Page 395 - DMTH404_STATISTICS

Unit 29: Estimation of Parameters: Criteria for Estimates

Notes
2 2
= å (X - X ) + ( n X - ) (last term is zero)
i
nS= 2 + ( n X - ) 2

1 2 n n 2
2
Therefore, we can write - 2 å (X - )  = - S - (X - ) .
i
2 2 2 2 2
Hence ( f X 1 ; , ) ´ ( f X 2 ; , ) .... ´ ´ ( f X n ; , )



n n n 2 n 2 n n
2
æ 1 ö - 2 S - 2 (X - ) - 2 (X - ) æ 1 ö - 2 S 2
= ç ÷ e 2 2 = e 2 ´ ç ÷ e 2
è  2 ø è  2 ø
= g ( , ,X   ) ( ,h S ´ 2 )
2
Since h is independent of m, therefore X is a sufficient estimator of . However, S is not
2
sufficient estimator of  because g is not independent of .
1 2
2
Further, if we define S = å (X -  ) , then
n i
n n 2
æ 1 ö - 2 S
( f X 1 ; , ) ´ ( f X 2 ; , ) .... ´ ´ ( f X n ; , )  = ç è  2 ø ÷ e 2


2
2
Thus, the newly defined S becomes a sufficient estimator of  . We note that h(X , X , ...... X ) =
1 2 n
1 in this case.
1 2
2
The above result suggests that if m is known, then we should use S = å (X - )  rather than
n i
1 2
2
S = å (X - X ) because former is better estimator of s .
2
n i
29.2.5 Methods of Point Estimation
Given various criteria of a good estimator, the next logical step is to obtain an estimator possessing
some or all of the above properties.
There are several methods of obtaining a point estimator of the population parameter. For
example, we can use the method of maximum likelihood, method of least squares, method of
minimum variance, method of minimum c 2 , method of moments, etc. We shall, however, use
the most popular method of maximum likelihood.

Method of Maximum Likelihood

Let X , X , ...... X be a random sample of n independent observations from a population with
1 2 n
probability density function (or p.m.f.) f(X; ), where  is unknown parameter for which we
desire to find an estimator.
Since X , X , ...... X are independent random variables, their joint probability function or the
1 2 n
probability of obtaining the given sample, termed as likelihood function, is given by

LOVELY PROFESSIONAL UNIVERSITY 387