Maximum Likelihood Estimation of Normal
Distribution
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The probability density (likelihood) of a single variable
drawn from a Normal distribution is:
Thus the likelihood of independent
draws from
this distribution is:
To identify the values of and that maximise it is easiest to
identify the maximum of the log likelihood (as the two will be maximised for
the same values of and since is
a monotonically increasing function of ).
The log likelihood is:
This is maximised when and
,
i.e. when:
The maximum likelihood estimator, ,
of the mean is thus the average of the
observations, .
It is possible to show that this is also the minimum variance unbiased
estimator of . The maximum likelihood estimator,
,
of is
the population standard deviation, of
the which
can be determined using the MnPopulationStdev
web function. Please note that whilst is
an unbiased estimator of , is
a biased estimator of . The minimum variance unbiased
estimator of is
the sample variance (i.e. square of the sample standard
deviation). The sample standard deviation is: