proposed
measures allow to simultaneously assess the influence of several observations
on the parameter estimates. This type of influence will be referred to as joint
influence. Furthermore, it makes it possible to evaluate the influence that the
kth observation has on the parameter estimates after another observation, say
observation i, has been deleted. The type of influence that the kth observation
has on the parameter estimates after the deletion of the ith observation is
called conditional influence.

2. The influence measure DIM
Consider the following nonlinear model with an additive error term
= (, ) + , (1)
where is the -vector of responses, the known matrix ∶ × comprises
explanatory variables, is a q-vector of unknown parameters, the vector of
2
random errors ~( , ), and denote the -vector with all elements

equal to zero and the identity matrix of size , respectively. The function is
assumed to be twice differentiable in , and
83 | I S I W S C 2 0 1 9