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edited Nov 29 at 22:30

mathematurgist

Corrected a notational error in the final equation where N was replaced with T.

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edited Nov 28 at 17:11

mathematurgist

Unbiased Variance Estimator

Let $x_1 , \ldots, x_N$ be iid sampled from X. Let Y(N) denote the N-mean estimator given by

$$ Y(N) = \frac{1}{N} \sum_{i=1}^N x_i $$

Let v(N) denote the unbiased N-variance estimator

$$ v(N) = \frac{1}{N-1} \sum_{i=1}^N (x_i - Y(N))^2 $$

Unbiased Variance Estimator with reused samples

I draw d additional samples $x_{N+1} , \ldots, x_{N+d}$ that are iid sampled from X. Similarly, let $Y(N+d)$ denote the $(N+d)$-mean estimator (with reused samples).

$$ Y(N+d) = \frac{1}{N+d} \sum_{i=1}^{(N+d)} x_i$$

Let v(N+d) denote the unbiased (N+d)-variance estimator with reused samples,

$$v(N+d) = \frac{1}{N+d-1} \sum_{i=1}^{(N+d)} (x_i - Y(N+d))^2$$

Covariance of the Variance Estimators with shared samples

My question is, what is the Covariance of these two estimators? $$\mathrm{Cov}[ v(N), v(N+d) ] = ??? $$

From E. Benhamou. “A few properties of sample variance" equation 10, I know that the variance of the estimators is given by $$\mathrm{Var}[ v(N) ] = \frac{1}{T}\left( m_4 - \frac{T-3}{T-1} m_2^2 \right)$$$$\mathrm{Var}[ v(N) ] = \frac{1}{N}\left( m_4 - \frac{N-3}{N-1} m_2^2 \right)$$ where $m_2$ and $m_4$ are the second-order and fourth-order moments of X. I tried to use the fact that v is a U-statistic but I ended up with a quadruple summation that I wasn't sure how to simplify. Is this a standard result? Can I pull a citation from somewhere or is there something that I can draw from?

Unbiased Variance Estimator

Let $x_1 , \ldots, x_N$ be iid sampled from X. Let Y(N) denote the N-mean estimator given by

$$ Y(N) = \frac{1}{N} \sum_{i=1}^N x_i $$

Let v(N) denote the unbiased N-variance estimator

$$ v(N) = \frac{1}{N-1} \sum_{i=1}^N (x_i - Y(N))^2 $$

Unbiased Variance Estimator with reused samples

I draw d additional samples $x_{N+1} , \ldots, x_{N+d}$ that are iid sampled from X. Similarly, let $Y(N+d)$ denote the $(N+d)$-mean estimator (with reused samples).

$$ Y(N+d) = \frac{1}{N+d} \sum_{i=1}^{(N+d)} x_i$$

Let v(N+d) denote the unbiased (N+d)-variance estimator with reused samples,

$$v(N+d) = \frac{1}{N+d-1} \sum_{i=1}^{(N+d)} (x_i - Y(N+d))^2$$

Covariance of the Variance Estimators with shared samples

My question is, what is the Covariance of these two estimators? $$\mathrm{Cov}[ v(N), v(N+d) ] = ??? $$

From E. Benhamou. “A few properties of sample variance" equation 10, I know that the variance of the estimators is given by $$\mathrm{Var}[ v(N) ] = \frac{1}{T}\left( m_4 - \frac{T-3}{T-1} m_2^2 \right)$$ where $m_2$ and $m_4$ are the second-order and fourth-order moments of X. I tried to use the fact that v is a U-statistic but I ended up with a quadruple summation that I wasn't sure how to simplify. Is this a standard result? Can I pull a citation from somewhere or is there something that I can draw from?

Unbiased Variance Estimator

Let $x_1 , \ldots, x_N$ be iid sampled from X. Let Y(N) denote the N-mean estimator given by

$$ Y(N) = \frac{1}{N} \sum_{i=1}^N x_i $$

Let v(N) denote the unbiased N-variance estimator

$$ v(N) = \frac{1}{N-1} \sum_{i=1}^N (x_i - Y(N))^2 $$

Unbiased Variance Estimator with reused samples

I draw d additional samples $x_{N+1} , \ldots, x_{N+d}$ that are iid sampled from X. Similarly, let $Y(N+d)$ denote the $(N+d)$-mean estimator (with reused samples).

$$ Y(N+d) = \frac{1}{N+d} \sum_{i=1}^{(N+d)} x_i$$

Let v(N+d) denote the unbiased (N+d)-variance estimator with reused samples,

$$v(N+d) = \frac{1}{N+d-1} \sum_{i=1}^{(N+d)} (x_i - Y(N+d))^2$$

Covariance of the Variance Estimators with shared samples

My question is, what is the Covariance of these two estimators? $$\mathrm{Cov}[ v(N), v(N+d) ] = ??? $$

From E. Benhamou. “A few properties of sample variance" equation 10, I know that the variance of the estimators is given by $$\mathrm{Var}[ v(N) ] = \frac{1}{N}\left( m_4 - \frac{N-3}{N-1} m_2^2 \right)$$ where $m_2$ and $m_4$ are the second-order and fourth-order moments of X. I tried to use the fact that v is a U-statistic but I ended up with a quadruple summation that I wasn't sure how to simplify. Is this a standard result? Can I pull a citation from somewhere or is there something that I can draw from?

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asked Nov 28 at 15:49

mathematurgist

Covariance of Unbiased Sample Variance Estimators with Overlapping Samples

Unbiased Variance Estimator

Let $x_1 , \ldots, x_N$ be iid sampled from X. Let Y(N) denote the N-mean estimator given by

$$ Y(N) = \frac{1}{N} \sum_{i=1}^N x_i $$

Let v(N) denote the unbiased N-variance estimator

$$ v(N) = \frac{1}{N-1} \sum_{i=1}^N (x_i - Y(N))^2 $$

Unbiased Variance Estimator with reused samples

I draw d additional samples $x_{N+1} , \ldots, x_{N+d}$ that are iid sampled from X. Similarly, let $Y(N+d)$ denote the $(N+d)$-mean estimator (with reused samples).

$$ Y(N+d) = \frac{1}{N+d} \sum_{i=1}^{(N+d)} x_i$$

Let v(N+d) denote the unbiased (N+d)-variance estimator with reused samples,

$$v(N+d) = \frac{1}{N+d-1} \sum_{i=1}^{(N+d)} (x_i - Y(N+d))^2$$

Covariance of the Variance Estimators with shared samples

My question is, what is the Covariance of these two estimators? $$\mathrm{Cov}[ v(N), v(N+d) ] = ??? $$

From E. Benhamou. “A few properties of sample variance" equation 10, I know that the variance of the estimators is given by $$\mathrm{Var}[ v(N) ] = \frac{1}{T}\left( m_4 - \frac{T-3}{T-1} m_2^2 \right)$$ where $m_2$ and $m_4$ are the second-order and fourth-order moments of X. I tried to use the fact that v is a U-statistic but I ended up with a quadruple summation that I wasn't sure how to simplify. Is this a standard result? Can I pull a citation from somewhere or is there something that I can draw from?