Page 398 - DMTH404_STATISTICS

Statistics

Notes 1 1
ö
= å æ ç - log - log 2 - 2 å (X - )  2
÷
è 2 ø 2 i
n 1 2
= - n log - log 2 - 2 å (X -  ) .... (1)
2 2 i
(i) MLE of m

¶ log L 1 å X i
2 ×
= 2 å (X -  ) 0 or = å (X -  ) 0 = Þ  = X
ˆ =
i
i
¶ 2 n
(ii) MLE of s 2
2
Rewriting equation (1) as a function of s , we get
n n 1 2
2
log L = - log - log 2 - 2 å (X -  )
i
2 2 2
¶ log L n å (X - )  2 2
2
-
 = - + i = 0 or n + å (X - )  = 0
¶ 2 2 2 2 4 i
2
å (X - ) 
ˆ =
Þ  2 i
n
29.3 Interval Estimation
Using point estimation, it is possible to provide a single quantity as an estimator of a parameter.
Any point estimator, even if it satisfies all the characteristics of a good estimator, has a limitation
that it provides no information about the magnitude of errors due to sampling. This problem is
taken care of by the method of interval estimation, that gives a range of the estimator of the
parameter.

The method of interval estimation is based upon the sampling distribution of an estimator. The
standard error of the estimator is used in the construction of an interval so that the probability
of the parameter lying within the interval can be specified.
Given a random sample of n observations X , X , ...... X , we can find two values l and l such that
1 2 n 1 2
the probability of population parameter q lying between l and l is (say) h. Using symbols, we
1 2
can write P(l £ q £ l ) = h.
1 2
Such an interval is termed as a Confidence Interval for q and the two limits l and l are termed
1 2
as Confidential or Fiducial Limits. The percentage probability or confidence is termed as the
Level of Confidence or Confidence Coefficient of the interval. For example, the level of confidence
of the above interval is 100h%. The level of confidence implies that if a large number of random
samples are taken from a population and confidence intervals are constructed for each, then
100h% of these intervals are expected to contain the population parameter q. Alternatively, a 100
h% confidence interval implies that we are 100 h% confident that the population parameter q lies
between l and l .
1 2
As compared to point estimation, the interval estimation is better because it takes into account
the variability of the estimator in addition to its single value and thus, provides a range of
values. Unlike point estimation, interval estimation indicates that estimation is an uncertain
process.

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