Page 28 - Special Topic Session (STS) - Volume 1
P. 28
STS346 A.H.M. Rahmatullah Imon
The − plot may be effective in the identification of single outlier but it
may be ineffective in the presence of multiple outliers unless we remove a
group of suspect outliers prior fitting the model. Denote a set of cases
‘remaining’ in the analysis by and a set of cases ‘deleted’ by . Also suppose
that R contains ( − ) cases after < ( − ) cases in are deleted. Without
loss of generality, assume that these observations are the last rows of and
so that we can partition the matrices as
= [ ] , = [ ] , = [ ]
−1
−1
where = ( ) and = ( ) and symmetric
−1
matrices of order ( − ) and respectively, and = ( ) is an
( − ) × matrix. However, ( ) −1 can be expressed as
−1
−1
−1
( ) −1 = ( − ) −1 = ( ) −1 + ( ) ( − ) ( ) (4)
where is an identity matrix of order . Using (4), Imon (2002) defined a
group deleted version of high leverage points called generalized potentials
defined as
(−)
ℎ ∈
∗
= 1 − ℎ (−) (5)
{ ℎ (−) ∈
−1
where ℎ (−) = ( ) , = 1,2, … … , . In other words ℎ (−) is the −
ℎ diagonal element of ( ) matrix. The vector of estimated
−1
parameters after the deletion of observations, denoted by ̂ (−) , is obtained
using (4) as
̂
−1
−1
−1
̂ (−) = ( ) = − ( ) ( − ) ̂ (6)
̂
where ̂ = − . Using (4), (5) and (6), Imon (2005) introduced a
group deleted version of resdiuals called generalized Studentized
residuals (GSR) defined as
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