Normal Distribution - variance of minimal MSE (mean squared error)
$begingroup$
We are given $n$ normally distributed random variables $X_1,ldots, X_n$ with mean $mu$ and variance $sigma^2$. We have
begin{align}
S^2 = C sum_{i=1}^n (x_i - bar{x})^2
end{align}
We want to find $C$, such that the variance has minimal MSE. Now, I already have that,
begin{align}
E(S^2)=C , (n-1) ,sigma ^2,
end{align}
but I don't see how I would calculate $operatorname{var}(S^2)$ to find the $operatorname{MSE}(S^2)$ (and minimize it with respect to $C$). Thank you in advance.
probability-theory statistics optimization random-variables normal-distribution
asked Dec 17 '18 at 19:04
F. MoeF. Moe
205
$endgroup$
add a comment |
$begingroup$
We are given $n$ normally distributed random variables $X_1,ldots, X_n$ with mean $mu$ and variance $sigma^2$. We have
begin{align}
S^2 = C sum_{i=1}^n (x_i - bar{x})^2
end{align}
We want to find $C$, such that the variance has minimal MSE. Now, I already have that,
begin{align}
E(S^2)=C , (n-1) ,sigma ^2,
end{align}
but I don't see how I would calculate $operatorname{var}(S^2)$ to find the $operatorname{MSE}(S^2)$ (and minimize it with respect to $C$). Thank you in advance.
probability-theory statistics optimization random-variables normal-distribution
asked Dec 17 '18 at 19:04
F. MoeF. Moe
205
$endgroup$
1
$begingroup$
You can use the fact that $S^2/Csigma^2sim chi^2_{(n-1)}$ to compute the variance of $S^2$.
$endgroup$
– Foobaz John
Dec 17 '18 at 19:16
$begingroup$
True. So I get for the variance var$(S^2)=2 , C^2, sigma ^2 , (n-1)$ and therefore for the MSE to be minimal $C$ should be $C=(n+1)^{-1}$. Thank you for the hint!
$endgroup$
– F. Moe
Dec 17 '18 at 19:39
1
$begingroup$
Related: math.stackexchange.com/questions/2860289/….
$endgroup$
– StubbornAtom
Dec 17 '18 at 19:54
add a comment |
$begingroup$
We are given $n$ normally distributed random variables $X_1,ldots, X_n$ with mean $mu$ and variance $sigma^2$. We have
begin{align}
S^2 = C sum_{i=1}^n (x_i - bar{x})^2
end{align}
We want to find $C$, such that the variance has minimal MSE. Now, I already have that,
begin{align}
E(S^2)=C , (n-1) ,sigma ^2,
end{align}
but I don't see how I would calculate $operatorname{var}(S^2)$ to find the $operatorname{MSE}(S^2)$ (and minimize it with respect to $C$). Thank you in advance.
probability-theory statistics optimization random-variables normal-distribution
asked Dec 17 '18 at 19:04
F. MoeF. Moe
205
$endgroup$
We are given $n$ normally distributed random variables $X_1,ldots, X_n$ with mean $mu$ and variance $sigma^2$. We have
begin{align}
S^2 = C sum_{i=1}^n (x_i - bar{x})^2
end{align}
We want to find $C$, such that the variance has minimal MSE. Now, I already have that,
begin{align}
E(S^2)=C , (n-1) ,sigma ^2,
end{align}
but I don't see how I would calculate $operatorname{var}(S^2)$ to find the $operatorname{MSE}(S^2)$ (and minimize it with respect to $C$). Thank you in advance.
probability-theory statistics optimization random-variables normal-distribution
probability-theory statistics optimization random-variables normal-distribution
asked Dec 17 '18 at 19:04
F. MoeF. Moe
205
asked Dec 17 '18 at 19:04
F. MoeF. Moe
205
edited Dec 17 '18 at 19:54
user593746
asked Dec 17 '18 at 19:04
F. MoeF. Moe
205
asked Dec 17 '18 at 19:04
F. MoeF. Moe
205
asked Dec 17 '18 at 19:04
F. MoeF. Moe
205
205
1
$begingroup$
You can use the fact that $S^2/Csigma^2sim chi^2_{(n-1)}$ to compute the variance of $S^2$.
$endgroup$
– Foobaz John
Dec 17 '18 at 19:16
$begingroup$
True. So I get for the variance var$(S^2)=2 , C^2, sigma ^2 , (n-1)$ and therefore for the MSE to be minimal $C$ should be $C=(n+1)^{-1}$. Thank you for the hint!
$endgroup$
– F. Moe
Dec 17 '18 at 19:39
1
$begingroup$
Related: math.stackexchange.com/questions/2860289/….
$endgroup$
– StubbornAtom
Dec 17 '18 at 19:54
add a comment |
1
$begingroup$
You can use the fact that $S^2/Csigma^2sim chi^2_{(n-1)}$ to compute the variance of $S^2$.
$endgroup$
– Foobaz John
Dec 17 '18 at 19:16
$begingroup$
True. So I get for the variance var$(S^2)=2 , C^2, sigma ^2 , (n-1)$ and therefore for the MSE to be minimal $C$ should be $C=(n+1)^{-1}$. Thank you for the hint!
$endgroup$
– F. Moe
Dec 17 '18 at 19:39
1
$begingroup$
Related: math.stackexchange.com/questions/2860289/….
$endgroup$
– StubbornAtom
Dec 17 '18 at 19:54
1
1
$begingroup$
You can use the fact that $S^2/Csigma^2sim chi^2_{(n-1)}$ to compute the variance of $S^2$.
$endgroup$
– Foobaz John
Dec 17 '18 at 19:16
$begingroup$
You can use the fact that $S^2/Csigma^2sim chi^2_{(n-1)}$ to compute the variance of $S^2$.
$endgroup$
– Foobaz John
Dec 17 '18 at 19:16
$begingroup$
True. So I get for the variance var$(S^2)=2 , C^2, sigma ^2 , (n-1)$ and therefore for the MSE to be minimal $C$ should be $C=(n+1)^{-1}$. Thank you for the hint!
$endgroup$
– F. Moe
Dec 17 '18 at 19:39
$begingroup$
True. So I get for the variance var$(S^2)=2 , C^2, sigma ^2 , (n-1)$ and therefore for the MSE to be minimal $C$ should be $C=(n+1)^{-1}$. Thank you for the hint!
$endgroup$
– F. Moe
Dec 17 '18 at 19:39
1
1
$begingroup$
Related: math.stackexchange.com/questions/2860289/….
$endgroup$
– StubbornAtom
Dec 17 '18 at 19:54
$begingroup$
Related: math.stackexchange.com/questions/2860289/….
$endgroup$
– StubbornAtom
Dec 17 '18 at 19:54
add a comment |
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$begingroup$
You can use the fact that $S^2/Csigma^2sim chi^2_{(n-1)}$ to compute the variance of $S^2$.
$endgroup$
– Foobaz John
Dec 17 '18 at 19:16
$begingroup$
True. So I get for the variance var$(S^2)=2 , C^2, sigma ^2 , (n-1)$ and therefore for the MSE to be minimal $C$ should be $C=(n+1)^{-1}$. Thank you for the hint!
$endgroup$
– F. Moe
Dec 17 '18 at 19:39
1
$begingroup$
Related: math.stackexchange.com/questions/2860289/….
$endgroup$
– StubbornAtom
Dec 17 '18 at 19:54