Confusion regarding OPEF and CRLB
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I am a little confused on this ,
We know that under suitable regularity conditions,the Cramer-Rao lower bound is attained by the variance of an unbiased estimator $T(X)$ of $g(theta)$ iff the family of distributions of $x$ is an One parameter exponential family.
Now, suppose , $X sim P(lambda)$.
Clearly, the family $(P(lambda): lambda>0)$ is an OPEF and it satisfies all the regularity conditions,then for any unbiased estimator $T$ of $e^{-lambda}$, $V(T)$ does not attain Cramer Rao lower bound.
Is this not contradictory?
Or am I missing something?
Help!
probability probability-theory statistics probability-distributions parameter-estimation
asked Dec 5 '18 at 15:21
Legend KillerLegend Killer
1,6061523
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add a comment |
$begingroup$
I am a little confused on this ,
We know that under suitable regularity conditions,the Cramer-Rao lower bound is attained by the variance of an unbiased estimator $T(X)$ of $g(theta)$ iff the family of distributions of $x$ is an One parameter exponential family.
Now, suppose , $X sim P(lambda)$.
Clearly, the family $(P(lambda): lambda>0)$ is an OPEF and it satisfies all the regularity conditions,then for any unbiased estimator $T$ of $e^{-lambda}$, $V(T)$ does not attain Cramer Rao lower bound.
Is this not contradictory?
Or am I missing something?
Help!
probability probability-theory statistics probability-distributions parameter-estimation
asked Dec 5 '18 at 15:21
Legend KillerLegend Killer
1,6061523
$endgroup$
$begingroup$
The condition you mention at the beginning is definitely not 'iff', as shown by your example. Starting from the equality condition of the Cramer-Rao inequality one gets OPEF, not the other way around. And indeed, variance of an unbiased estimator/UMVUE need not attain CRLB in general.
$endgroup$
– StubbornAtom
Dec 5 '18 at 19:34
add a comment |
$begingroup$
I am a little confused on this ,
We know that under suitable regularity conditions,the Cramer-Rao lower bound is attained by the variance of an unbiased estimator $T(X)$ of $g(theta)$ iff the family of distributions of $x$ is an One parameter exponential family.
Now, suppose , $X sim P(lambda)$.
Clearly, the family $(P(lambda): lambda>0)$ is an OPEF and it satisfies all the regularity conditions,then for any unbiased estimator $T$ of $e^{-lambda}$, $V(T)$ does not attain Cramer Rao lower bound.
Is this not contradictory?
Or am I missing something?
Help!
probability probability-theory statistics probability-distributions parameter-estimation
asked Dec 5 '18 at 15:21
Legend KillerLegend Killer
1,6061523
$endgroup$
I am a little confused on this ,
We know that under suitable regularity conditions,the Cramer-Rao lower bound is attained by the variance of an unbiased estimator $T(X)$ of $g(theta)$ iff the family of distributions of $x$ is an One parameter exponential family.
Now, suppose , $X sim P(lambda)$.
Clearly, the family $(P(lambda): lambda>0)$ is an OPEF and it satisfies all the regularity conditions,then for any unbiased estimator $T$ of $e^{-lambda}$, $V(T)$ does not attain Cramer Rao lower bound.
Is this not contradictory?
Or am I missing something?
Help!
probability probability-theory statistics probability-distributions parameter-estimation
probability probability-theory statistics probability-distributions parameter-estimation
asked Dec 5 '18 at 15:21
Legend KillerLegend Killer
1,6061523
asked Dec 5 '18 at 15:21
Legend KillerLegend Killer
1,6061523
edited Dec 5 '18 at 17:34
Legend Killer
asked Dec 5 '18 at 15:21
Legend KillerLegend Killer
1,6061523
asked Dec 5 '18 at 15:21
Legend KillerLegend Killer
1,6061523
asked Dec 5 '18 at 15:21
Legend KillerLegend Killer
1,6061523
1,6061523
$begingroup$
The condition you mention at the beginning is definitely not 'iff', as shown by your example. Starting from the equality condition of the Cramer-Rao inequality one gets OPEF, not the other way around. And indeed, variance of an unbiased estimator/UMVUE need not attain CRLB in general.
$endgroup$
– StubbornAtom
Dec 5 '18 at 19:34
add a comment |
$begingroup$
The condition you mention at the beginning is definitely not 'iff', as shown by your example. Starting from the equality condition of the Cramer-Rao inequality one gets OPEF, not the other way around. And indeed, variance of an unbiased estimator/UMVUE need not attain CRLB in general.
$endgroup$
– StubbornAtom
Dec 5 '18 at 19:34
$begingroup$
The condition you mention at the beginning is definitely not 'iff', as shown by your example. Starting from the equality condition of the Cramer-Rao inequality one gets OPEF, not the other way around. And indeed, variance of an unbiased estimator/UMVUE need not attain CRLB in general.
$endgroup$
– StubbornAtom
Dec 5 '18 at 19:34
$begingroup$
The condition you mention at the beginning is definitely not 'iff', as shown by your example. Starting from the equality condition of the Cramer-Rao inequality one gets OPEF, not the other way around. And indeed, variance of an unbiased estimator/UMVUE need not attain CRLB in general.
$endgroup$
– StubbornAtom
Dec 5 '18 at 19:34
add a comment |
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$begingroup$
The condition you mention at the beginning is definitely not 'iff', as shown by your example. Starting from the equality condition of the Cramer-Rao inequality one gets OPEF, not the other way around. And indeed, variance of an unbiased estimator/UMVUE need not attain CRLB in general.
$endgroup$
– StubbornAtom
Dec 5 '18 at 19:34