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STS346 A.H.M. Rahmatullah Imon
− ̂ (−) ∈
(−)
̂ (−) √(1 − ℎ )
∗
= (7)
− ̂ (−)
∈
̂ (−) √(1 − ℎ (−) )
{
̂ (−)
where ̂ (−) = and ̂ (−) are the fitted values of and the scale
parameter respectively after the omission of the suspected outlier group
indexed by . Although the expression of generalized potentials is available
for any arbitrary set of deleted cases, , the choice of such a set is clearly
important since the omission of this group determines the weights for the
whole set. We call an observation outlier when its corresponding generalized
Studentized residual value exceeds 3 in absolute value. No such value exists
for generalized potentials. We follow Hadi (1992) to declare an observation as
∗
a high leverage point if its corresponding exceeds a threshold given as
∗
∗
∗
> ( ) + 3 ( ). (8)
where MAD stands for the median absolute deviation.
These above results enable us to define a simple graphical display of
classifying group deleted leverages and residuals for possible identification of
them. Generalized potentials are used as leverages and the generalized
Studentized residuals as deletion residuals in a ‘generalized potentials –
generalized Studentized residuals (GPGSR)’ plot. Since the high leverage
points need not to be outliers and outliers may not be points of high leverage
we may expect different deletion sets D from the computation of these two
quantities. Since D is the group of suspected outliers we prefer to include all
observations considered to be suspect either along the y dimension or along
the dimension. We employ the blocked adaptive computationally-efficient
outlier nominators (BACON) proposed by Billor et al. (2000) as a classifier.
Another possibility could be the application of support vector regression for
the same, especially when the data is big. The main advantage of the GPGSR
plot is that it is suitable for the data where masking (false negative) and/or
swamping (false positive) make single case diagnostic plots misleading. This
plot, unlike the L-R plot retains the signs of residuals, which can be very
important when their interpretation is concerned. Since the bulk of the cases
will be associated with low leverage and small residuals, most of the pairs
∗
( , ) will cluster near the origin (0, 0). The unusual cases will have either
∗
high leverages or large residual components and will tend to be separated
from the bulk of the cases. High leverage cases will be located at the right
corner of the plot and observations with large residuals will be located either
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