What lemma for product of derivatives equals the n-derivative?
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Why must one require regularity in order for Fisher information to be $Ebigg( frac{partial^2 l(theta, X)}{partial theta^2} bigg)$?
Rather than
$Ebigg( frac{partial l(theta, X)}{partial theta}frac{partial l(theta, X)}{partial theta}^T bigg)$
Since my notes say that "under sufficient regularity conditions", then:
$I(theta)=Ebigg( frac{partial^2 l(theta, X)}{partial theta^2} bigg)$
However, what's the lemma that says that the product equals 2nd derivative?
derivatives fisher-information
asked Nov 21 at 9:23
mavavilj
2,6501932
add a comment |
up vote
0
down vote
favorite
Why must one require regularity in order for Fisher information to be $Ebigg( frac{partial^2 l(theta, X)}{partial theta^2} bigg)$?
Rather than
$Ebigg( frac{partial l(theta, X)}{partial theta}frac{partial l(theta, X)}{partial theta}^T bigg)$
Since my notes say that "under sufficient regularity conditions", then:
$I(theta)=Ebigg( frac{partial^2 l(theta, X)}{partial theta^2} bigg)$
However, what's the lemma that says that the product equals 2nd derivative?
derivatives fisher-information
asked Nov 21 at 9:23
mavavilj
2,6501932
add a comment |
up vote
0
down vote
favorite
up vote
0
down vote
favorite
Why must one require regularity in order for Fisher information to be $Ebigg( frac{partial^2 l(theta, X)}{partial theta^2} bigg)$?
Rather than
$Ebigg( frac{partial l(theta, X)}{partial theta}frac{partial l(theta, X)}{partial theta}^T bigg)$
Since my notes say that "under sufficient regularity conditions", then:
$I(theta)=Ebigg( frac{partial^2 l(theta, X)}{partial theta^2} bigg)$
However, what's the lemma that says that the product equals 2nd derivative?
derivatives fisher-information
asked Nov 21 at 9:23
mavavilj
2,6501932
Why must one require regularity in order for Fisher information to be $Ebigg( frac{partial^2 l(theta, X)}{partial theta^2} bigg)$?
Rather than
$Ebigg( frac{partial l(theta, X)}{partial theta}frac{partial l(theta, X)}{partial theta}^T bigg)$
Since my notes say that "under sufficient regularity conditions", then:
$I(theta)=Ebigg( frac{partial^2 l(theta, X)}{partial theta^2} bigg)$
However, what's the lemma that says that the product equals 2nd derivative?
derivatives fisher-information
derivatives fisher-information
asked Nov 21 at 9:23
mavavilj
2,6501932
asked Nov 21 at 9:23
mavavilj
2,6501932
edited Nov 21 at 9:47
asked Nov 21 at 9:23
mavavilj
2,6501932
asked Nov 21 at 9:23
mavavilj
2,6501932
asked Nov 21 at 9:23
mavavilj
2,6501932
2,6501932
add a comment |
add a comment |
1 Answer
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My impression is that you got something wrong here. Note that $$frac{partial^2 l(theta, X)}{partial theta^2} $$ is a second derivative, while $$frac{partial l(theta, X)}{partial theta}frac{partial l(theta, X)}{partial theta}^T$$ is a product of two first-order derivatives. As such, they are surely not equal in general: consider for example the function $$l(theta,X) = theta cdot X.$$
Try reading the wikipedia page on the Fisher information and compare it with your notes.
answered Nov 21 at 9:44
user159517
4,248930
Then what's a lemma that says that they can be equal?
– mavavilj
Nov 21 at 9:46
@mavavilj I wouldn't look for such a lemma (it would mean that $l$ has to satisfy a certain differential equation for all $X$, which is a very strict limitation), I'd rather try to understand what was meant in your notes. What is the definition of $l$?
– user159517
Nov 21 at 9:49
The wikipedia says same as my notes: "If log f(x; θ) is twice differentiable with respect to θ, and under certain regularity conditions,[4] then the Fisher information may also be written as". But I want to know what the lemma is that allows for that equality.
– mavavilj
Nov 21 at 9:52
Cramer-Rao bound? stat.tamu.edu/~suhasini/teaching613/inference.pdf (Theorem 1.1)
– mavavilj
Nov 21 at 9:53
add a comment |
1 Answer
1
active
oldest
votes
1 Answer
1
active
oldest
votes
active
oldest
votes
active
oldest
votes
up vote
0
down vote
My impression is that you got something wrong here. Note that $$frac{partial^2 l(theta, X)}{partial theta^2} $$ is a second derivative, while $$frac{partial l(theta, X)}{partial theta}frac{partial l(theta, X)}{partial theta}^T$$ is a product of two first-order derivatives. As such, they are surely not equal in general: consider for example the function $$l(theta,X) = theta cdot X.$$
Try reading the wikipedia page on the Fisher information and compare it with your notes.
answered Nov 21 at 9:44
user159517
4,248930
Then what's a lemma that says that they can be equal?
– mavavilj
Nov 21 at 9:46
@mavavilj I wouldn't look for such a lemma (it would mean that $l$ has to satisfy a certain differential equation for all $X$, which is a very strict limitation), I'd rather try to understand what was meant in your notes. What is the definition of $l$?
– user159517
Nov 21 at 9:49
The wikipedia says same as my notes: "If log f(x; θ) is twice differentiable with respect to θ, and under certain regularity conditions,[4] then the Fisher information may also be written as". But I want to know what the lemma is that allows for that equality.
– mavavilj
Nov 21 at 9:52
Cramer-Rao bound? stat.tamu.edu/~suhasini/teaching613/inference.pdf (Theorem 1.1)
– mavavilj
Nov 21 at 9:53
add a comment |
up vote
0
down vote
My impression is that you got something wrong here. Note that $$frac{partial^2 l(theta, X)}{partial theta^2} $$ is a second derivative, while $$frac{partial l(theta, X)}{partial theta}frac{partial l(theta, X)}{partial theta}^T$$ is a product of two first-order derivatives. As such, they are surely not equal in general: consider for example the function $$l(theta,X) = theta cdot X.$$
Try reading the wikipedia page on the Fisher information and compare it with your notes.
answered Nov 21 at 9:44
user159517
4,248930
Then what's a lemma that says that they can be equal?
– mavavilj
Nov 21 at 9:46
@mavavilj I wouldn't look for such a lemma (it would mean that $l$ has to satisfy a certain differential equation for all $X$, which is a very strict limitation), I'd rather try to understand what was meant in your notes. What is the definition of $l$?
– user159517
Nov 21 at 9:49
The wikipedia says same as my notes: "If log f(x; θ) is twice differentiable with respect to θ, and under certain regularity conditions,[4] then the Fisher information may also be written as". But I want to know what the lemma is that allows for that equality.
– mavavilj
Nov 21 at 9:52
Cramer-Rao bound? stat.tamu.edu/~suhasini/teaching613/inference.pdf (Theorem 1.1)
– mavavilj
Nov 21 at 9:53
add a comment |
up vote
0
down vote
up vote
0
down vote
My impression is that you got something wrong here. Note that $$frac{partial^2 l(theta, X)}{partial theta^2} $$ is a second derivative, while $$frac{partial l(theta, X)}{partial theta}frac{partial l(theta, X)}{partial theta}^T$$ is a product of two first-order derivatives. As such, they are surely not equal in general: consider for example the function $$l(theta,X) = theta cdot X.$$
Try reading the wikipedia page on the Fisher information and compare it with your notes.
answered Nov 21 at 9:44
user159517
4,248930
My impression is that you got something wrong here. Note that $$frac{partial^2 l(theta, X)}{partial theta^2} $$ is a second derivative, while $$frac{partial l(theta, X)}{partial theta}frac{partial l(theta, X)}{partial theta}^T$$ is a product of two first-order derivatives. As such, they are surely not equal in general: consider for example the function $$l(theta,X) = theta cdot X.$$
Try reading the wikipedia page on the Fisher information and compare it with your notes.
answered Nov 21 at 9:44
user159517
4,248930
edited Nov 21 at 9:51
answered Nov 21 at 9:44
user159517
4,248930
answered Nov 21 at 9:44
user159517
4,248930
answered Nov 21 at 9:44
user159517
4,248930
4,248930
Then what's a lemma that says that they can be equal?
– mavavilj
Nov 21 at 9:46
@mavavilj I wouldn't look for such a lemma (it would mean that $l$ has to satisfy a certain differential equation for all $X$, which is a very strict limitation), I'd rather try to understand what was meant in your notes. What is the definition of $l$?
– user159517
Nov 21 at 9:49
The wikipedia says same as my notes: "If log f(x; θ) is twice differentiable with respect to θ, and under certain regularity conditions,[4] then the Fisher information may also be written as". But I want to know what the lemma is that allows for that equality.
– mavavilj
Nov 21 at 9:52
Cramer-Rao bound? stat.tamu.edu/~suhasini/teaching613/inference.pdf (Theorem 1.1)
– mavavilj
Nov 21 at 9:53
add a comment |
Then what's a lemma that says that they can be equal?
– mavavilj
Nov 21 at 9:46
@mavavilj I wouldn't look for such a lemma (it would mean that $l$ has to satisfy a certain differential equation for all $X$, which is a very strict limitation), I'd rather try to understand what was meant in your notes. What is the definition of $l$?
– user159517
Nov 21 at 9:49
The wikipedia says same as my notes: "If log f(x; θ) is twice differentiable with respect to θ, and under certain regularity conditions,[4] then the Fisher information may also be written as". But I want to know what the lemma is that allows for that equality.
– mavavilj
Nov 21 at 9:52
Cramer-Rao bound? stat.tamu.edu/~suhasini/teaching613/inference.pdf (Theorem 1.1)
– mavavilj
Nov 21 at 9:53
Then what's a lemma that says that they can be equal?
– mavavilj
Nov 21 at 9:46
Then what's a lemma that says that they can be equal?
– mavavilj
Nov 21 at 9:46
@mavavilj I wouldn't look for such a lemma (it would mean that $l$ has to satisfy a certain differential equation for all $X$, which is a very strict limitation), I'd rather try to understand what was meant in your notes. What is the definition of $l$?
– user159517
Nov 21 at 9:49
@mavavilj I wouldn't look for such a lemma (it would mean that $l$ has to satisfy a certain differential equation for all $X$, which is a very strict limitation), I'd rather try to understand what was meant in your notes. What is the definition of $l$?
– user159517
Nov 21 at 9:49
The wikipedia says same as my notes: "If log f(x; θ) is twice differentiable with respect to θ, and under certain regularity conditions,[4] then the Fisher information may also be written as". But I want to know what the lemma is that allows for that equality.
– mavavilj
Nov 21 at 9:52
The wikipedia says same as my notes: "If log f(x; θ) is twice differentiable with respect to θ, and under certain regularity conditions,[4] then the Fisher information may also be written as". But I want to know what the lemma is that allows for that equality.
– mavavilj
Nov 21 at 9:52
Cramer-Rao bound? stat.tamu.edu/~suhasini/teaching613/inference.pdf (Theorem 1.1)
– mavavilj
Nov 21 at 9:53
Cramer-Rao bound? stat.tamu.edu/~suhasini/teaching613/inference.pdf (Theorem 1.1)
– mavavilj
Nov 21 at 9:53
add a comment |
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