Page 38 - Special Topic Session (STS) - Volume 1
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STS346 Abu Sayed M. et al.
1
ˆ (x i X ˆ i ) 2 (y i ˆ X ˆ ˆ i ) 2
2
n 2
and we obtain the estimated values of X as
ˆ
X i x i ˆ ( y i ) ˆ
( )
ˆ 2
y ˆ ˆ x and
where
X ˆ y ˆ X
ˆ
ˆ
i i i
X i 2
ˆ
ˆ
ˆ
ˆ
( 2 )( S xy S yy n 3 x 2 n ˆ x 2 )
ˆ
ˆ
2 y i 2 2 ˆ S xy 2 S yy n 4 x 2 2n ˆ 2 x 2
ˆ
gives S ( S S ) S 0 and that implies
ˆ 2
xy xx yy xy
S )
2
S ( S ( S ) 4 S 2
ˆ
yy xx yy xx xy
S 2 xy
y x
where, y i , x i ,S y 2 yn 2 ,S x 2 xn 2 and
n n yy i xx i
S xy x i y i xn y
Identification of High Leverage Points in LFRM
In this section we suggest a procedure for the identification of high
leverage points in linear functional relation model. From a set of observed x
and y (both assumed to be measured with error), we have estimated the
fixed-X
ˆ
X x i ˆ ( y i ) ˆ
i
( )
ˆ 2
Since we have a single explanatory variable here, the formula (5) for the
computation of leverage values can be simplified as
2
ˆ
ˆ
ˆ
T
w ˆ x ( X T X) 1 x ˆ = 1 X ˆ i X ˆ
ii
i
i
n n 2
X ˆ i X ˆ
i 1
Since the above formula contains mean and sum of squares of means which
could be very sensitive to high leverage points. For this reason we propose a
new formula for the leverages analogous to formula (20), but here the non-
robust components are replaced by their corresponding robust alternatives.
Hence the formula is
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