Estimation of parameters of a normal distribution
Here we show that the knowledge of the distribution of for linear regression allows one to do without long calculations contained in the guide ST 2134 by J. Abdey.
Theorem. Let be independent observations from
. 1)
is distributed as
2) The estimators
and
are independent. 3)
4)
5)
converges in distribution to
Proof. We can write where
is distributed as
Putting
and
(a vector of ones) we satisfy (1) and (2). Since
we have
Further,
and
Thus 1) and 2) follow from results for linear regression.
3) For a normal variable its moment generating function is
(see Guide ST2133, 2021, p.88). For the standard normal we get
Applying the general property (same guide, p.84) we see that
Therefore
4) By independence of standard normals
5) By standardizing we have
and this converges in distribution to
by the central limit theorem.
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