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Statistics

Notes 29.1 Theory of Estimation

Let X be a random variable with probability density function (or probability mass function)
f(X ;  ,  , ....  ), where  ,  , ....  are k parameters of the population.
1 2 k 1 2 k
Given a random sample X , X , ...... X from this population, we may be interested in estimating
1 2 n
one or more of the k parameters  ,  , ......  . In order to be specific, let X be a normal variate so
1 2 k
that its probability density function can be written as N(X : , ). We may be interested in
estimating m or s or both on the basis of random sample obtained from this population.
It should be noted here that there can be several estimators of a parameter, e.g., we can have any
of the sample mean, median, mode, geometric mean, harmonic mean, etc., as an estimator of
1 2 1 2
population mean . Similarly, we can use either S = å (X - X ) or s = å (X - X ) as
i
i
n n - 1
an estimator of population standard deviation s. This method of estimation, where single statistic
like Mean, Median, Standard deviation, etc. is used as an estimator of population parameter, is
known as Point Estimation. Contrary to this it is possible to estimate an interval in which the
value of parameter is expected to lie. Such a procedure is known as Interval Estimation. The
estimated interval is often termed as Confidence Interval.

29.2 Point Estimation

As mentioned above, there can be more than one estimators of a population parameter. Therefore,
it becomes necessary to determine a good estimator out of a number of available estimators. We
may recall that an estimator, a function of random variables X , X , ...... X , is a random variable.
1 2 n
Therefore, we can say that a good estimator is one whose distribution is more concentrated
around the population parameter. R. A. Fisher has given the following properties of a good
estimators. These are:
(i) Unbiasedness (ii) Consistency (iii) Efficiency (iv) Sufficiency.

29.2.1 Unbiasedness

An estimator t (X , X , ...... X ) is said to be an unbiased estimator of a parameter q if E( t ) = .
1 2 n
If E( t )  q, then t is said to be a biased estimator of . The magnitude of bias = E( t ) – . We have
E X
seen in § 20.2 that ( ) = , therefore, X is said to be an unbiased estimator of population mean
n - 1 1 2
2
2
E S
m. Further, refer to § 20.4.1, we note that ( ) =  × 2 , where S = å (X - X ) . Therefore,
n n i
æ n - 1 ö 1
2 2 2 2
1  = -
S is a biased estimator of s . The magnitude of bias = ç - ÷  .
è n ø n
1 2
2
2
2
Contrary to this, if we define s = å (X - X ) , we have seen in § 20.4.1 that E(s ) =  . Thus,
n - 1 i
s is an unbiased estimator of s . Also from § 20.3.1 we note that E(p) = , therefore, p is an
2
2
unbiased estimator of .
29.2.2 Consistency
It is desirable to have an estimator, with a probability distribution, that comes closer and closer
to the population parameter as the sample size is increased. An estimator possessing this property

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