Mixing
Equivalence of two MLE estimators with a constraint
$begingroup$
Consider the following objective function $Sleft( theta ,w|Lambda right)
.$ The value of $Sleft( theta ,w|Lambda right) $ depends on the nuisance
parameters $Lambda .$ However, the first order conditions of $Sleft(
theta ,w|Lambda right) $ is $
[
frac{partial Sleft( theta ,w|Lambda right) }{partial theta }=0text{
and }frac{partial Sleft( theta ,w|Lambda right) }{partial w}=0,
]%
$which will give $w^{T}Lambda =0$ (assume the multiplication is admissible)$.
$ Substituting $w^{T}Lambda =0$ back to the origirinal objective function,
it becomes $Sleft( theta |wright) ,$ which no longer a function of $%
Lambda .$ My question is whether the solution of $arg min Sleft( theta
,w|Lambda right) $ and $arg min Sleft( theta |wright) $ are
equivalent for $theta $ and $w,$ i.e., let
begin{eqnarray*}
left( hat{theta},hat{w}right) &=&underset{theta ,w}{arg min }%
Sleft( theta ,w|Lambda right) \
left( hat{theta}^{ast },hat{w}^{ast }right) &=&underset{theta ,w}{%
arg min }Sleft( theta ,w|Lambda right) text{ subject to }w^{T}Lambda
=0,
end{eqnarray*}
then is it$%
[
hat{theta}=hat{theta}^{ast }text{ and }hat{w}=hat{w}^{ast }.
]%
$Any reference on this? Thanks
maximum-likelihood constraints
asked Feb 1 at 3:31
Charles ChouCharles Chou
869
$endgroup$
add a comment |
$begingroup$
Consider the following objective function $Sleft( theta ,w|Lambda right)
.$ The value of $Sleft( theta ,w|Lambda right) $ depends on the nuisance
parameters $Lambda .$ However, the first order conditions of $Sleft(
theta ,w|Lambda right) $ is $
[
frac{partial Sleft( theta ,w|Lambda right) }{partial theta }=0text{
and }frac{partial Sleft( theta ,w|Lambda right) }{partial w}=0,
]%
$which will give $w^{T}Lambda =0$ (assume the multiplication is admissible)$.
$ Substituting $w^{T}Lambda =0$ back to the origirinal objective function,
it becomes $Sleft( theta |wright) ,$ which no longer a function of $%
Lambda .$ My question is whether the solution of $arg min Sleft( theta
,w|Lambda right) $ and $arg min Sleft( theta |wright) $ are
equivalent for $theta $ and $w,$ i.e., let
begin{eqnarray*}
left( hat{theta},hat{w}right) &=&underset{theta ,w}{arg min }%
Sleft( theta ,w|Lambda right) \
left( hat{theta}^{ast },hat{w}^{ast }right) &=&underset{theta ,w}{%
arg min }Sleft( theta ,w|Lambda right) text{ subject to }w^{T}Lambda
=0,
end{eqnarray*}
then is it$%
[
hat{theta}=hat{theta}^{ast }text{ and }hat{w}=hat{w}^{ast }.
]%
$Any reference on this? Thanks
maximum-likelihood constraints
asked Feb 1 at 3:31
Charles ChouCharles Chou
869
$endgroup$
add a comment |
$begingroup$
Consider the following objective function $Sleft( theta ,w|Lambda right)
.$ The value of $Sleft( theta ,w|Lambda right) $ depends on the nuisance
parameters $Lambda .$ However, the first order conditions of $Sleft(
theta ,w|Lambda right) $ is $
[
frac{partial Sleft( theta ,w|Lambda right) }{partial theta }=0text{
and }frac{partial Sleft( theta ,w|Lambda right) }{partial w}=0,
]%
$which will give $w^{T}Lambda =0$ (assume the multiplication is admissible)$.
$ Substituting $w^{T}Lambda =0$ back to the origirinal objective function,
it becomes $Sleft( theta |wright) ,$ which no longer a function of $%
Lambda .$ My question is whether the solution of $arg min Sleft( theta
,w|Lambda right) $ and $arg min Sleft( theta |wright) $ are
equivalent for $theta $ and $w,$ i.e., let
begin{eqnarray*}
left( hat{theta},hat{w}right) &=&underset{theta ,w}{arg min }%
Sleft( theta ,w|Lambda right) \
left( hat{theta}^{ast },hat{w}^{ast }right) &=&underset{theta ,w}{%
arg min }Sleft( theta ,w|Lambda right) text{ subject to }w^{T}Lambda
=0,
end{eqnarray*}
then is it$%
[
hat{theta}=hat{theta}^{ast }text{ and }hat{w}=hat{w}^{ast }.
]%
$Any reference on this? Thanks
maximum-likelihood constraints
asked Feb 1 at 3:31
Charles ChouCharles Chou
869
$endgroup$
Consider the following objective function $Sleft( theta ,w|Lambda right)
.$ The value of $Sleft( theta ,w|Lambda right) $ depends on the nuisance
parameters $Lambda .$ However, the first order conditions of $Sleft(
theta ,w|Lambda right) $ is $
[
frac{partial Sleft( theta ,w|Lambda right) }{partial theta }=0text{
and }frac{partial Sleft( theta ,w|Lambda right) }{partial w}=0,
]%
$which will give $w^{T}Lambda =0$ (assume the multiplication is admissible)$.
$ Substituting $w^{T}Lambda =0$ back to the origirinal objective function,
it becomes $Sleft( theta |wright) ,$ which no longer a function of $%
Lambda .$ My question is whether the solution of $arg min Sleft( theta
,w|Lambda right) $ and $arg min Sleft( theta |wright) $ are
equivalent for $theta $ and $w,$ i.e., let
begin{eqnarray*}
left( hat{theta},hat{w}right) &=&underset{theta ,w}{arg min }%
Sleft( theta ,w|Lambda right) \
left( hat{theta}^{ast },hat{w}^{ast }right) &=&underset{theta ,w}{%
arg min }Sleft( theta ,w|Lambda right) text{ subject to }w^{T}Lambda
=0,
end{eqnarray*}
then is it$%
[
hat{theta}=hat{theta}^{ast }text{ and }hat{w}=hat{w}^{ast }.
]%
$Any reference on this? Thanks
maximum-likelihood constraints
maximum-likelihood constraints
asked Feb 1 at 3:31
Charles ChouCharles Chou
869
asked Feb 1 at 3:31
Charles ChouCharles Chou
869
asked Feb 1 at 3:31
Charles ChouCharles Chou
869
asked Feb 1 at 3:31
Charles ChouCharles Chou
869
asked Feb 1 at 3:31
Charles ChouCharles Chou
869
869
add a comment |
add a comment |
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