Page 394 - DMTH404_STATISTICS

Statistics

Notes Let X , X , ...... X be a random sample of n independent observations from a population with
1 2 n
p.d.f. (or p.m.f.) given by f(X;  ,  ), where q and q are two parameters. The joint probability
1 2 1 2
distribution of X , X , ...... X , denoted by L(X;  ,  ) is given by :
1 2 n 1 2
L(X;  ,  ) = f(X ;  ,  ) × f(X ;  ,  ) × ...... × f(X ;  ,  )
1 2 1 1 2 2 1 2 n 1 2
An estimator t is said to be sufficient for q if the conditional p.d.f. (or p.m.f.) of X , X , ...... X
1 1 2 n
given t is independent of q , i.e.,
1


( f X 1 ; , 2 ) ´ ( f X 2 ; , 2 ) .... ´ ´ ( f X n ; ,  2 ) = ( h X X , .... X ) , where g(t, q ) is p.d.f.

1
1
1
,
g ( ,t  1 ) 1 2 n 1
(or p.m.f.) of t and h is a function of sample values that is independent of  . We may note that
1
each of the functions g(t,  ) and h(X , X , ...... X ) may or may not be function of  .
1 1 2 n 2
Alternatively, we can write the sufficiency condition as
f(X ;  ,  ) × f(X ;  ,  ) × ...... × f(X ;  ,  ) = g(t, q ) × h(X , X , ...... X ), which implies that if the
1 1 2 2 1 2 n 1 2 1 1 2 n
joint p.d.f. (or p.m.f.) of X , X , ...... X can be written as a function of t and  multiplied by a
1 2 n 1
function independent of  , then t is sufficient estimator of  .
1 1
Sufficient estimators are the most desirable but are not very commonly available. The following
points must be noted about sufficient estimators:
1. A sufficient estimator is always consistent.
2. A sufficient estimator is most efficient if an efficient estimator exists.
3. A sufficient estimator may or may not be unbiased.

Example 1: If X , X , ...... X is a sample of n independent observations from a normal
1 2 n
population with mean m and variance s , show that X is a sufficient estimator of m but
2
1 2
2
S = å (X - X ) is not sufficient estimator of s .
2
i
n
Solution.
The probability density function of a normal variate is given by
1 - 1 (X - ) 2
)
f ( ; ,X   = e 2 2
 2
Thus, the joint probability density function of X , X , ...... X is given by
1 2 n
n
1
-
å
f X ; ,g b 2  ´ f X ; ,g = F G  H 1 I J n - 2 2 H F X i  I K 2
n b
1 b
2 K
f X ; ,g ´ ....
e
=
´

i 1

We can write X -  = (X - X ) (X + - ) .
i i
Squaring both sides and taking sum over n observations, we get
2
2
å (X - )  2 = å (X - X ) + å (X - ) + å (X - X )(X - )
2
i
i
i
2 2
= å (X - X ) + ( n X - ) + ( 2 X - å i X )
) (X -
i
386 LOVELY PROFESSIONAL UNIVERSITY