Deriving UMVUE for $musigma^k$ when both $mu$ and $sigma$ are unknown and are the parameters of a normal...
Let $X_1,...,X_n$ be iid $N(mu,sigma^2)$, and define $f$ as $f(
theta)=f(mu,sigma)=musigma^k$. I'm attempting to find the UMVUE for $f(theta)$ via the Lehmann-Scheffe approach, i.e. I'm calculating $E[T|U=u]$ where $T(X)$ is an unbiased estimator for $f(theta)$ and $U(X)$ is complete-sufficient for $theta$.
Thus far, I have found a suitable complete-sufficient statistic: $$U(X)=(sum_{i=1}^nX_i,sum_{i=1}^nX_i^2). $$ It is complete due to the properties of the 2-parameter exponential family. However, I'm struggling to find any sort of unbiased estimator for $f(theta)$, regardless of whether or not it depends on $X$ through $U(X)$. I have a the feeling that the estimator involves a chi-squared distribution, because the parametric function looks to me like a chi-squared moment, but I'm not really sure of the logic involved.
statistics statistical-inference estimation parameter-estimation
asked Nov 25 at 15:33
DavidS
337111
add a comment |
Let $X_1,...,X_n$ be iid $N(mu,sigma^2)$, and define $f$ as $f(
theta)=f(mu,sigma)=musigma^k$. I'm attempting to find the UMVUE for $f(theta)$ via the Lehmann-Scheffe approach, i.e. I'm calculating $E[T|U=u]$ where $T(X)$ is an unbiased estimator for $f(theta)$ and $U(X)$ is complete-sufficient for $theta$.
Thus far, I have found a suitable complete-sufficient statistic: $$U(X)=(sum_{i=1}^nX_i,sum_{i=1}^nX_i^2). $$ It is complete due to the properties of the 2-parameter exponential family. However, I'm struggling to find any sort of unbiased estimator for $f(theta)$, regardless of whether or not it depends on $X$ through $U(X)$. I have a the feeling that the estimator involves a chi-squared distribution, because the parametric function looks to me like a chi-squared moment, but I'm not really sure of the logic involved.
statistics statistical-inference estimation parameter-estimation
asked Nov 25 at 15:33
DavidS
337111
Find an unbiased estimator of $sigma^k$ based on $S$, the sample sd with divisor $n-1$. Since $E(bar X)=mu$, by independence of $bar X$ and $S$, you would get the UMVUE.
– StubbornAtom
Nov 25 at 15:44
add a comment |
Let $X_1,...,X_n$ be iid $N(mu,sigma^2)$, and define $f$ as $f(
theta)=f(mu,sigma)=musigma^k$. I'm attempting to find the UMVUE for $f(theta)$ via the Lehmann-Scheffe approach, i.e. I'm calculating $E[T|U=u]$ where $T(X)$ is an unbiased estimator for $f(theta)$ and $U(X)$ is complete-sufficient for $theta$.
Thus far, I have found a suitable complete-sufficient statistic: $$U(X)=(sum_{i=1}^nX_i,sum_{i=1}^nX_i^2). $$ It is complete due to the properties of the 2-parameter exponential family. However, I'm struggling to find any sort of unbiased estimator for $f(theta)$, regardless of whether or not it depends on $X$ through $U(X)$. I have a the feeling that the estimator involves a chi-squared distribution, because the parametric function looks to me like a chi-squared moment, but I'm not really sure of the logic involved.
statistics statistical-inference estimation parameter-estimation
asked Nov 25 at 15:33
DavidS
337111
Let $X_1,...,X_n$ be iid $N(mu,sigma^2)$, and define $f$ as $f(
theta)=f(mu,sigma)=musigma^k$. I'm attempting to find the UMVUE for $f(theta)$ via the Lehmann-Scheffe approach, i.e. I'm calculating $E[T|U=u]$ where $T(X)$ is an unbiased estimator for $f(theta)$ and $U(X)$ is complete-sufficient for $theta$.
Thus far, I have found a suitable complete-sufficient statistic: $$U(X)=(sum_{i=1}^nX_i,sum_{i=1}^nX_i^2). $$ It is complete due to the properties of the 2-parameter exponential family. However, I'm struggling to find any sort of unbiased estimator for $f(theta)$, regardless of whether or not it depends on $X$ through $U(X)$. I have a the feeling that the estimator involves a chi-squared distribution, because the parametric function looks to me like a chi-squared moment, but I'm not really sure of the logic involved.
statistics statistical-inference estimation parameter-estimation
statistics statistical-inference estimation parameter-estimation
asked Nov 25 at 15:33
DavidS
337111
asked Nov 25 at 15:33
DavidS
337111
asked Nov 25 at 15:33
DavidS
337111
asked Nov 25 at 15:33
DavidS
337111
asked Nov 25 at 15:33
DavidS
337111
337111
Find an unbiased estimator of $sigma^k$ based on $S$, the sample sd with divisor $n-1$. Since $E(bar X)=mu$, by independence of $bar X$ and $S$, you would get the UMVUE.
– StubbornAtom
Nov 25 at 15:44
add a comment |
Find an unbiased estimator of $sigma^k$ based on $S$, the sample sd with divisor $n-1$. Since $E(bar X)=mu$, by independence of $bar X$ and $S$, you would get the UMVUE.
– StubbornAtom
Nov 25 at 15:44
Find an unbiased estimator of $sigma^k$ based on $S$, the sample sd with divisor $n-1$. Since $E(bar X)=mu$, by independence of $bar X$ and $S$, you would get the UMVUE.
– StubbornAtom
Nov 25 at 15:44
Find an unbiased estimator of $sigma^k$ based on $S$, the sample sd with divisor $n-1$. Since $E(bar X)=mu$, by independence of $bar X$ and $S$, you would get the UMVUE.
– StubbornAtom
Nov 25 at 15:44
add a comment |
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Find an unbiased estimator of $sigma^k$ based on $S$, the sample sd with divisor $n-1$. Since $E(bar X)=mu$, by independence of $bar X$ and $S$, you would get the UMVUE.
– StubbornAtom
Nov 25 at 15:44