Page 99 - Contributed Paper Session (CPS) - Volume 8

CPS2204 T. von R. et al.
2
−
where ~ (, ()) , (): × is a diagonal weight matrix, with

diagonal elements = ( , … , ) and where 0 < ≤ 1, for = 1, … , .

1
Also, let K be the subset containing the indices of the observations for which
we would like to assess influence.

We present the following definition

Definition 3.1. The diagnostic measure for assessing the influence of the
observations with indices specified in the subset K, on the parameter estimate
̂
, is defined as the following derivative
̂
()
̂ = | , (12)
,
=
where ∶ × 1 is a vector with nonzero entries in the rows with indices in K,
where ‖‖ = √ = 1 and where () is the weighted least squares estimate
̂

of , which is a function of the weight .

̂
̂
If → , then () → , the unweighted least squares estimate.

To derive the ̂ , for assessing the influence of multiple observations
,
simultaneously on the parameter estimates, we need the weighted least
squares estimate of in (11). The weighted least squares criterion, which
should be minimized, is given by

() = ( − (, )) ()( − (, )).

There is generally no explicit solution to the normal equations and iterative
methods are used to find an estimate. The obtained estimate of is a function
̂
of the weights, , and is denoted (). The next theorem provides an explicit
expression of the ̂ defined in (12).
,

Theorem 3.1. Let ̂ be given in Definition 3.1. Then
,
−1
̂
̂
̂
̂

∗
̂ = ( ⊗ ()) (() () − ()(⨂ )) ,

,

provided that the inverse exists.

In the expression above, : × is a matrix with row vectors ,
∗
2

= ⊗ , = 1, . . . , , (13)

where is the th column of the identity matrix of size n. The quantities

̂
̂
, () and () are defined in (4), (5) and (6), respectively.

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