Page 97 - Contributed Paper Session (CPS) - Volume 8

CPS2204 T. von R. et al.

̂ = ( )| , (7)

,
=1,
̂
Where ( ) is the weighted least squares estimate of , given

̂
̂ 1 ̂ 1 ̂ −1 , +1 , … ) .
̂

= ( , … ,

̂
̂
Observe that, in Definition 2.2, if ωk → 1, then ( ) → , the unweighted

least squares estimate.
The weighted least squares criterion and the normal equation for the case
when a single parameter is estimated from the perturbed model are given by

̂
̂
̂
̂
( ) = ( − (, , )) ( )( − (, , )),

and
̂ ̂
(, , ) ̂ ̂ (8)
1

1

( ) ( − (, , ( ))) = 0.
In the next theorem, an explicit expression of the marginal-parameter
influence diagnostic ̂ defined in (7) will be provided.
,

Theorem 2.2. Let ̂ be given in Definition 2.2. Then
,
̂
̂
̂
̂
−1
̂ = ( )(( ) ( ) − ( )) ,

,
̂
̂
provided that the inverse exists, where = ( ) = − (, , ),

1

̂
̂
(, , )

1
̂
̂
̂
( ) = ( ( ), … , ( )) = | , 1 × , (9)

1

=
̂

̂
̂
̂
2
( ) (, , )

1
̂
( ) = | = | , 1 × . (10)

̂
= ̂ 2 = ̂
A note on ̂ and ̂
,
,

When deriving the influence measures and studying the single
observations’ influence on the parameter estimates we observe some
interesting aspects of influence analysis in nonlinear regression. A benefit of
using the differentiation approach, where we compute derivatives of various
quantities with respect to and evaluate the derivatives at = 1, is that no

additional iterations for computing the parameter estimates are needed. An
alternative way of using the differentiation approach is to evaluate the same
derivatives as → 0. If this approach were to be used instead, the explicit

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