Page 98 - Contributed Paper Session (CPS) - Volume 8

CPS2204 T. von R. et al.

expressions of ̂ and ̂ would be functions of the parameter
,
,
estimates with weights attached. As an example, consider the following
derivative
̂
(, ( )) | = (( )),

̂
→0
where → 0. This means that we would need to compute a parameter

estimate for each k and additional iterations are needed. On the contrary, with
the new proposed method in this thesis

̂
(, ( )) | = (),

̂
=1
which is the derivative of the expectation function from the unperturbed
model (1) and hence, no additional iterations are needed.
We can further make a comparison between the proposed measure,
̂ , and the nonlinear version of Cook’s distance and given by
,

̂
where q is the number of parameters in the model, () is the estimate of
̂
when the kth observation is excluded from the calculations and () is
defined in (5). The nonlinear version of Cook’s distance is based on case-
deletion. A consequence of this is that re-estimation of the parameters is
needed for every observation we are interested in. Thus, the nonlinear version
of Cook’s distance demands additional iterations when estimating the
parameters, which is avoided using our measure ̂ .
,

3. Assessment of influence of multiple observations
Thus far, we have discussed the differentiation approach to the detection
of single influential observations. However, in practice it is likely that a data
set contains more than one influential observation. Influence analysis
concerning multiple observations is a more challenging problem since
multiple influential observations can be more difficult to detect. We will
borrow the idea of using the” directional” derivative and define the influence
̂
measure ̂ for assessing the influence of multiple observations on .
β,

a. Joint influence in nonlinear regression
Consider the following perturbed nonlinear model
= (, ) + , (11)

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