The expected-value of the square of Sample Variance.
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Suppose $X_1, cdots, X_n$ are i.d.d. samples from population $X sim N(mu,sigma^2)$, and the sample variance is denoted by
$T = sum_{i = 1}^n frac{(X_i - overline{X})^2}{n}$.
I am curious about the expected-value of $T^2$, which is the square of $T$.
Apparently the key problem is what the distribution of $T^2$ is ?
According to my intuition, it may be some kind of F-distribution, but how to prove it ,especially to solve the cross term is the biggest problem that I have encountered.
probability expected-value
asked 18 hours ago
Maxius Xu
62
add a comment |
up vote
0
down vote
favorite
Suppose $X_1, cdots, X_n$ are i.d.d. samples from population $X sim N(mu,sigma^2)$, and the sample variance is denoted by
$T = sum_{i = 1}^n frac{(X_i - overline{X})^2}{n}$.
I am curious about the expected-value of $T^2$, which is the square of $T$.
Apparently the key problem is what the distribution of $T^2$ is ?
According to my intuition, it may be some kind of F-distribution, but how to prove it ,especially to solve the cross term is the biggest problem that I have encountered.
probability expected-value
asked 18 hours ago
Maxius Xu
62
$S^2=frac{1}{n-1}sum_{i=1}^n (X_i-overline X)^2implies S_n^2=frac{n-1}{n}S^2implies E(S_n^2)=frac{n-1}{n}sigma^2$
– StubbornAtom
18 hours ago
en.wikipedia.org/wiki/… Both normal case, and the general case are given, for the variance of the sample variance. Then you add back $sigma^4$ to obtain the desired moment. Usually we denote $S^2$ to be the sample variance and $S$ for the sample standard deviation.
– BGM
17 hours ago
What I want is the "square" of the sample variance.
– Maxius Xu
14 hours ago
add a comment |
up vote
0
down vote
favorite
up vote
0
down vote
favorite
Suppose $X_1, cdots, X_n$ are i.d.d. samples from population $X sim N(mu,sigma^2)$, and the sample variance is denoted by
$T = sum_{i = 1}^n frac{(X_i - overline{X})^2}{n}$.
I am curious about the expected-value of $T^2$, which is the square of $T$.
Apparently the key problem is what the distribution of $T^2$ is ?
According to my intuition, it may be some kind of F-distribution, but how to prove it ,especially to solve the cross term is the biggest problem that I have encountered.
probability expected-value
asked 18 hours ago
Maxius Xu
62
Suppose $X_1, cdots, X_n$ are i.d.d. samples from population $X sim N(mu,sigma^2)$, and the sample variance is denoted by
$T = sum_{i = 1}^n frac{(X_i - overline{X})^2}{n}$.
I am curious about the expected-value of $T^2$, which is the square of $T$.
Apparently the key problem is what the distribution of $T^2$ is ?
According to my intuition, it may be some kind of F-distribution, but how to prove it ,especially to solve the cross term is the biggest problem that I have encountered.
probability expected-value
probability expected-value
asked 18 hours ago
Maxius Xu
62
asked 18 hours ago
Maxius Xu
62
edited 14 hours ago
asked 18 hours ago
Maxius Xu
62
asked 18 hours ago
Maxius Xu
62
asked 18 hours ago
Maxius Xu
62
62
$S^2=frac{1}{n-1}sum_{i=1}^n (X_i-overline X)^2implies S_n^2=frac{n-1}{n}S^2implies E(S_n^2)=frac{n-1}{n}sigma^2$
– StubbornAtom
18 hours ago
en.wikipedia.org/wiki/… Both normal case, and the general case are given, for the variance of the sample variance. Then you add back $sigma^4$ to obtain the desired moment. Usually we denote $S^2$ to be the sample variance and $S$ for the sample standard deviation.
– BGM
17 hours ago
What I want is the "square" of the sample variance.
– Maxius Xu
14 hours ago
add a comment |
$S^2=frac{1}{n-1}sum_{i=1}^n (X_i-overline X)^2implies S_n^2=frac{n-1}{n}S^2implies E(S_n^2)=frac{n-1}{n}sigma^2$
– StubbornAtom
18 hours ago
en.wikipedia.org/wiki/… Both normal case, and the general case are given, for the variance of the sample variance. Then you add back $sigma^4$ to obtain the desired moment. Usually we denote $S^2$ to be the sample variance and $S$ for the sample standard deviation.
– BGM
17 hours ago
What I want is the "square" of the sample variance.
– Maxius Xu
14 hours ago
$S^2=frac{1}{n-1}sum_{i=1}^n (X_i-overline X)^2implies S_n^2=frac{n-1}{n}S^2implies E(S_n^2)=frac{n-1}{n}sigma^2$
– StubbornAtom
18 hours ago
$S^2=frac{1}{n-1}sum_{i=1}^n (X_i-overline X)^2implies S_n^2=frac{n-1}{n}S^2implies E(S_n^2)=frac{n-1}{n}sigma^2$
– StubbornAtom
18 hours ago
en.wikipedia.org/wiki/… Both normal case, and the general case are given, for the variance of the sample variance. Then you add back $sigma^4$ to obtain the desired moment. Usually we denote $S^2$ to be the sample variance and $S$ for the sample standard deviation.
– BGM
17 hours ago
en.wikipedia.org/wiki/… Both normal case, and the general case are given, for the variance of the sample variance. Then you add back $sigma^4$ to obtain the desired moment. Usually we denote $S^2$ to be the sample variance and $S$ for the sample standard deviation.
– BGM
17 hours ago
What I want is the "square" of the sample variance.
– Maxius Xu
14 hours ago
What I want is the "square" of the sample variance.
– Maxius Xu
14 hours ago
add a comment |
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$S^2=frac{1}{n-1}sum_{i=1}^n (X_i-overline X)^2implies S_n^2=frac{n-1}{n}S^2implies E(S_n^2)=frac{n-1}{n}sigma^2$
– StubbornAtom
18 hours ago
en.wikipedia.org/wiki/… Both normal case, and the general case are given, for the variance of the sample variance. Then you add back $sigma^4$ to obtain the desired moment. Usually we denote $S^2$ to be the sample variance and $S$ for the sample standard deviation.
– BGM
17 hours ago
What I want is the "square" of the sample variance.
– Maxius Xu
14 hours ago