Asymptotic Noramlity of Maximum Likelihood Estimator and the distribution of Lagrange Multiplier

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Asymptotic Noramlity of Maximum Likelihood Estimator and the
distribution of Lagrange Multiplier and Likelihood Ratio test statistic
This article gives the proof of the asymptotic normality of maximum likelihood estimator and the distribution of LM
and LR statistics, which are frequenltly used in application of econometrics.
TheoremLet yi(i = 1;;n)be the independently and identically distributed with probability density f(yi;0)
characterized by0.And definethe likelihood functionto be L(y;)=
n
i= 1 f(yi;0).Now letˆ bethe maximum
likelihood estimator;which maximizes L(y;0),such thatplimˆ = 0,where0 is the true value of parameter.
Then under the regurality conditions, ˆ has asymptotic normalityˆ
a
N(0;E[I (0)] 1);where E[I (0)] =
E0[ 2logL= 0 0
0].
Regularity Conditions
(i)The first,second and third derivatives oflog f(yi;0)with respect to0 exist,are continuous and finite for
almost all yi and for all.
(ii)The conditions to obtain the expectaions of the first and second derivatives oflog f(yi;0)are met.
(iii)For all values of,j 3log f(yi;0)=q jq kq l j is less than a function that has a finite expectation.
Proof
Now under some assumptions, we can interchange the operations of integration and differentiation.
0
Z
f(yi;0)dyi =
Z
0
f(yi;0)dyi
=
Z
f(yi;0)
log f(yi;0)
0
dyi
=
Z
log f(yi;0)
0
f(yi;0)dyi
= E0
h
log f(yi;0)
0
i
= 0:
(1)
In order to obtain the result above, we used the following.
f(yi;0)
0
=
exp(log f(yi;0))
0
=
exp(log f(yi;0))
log f(yi;0)
log f(yi;0)
0
= exp(log f(yi;