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STS346 Abu Sayed M. et al.
situations. In case 1, 5 low leverage cases (10%) are replaced by high leverage
points. In case 2 and case 3 we replace 20% and 30% low leverage points by
points of high leverages respectively. Now we compute the leverage values for
this data set. Here the cut-off point for rule 1 and 3 is 0.08 and for rule 2 and
4 is 0.12 as they were before. The cut-off points for rule 5 are 0.1052, 0.1024,
and 0.0845 for 10%, 20% and 30% high leverage points respectively. We
observed from that for the 10% contamination, the traditional leverage values
w ˆ can identify high leverage points successfully, but their performances tend
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to deteriorate with the increase in the level of contamination. For 20%
contamination it fails to identify 4 high leverage cases out of 10 and for the
30% contamination it fails to identify 10 out of 15 high leverage points. The
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newly proposed leverage measures w perform very well in this regard. All
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high leverage points are successfully identified irrespective of the level of
contamination.
In this section we report a Monte Carlo simulation which is designed to
investigate the performances of different measures of leverages in linear
functional relation model. For four different sample sizes, n = 20, 30, 50 and
100, we generated the X values from Uniform (20, 40). Here we consider three
different percentages, i.e., 10%, 20%, and 30% high leverage points. The X
value corresponding to the lowest high leverage value is then set at 100 and
the next values have an increment of 5 each. To generate a model like (2), we
then define x  X   i , where  is N (0, 1). The values of y are generated
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as y  20  2 X   , where  is also N (0, 1). For each different sample we
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apply all five leverage identification rules mentioned in section 4 and compute
the correct identification rate (IR) and the swamping rate (SR) in terms of
percentages. We run 10,000 simulations for each combination. When no high
leverage point exists, we observe from the above table that for n = 20, all
methods considered in the simulation perform well. However, rule 1, i.e., the
traditional leverage measure based on the 2M rule has about 5% swamping
rate. The newly proposed rule 4 performs the best as its swamping rate is the
lowest followed by rule 2, rule 5 and rule 3. The performance of all these rules
tend to improve with the increase in sample sizes but still rule 1 has relatively
very high swamping rate which clearly shows that the 2M rule is too prone to
declare low leverage points as points of high leverages. In case of 10% high
leverages, almost all methods perform very well. Each method maintains 100%
identification rate with low swamping rate. Only when n = 100, the
identification rate for rule 2 is 90%. But when 20% or 30% high leverage points
are present in the data both the 2M and the 3M rule break down. The rule 2,
i.e., the 3M performs worst as often its correct identification rate is 0%. The
performance of the rule 1 is also poor as it can identify around 13% cases

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