Page 324 - DMTH404_STATISTICS

Statistics

Notes particular value of X say X is called error in estimation of the i th observation on the assumption
i
of a particular line of regression. There will be similar type of errors for all the n observations.
We denote by e = Y - Y (i = 1,2,.....n), the error in estimation of the i th observation. As is
i i Ci
obvious from figure 23.1, e will be positive if the observed point lies above the line and will be
i
negative if the observed point lies below the line. Therefore, in order to obtain a figure of total
s
error, e ' are squared and added. Let S denote the sum of squares of these errors, i.e.,
i
n 2 n 2
Y -
S = å e = å ( i Y Ci ) .
i
i 1 i 1
=
=
Figure 23.1

Note The regression line can, alternatively, be written as a deviation of Y from Y
i ci
i.e. Y – Y = e or Y = Y + e or Y = a + bX + e . The component a + bX is known as the
i ci i i ci i i i i i
deterministic component and e is random component.
i
The value of S will be different for different lines of regression. A different line of regression
means a different pair of constants a and b. Thus, S is a function of a and b. We want to find such
values of a and b so that S is minimum. This method of finding the values of a and b is known as
the Method of Least Squares.

Rewrite the above equation as S = S(Y - a - bX ) ( Y = a + bX ).
2
i i Ci i
The necessary conditions for minima of S are
¶ S ¶ S ¶ S ¶ S
(i) = 0 and (ii) = 0 , where and are the partial derivatives of S w.r.t. a and b
a ¶ ¶ b a ¶ b ¶
respectively.

¶ S n
Now = - 2å ( i a bX i ) = 0
Y -
-
a ¶ i 1
=
n n n
-
or å ( i a bX i ) = å Y - na b å X = 0
-
Y -
i
i
=
=
i 1 i 1 i 1
=
n n
+
or å Y = na bå X i .... (1)
i
i 1 i 1
=
=
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