Page 39 - Special Topic Session (STS) - Volume 1

STS346 Abu Sayed M. et al.
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X
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w  1  | X  Med  |
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n nMAD X i
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It is easy to show that mean ( w ˆ ) = median ( w ) = 2/n.
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We consider several measures of the identification of high leverage points,
the twice the mean (2M) rule, the thrice-the-mean (3M) rule, and then
introduce a new cut-off point. Since it may not be easy to find the theoretical
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distribution of w and often excessive high leverage values can affect
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measures like mean and standard deviation, we define a confidence bound
type cut-off point
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w > Median( w ) + 3 MAD( w )
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which is analogous to forms used by Hadi (1992), Imon (2002,2005) and others.
In this paper, we consider five identification rules which are listed below:
Rule 1 (Classical 2M): w ˆ > 4/n
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Rule 2 (Classical 3M): w ˆ > 6/n
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Rule 3 (New 2M based on Median): w > 4/n
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Rule 4 (New 3M based on Median): w > 6/n
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Rule 5 (New Median based Cut-off point): w > Median( w ) + 3 MAD( w )
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We compare the performances of the above rules in terms of correct
identification of high leverage points and swamping rate of good leverages.

3. Results
We consider a real world data to investigate the performance of our
proposed method. In order to make the relationship as model (2), we assume
that measurement error can occur in either variable of these two examples.
The data is taken from Hand et al. (1994) where the data for 50 results of iron
content of crushed blast furnace slag measured by two different techniques,
which are chemical test (Y) and magnetic test (X). The X values are estimated
by the maximum likelihood formula (17). Now we compute the leverage values
for this data set. Here the cut-off point for rule 1 and 3 is 0.08, for rule 2 and
4 is 0.12 and rule 5 is 0.0975 respectively. We observed that the traditional
leverage values w ˆ do not identify any high leverage points, but the 2M rule
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swamps in 6 good cases. The newly proposed leverage measures w do not
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identify any high leverage points but the 2M rule swamps in 1 good case. The
3M rule does not identify any high leverage point for both of these two
leverage measures. We observe exactly the same performance from the rule
based on the new cut-off point as well. Now we modified the original iron in
slag data by inserting few high leverage points. We consider three different

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