Page 404 - DMTH404_STATISTICS

Statistics

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estimator of population mean . Similarly, we can use either S = å (X - X ) or
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s = å (X - X ) as an estimator of population standard deviation s. This method of
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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.
 The maximum likelihood estimators are consistent.
 The maximum likelihood estimators are not necessarily unbiased. If a maximum likelihood
estimator is biased, then by slight modifications it can be converted into an unbiased
estimator.
 If a maximum likelihood estimator is unbiased, then it will also be most efficient.

 A maximum likelihood estimator is sufficient provided sufficient estimator exists.
 The maximum likelihood estimators are invariant under functional transformation, i.e., if
t is a maximum likelihood estimator of , then f(t) would be maximum likelihood estimator
of f().

29.5 Keywords

Estimation: It is a procedure by which sample information is used to estimate the numerical
magnitude of one or more parameters of the population.
Cramer Rao Inequality: This inequality gives the minimum possible value of the variance of an
unbiased estimator.

Estimator: An estimator t is said to be a sufficient estimator of parameter  if it utilises all the
information given in the sample about .

29.6 Self Assessment

1. State whether the following statements are True or False:
(i) Sample mean is an unbiased estimator of population mean.
(ii) Sample standard deviation is an unbiased estimator of population standard deviation.
(iii) An estimator whose variance tends to zero as sample size tends to infinity is called
a consistent estimator.
(iv) An efficient estimator may or may not be unbiased.
(v) A sufficient estimator is always consistent.
(vi) The width of the confidence interval depends upon the level of significance as well
as on the sample size.

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