Page 37 - Special Topic Session (STS) - Volume 1

STS346 Abu Sayed M. et al.
distances are also suggested to use as measures of leverages in the literature,
however, Rousseeuw and Leroy (1987) showed that Mahalanobis distance for
each of the points has a one-one relationship with w and do not yield any
ii
extra information in the leverage structure of a data point. Hadi (1992) pointed
out that traditionally used measures of leverages are not sensitive enough to
the high leverage points. He introduced a single case deleted leverage
measure, named as potential, which is believed to be more sensitive to the
high leverage point. Imon and Khan (2003b) showed that in the presence of
multiple high leverage points, observations are masked in such a way that
even potential values may not focus on all of them. As a remedy to this
problem, Imon (2002) proposed generalized potentials for the identification
of multiple high leverage points in linear regression. Further developments of
the generalized potentials are done by Habshah et al. (2009) and Bagheri et
al. (2002). As we already know that in linear functional relation model the
explanatory variable is measured with error, or in other words is not fixed, we
cannot readily apply the leverage measures discussed in section 2 since they
are designed for fixed explanatory variable. In this section we obtain the
estimated values of X so that these values can be used as fixed-X in the
subsequent studies. Let us assume,
2
2
E( i )  E( i )  ,0 var( i )   , var( i )   , i 
cov( i , j )  cov( i , j )  ,0 i  j
cov( , )  , 0  i, j
i j (6)
Model (2) is also known as the unreplicated linear functional relationship when
there is only one relationship between the two variables X and Y .There are
2
( n ) 4 parameters to be estimated, which are , , 2 , and X , X ,..., X
1 2 n
. Several methods of parameter estimation have been developed (Fuller, 1987)
but our primary interest is the maximum likelihood (ML) method. Let (2) and
(6) hold, and that  and  are independent normal variables, viz.
i
i
 i ~ N , 0 (   2 ) and  i ~ N , 0 (   ) (7)
2
Since X i are non-random variables,  2 x  0 and there are ( n ) 4

parameters, namely , , 2 , and the n values of X to be estimated. the
2
i
estimator of  is derived as
2

1 
ˆ
ˆ  2    (x  X ) 2   (y  ˆ   X ˆ ˆ ) 2

2n i i i i
consistent. Kendall and Stuart (1979) showed a consistent
which is not
estimator of  can be derived by multiplying 2n to (15), that is
2

n  2
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