Minimax estimator for mean of bounded distribution with squared error loss function
I have been studying statistical inference and came across a problem regarding minimax estimator that I don't have an idea how to solve
Question:
Suppose $X_1,...,X_n$ are i.i.d and follows distribution $F$, where $F$ is a distribution supported on the unit interval $[0,1]$. Find a minimax estimator for $g(F):=E_F(X_1)$ with respect to the squared error loss function and show that it is unique.
Attempt:
My attempt is as followed: Knowing that the loss function is a squared error loss, then we know that given a prior, let $mu=E_f(X_1)$, the Bayes estimator should be $E(mu|X)$ and is unique.
We then wish to find a prior such that makes the risk of the Bayes estimator to be a constant of $mu$. i.e.: Find a prior such that $E[(E(mu|X)-mu)^2]=C$, but as the specific form of $F$ is not known, I got stuck.
Am I doing it the right way or are my thoughts completely wrong? Many thanks and appreciation for the helps in advance!
bayesian estimation
asked Nov 25 at 9:07
dogthepeter
274
add a comment |
I have been studying statistical inference and came across a problem regarding minimax estimator that I don't have an idea how to solve
Question:
Suppose $X_1,...,X_n$ are i.i.d and follows distribution $F$, where $F$ is a distribution supported on the unit interval $[0,1]$. Find a minimax estimator for $g(F):=E_F(X_1)$ with respect to the squared error loss function and show that it is unique.
Attempt:
My attempt is as followed: Knowing that the loss function is a squared error loss, then we know that given a prior, let $mu=E_f(X_1)$, the Bayes estimator should be $E(mu|X)$ and is unique.
We then wish to find a prior such that makes the risk of the Bayes estimator to be a constant of $mu$. i.e.: Find a prior such that $E[(E(mu|X)-mu)^2]=C$, but as the specific form of $F$ is not known, I got stuck.
Am I doing it the right way or are my thoughts completely wrong? Many thanks and appreciation for the helps in advance!
bayesian estimation
asked Nov 25 at 9:07
dogthepeter
274
add a comment |
I have been studying statistical inference and came across a problem regarding minimax estimator that I don't have an idea how to solve
Question:
Suppose $X_1,...,X_n$ are i.i.d and follows distribution $F$, where $F$ is a distribution supported on the unit interval $[0,1]$. Find a minimax estimator for $g(F):=E_F(X_1)$ with respect to the squared error loss function and show that it is unique.
Attempt:
My attempt is as followed: Knowing that the loss function is a squared error loss, then we know that given a prior, let $mu=E_f(X_1)$, the Bayes estimator should be $E(mu|X)$ and is unique.
We then wish to find a prior such that makes the risk of the Bayes estimator to be a constant of $mu$. i.e.: Find a prior such that $E[(E(mu|X)-mu)^2]=C$, but as the specific form of $F$ is not known, I got stuck.
Am I doing it the right way or are my thoughts completely wrong? Many thanks and appreciation for the helps in advance!
bayesian estimation
asked Nov 25 at 9:07
dogthepeter
274
I have been studying statistical inference and came across a problem regarding minimax estimator that I don't have an idea how to solve
Question:
Suppose $X_1,...,X_n$ are i.i.d and follows distribution $F$, where $F$ is a distribution supported on the unit interval $[0,1]$. Find a minimax estimator for $g(F):=E_F(X_1)$ with respect to the squared error loss function and show that it is unique.
Attempt:
My attempt is as followed: Knowing that the loss function is a squared error loss, then we know that given a prior, let $mu=E_f(X_1)$, the Bayes estimator should be $E(mu|X)$ and is unique.
We then wish to find a prior such that makes the risk of the Bayes estimator to be a constant of $mu$. i.e.: Find a prior such that $E[(E(mu|X)-mu)^2]=C$, but as the specific form of $F$ is not known, I got stuck.
Am I doing it the right way or are my thoughts completely wrong? Many thanks and appreciation for the helps in advance!
bayesian estimation
bayesian estimation
asked Nov 25 at 9:07
dogthepeter
274
asked Nov 25 at 9:07
dogthepeter
274
asked Nov 25 at 9:07
dogthepeter
274
asked Nov 25 at 9:07
dogthepeter
274
asked Nov 25 at 9:07
dogthepeter
274
274
add a comment |
add a comment |
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