Linked Questions

5
votes
3answers
16k views

Prove the sample variance is an unbiased estimator

I'm trying to prove that the sample variance is an unbiased estimator. I know that I need to find the expected value of the sample variance estimator $$\sum_i\frac{(M_i - \bar{M})^2}{n-1}$$ but I get ...
2
votes
1answer
3k views

Variance of Sample Variance

Given $X_1,...,X_n$ iid to a certain distribution (not necessarily normal), with $\mathbb{E}(X_i)=\mu$ and $\mathbb{V}(X_i)=\sigma^2$, I'm trying to deduce the standard and mean squared error of the ...
1
vote
2answers
1k views

Covariance of sample central moments

What is the covariance matrix of the first four sample central moments? Assuming a zero mean and possibly without assuming normality. The covariance matrix of the first four sample raw moments for a ...
1
vote
1answer
2k views

Choosing constant to minimize mean square error

Can you please help me show the statement below? I am not exactly sure where to start.
1
vote
1answer
1k views

Showing that $ S_{n}^{2}$ converges to $ \sigma ^{2} $ in probability

Let $x_{1},...,x_{n} \sim F$ where the expected value of $F$ is $\mu$ and the variance is $\sigma^{2}$. $ S_{n}^{2}=\frac{1}{n-1}\sum_{i=1}^{n}\left(x_{i}-\overline{x}\right)^{2} $ converges in ...
2
votes
1answer
1k views

Convergence in probability of Sn / n

Can you please help verify if what I have done is correct for the question below? I applied Chebyshev's theorem in the first step, but I am worried if there are any mathematical errors or misapplied ...
0
votes
2answers
402 views

how to interpret the variance of a variance?

I understand how to interpret variance, but, what is the physical meaning behind the variance of a variance? Does it relate to error bounds at all (e.g. confidence interval)?
1
vote
2answers
486 views

Variance of Variance Estimation Simulation in Matlab

I am trying to verify, through numerical simulation, the expression for the variance of the variance estimation, namely: $$ \text{Var}(s^2) = \frac{2}{n \, \sigma^4} $$ where $n$ is the number of ...
1
vote
0answers
619 views

If $S'^2 = \frac{\sum_{i=1}^n (Y_i - \bar{Y})^2}{n}$ and $S^2 = \frac{\sum_{i=1}^n (Y_i - \bar{Y})^2}{n-1}$, find $V(S'^2)$.

If $S'^2 = \frac{\sum_{i=1}^n (Y_i - \bar{Y})^2}{n}$ and $S^2 = \frac{\sum_{i=1}^n (Y_i - \bar{Y})^2}{n-1}$ then $S'^2$ is a biased estimator of $σ^2$, but $S^2$ is an unbiased estimator of the same ...
2
votes
1answer
300 views

Variance of an unbiased estimator of central moments

Some days ago I asked this question: Unbiased estimators for the moments of 2 non-independent random variables Now, using the same framework employed for the previous question, I'm facing with the ...
1
vote
1answer
70 views

Why does the expected value of $\left[{1\over2}(X_i-X_j)^2-\sigma^2\right] \left[{1\over2}(X_i-X_k)^2-\sigma^2\right]$ equal $(\mu_4-\sigma^4)/4$?

Question Let $X_i,X_j,X_k$ be IID random variables with finite moments. In particular we denote $E[X_i] := \mu$, $E[(X_i - \mu)^4] := \mu_4$ and $Var[X_i] := \sigma^2$. Why does the expected value ...
1
vote
1answer
71 views

Determine the variance of estimator T

I'm having trouble figuring out how to find the variance of the following estimator. Let $X_1,X_2,...,X_n$ denote random sample from a population which has a normal distribution with unknown mean $\...
1
vote
1answer
51 views

Prove that $\hat{\sigma^2}=\frac{1}{n-1}\sum_{i=1}^n (X_i-\bar{X})^2$ is not an efficient estimator.

I am asked to show that the unbiased estimator $\hat{\sigma^2}=\frac{1}{n-1}\sum_{i=1}^n (X_i-\bar{X})^2$ is not efficient. So far I was able to show that the Rao-Cramer Lower Bound is $\frac{2\sigma^...
1
vote
1answer
45 views

Standard error of variance estimation

We know that a mean estimation is the following $\bar{x}=\frac{1}{n}\sum_{n=1}^{N}x_{n}$ and the standard error of the mean estimation is $Var(\bar{x})=\sigma^{2}/n$. Also the estimation of ...
1
vote
0answers
22 views

Is the variance of variance larger than the variance itself?

After performing a simulation I'm trying to analyse my results. Due to a limited number of measurements, my results are quite noisy. I notice that in calculating a mean value I have gotten somewhat ...