Unbiased Estimator of the Variance of the Sample Variance - Cross Validated most recent 30 from stats.stackexchange.com 2019-08-17T21:07:35Z https://stats.stackexchange.com/feeds/question/307537 http://www.creativecommons.org/licenses/by-sa/3.0/rdf https://stats.stackexchange.com/q/307537 4 Unbiased Estimator of the Variance of the Sample Variance Hiro https://stats.stackexchange.com/users/178794 2017-10-12T09:50:40Z 2017-10-13T15:48:59Z <p>At Mathematics Stack Exchange, user940 provided a general formula to calculate the variance of the sample variance based on the fourth central moment&nbsp;$\mu_4$ and the population variance&nbsp;$\sigma^2$ (<a href="https://math.stackexchange.com/questions/72975/variance-of-sample-variance">1</a>):</p> <p>$$\text{Var}(S^2)=\frac{\mu_4}{n}-\frac{\sigma^4(n-3)}{n(n-1)}$$</p> <p>I would be interested in an unbiased estimator for this, without knowing the population parameters&nbsp;$\mu_4$ and&nbsp;$\sigma^2$, but using the fourth and second sample central moment&nbsp;$m_4$ and&nbsp;$m_2$ (or the unbiased sample variance&nbsp;$S^2=\frac{n}{n-1}m_2$) instead.</p> https://stats.stackexchange.com/questions/307537/-/307576#307576 7 Answer by wolfies for Unbiased Estimator of the Variance of the Sample Variance wolfies https://stats.stackexchange.com/users/24905 2017-10-12T13:40:14Z 2017-10-13T15:48:59Z <p>The question is to find an unbiased estimator of: </p> <p>$$\text{Var}(S^2)=\frac{\mu_4}{n}-\frac{(n-3)}{n(n-1)} {\mu_2^2}$$</p> <p>... where $\mu_r$ denotes the $r^\text{th}$ central moment of the population. This requires finding unbiased estimators of $\mu_4$ and of $\mu_2^2$.</p> <p><strong>An unbiased estimator of $\mu_4$</strong></p> <p>By defn, an unbiased estimator of the $r^\text{th}$ central moment is the $r^\text{th}$ h-statistic: $$\mathbb{E}[h_r] = \mu_r$$ The $4^\text{th}$ h-statistic is given by:</p> <p><a href="https://i.stack.imgur.com/hQoqS.png" rel="nofollow noreferrer"><img src="https://i.stack.imgur.com/hQoqS.png" alt="enter image description here"></a> where:</p> <p>i) I am using the <code>HStatistic</code> function from the <em>mathStatica</em> package for <em>Mathematica</em> </p> <p>ii) $s_r$ denotes the $r^\text{th}$ power sum $$s_r=\sum _{i=1}^n X_i^r$$</p> <p>Alternative: The OP asked about finding an unbiased solution in terms of sample central moments $m_r=\frac{1}{n} \sum _{i=1}^n \left(X_i-\bar{X}\right)^r$. An unbiased estimator of $\mu_4$ in terms of $m_i$ is:</p> <p><a href="https://i.stack.imgur.com/DxYMs.png" rel="nofollow noreferrer"><img src="https://i.stack.imgur.com/DxYMs.png" alt="enter image description here"></a></p> <p><strong>An unbiased estimator of $\mu_2^2$</strong></p> <p>An unbiased estimator of a product of central moments (here, $\mu_2 \times \mu_2$)is known as a polyache (play on poly-h). An unbiased estimator of $\mu_2^2$ is given by:</p> <p><a href="https://i.stack.imgur.com/8TXrY.png" rel="nofollow noreferrer"><img src="https://i.stack.imgur.com/8TXrY.png" alt="enter image description here"></a></p> <p>where:</p> <p>i) I am using the <code>PolyH</code> function from the <em>mathStatica</em> package for <em>Mathematica</em> </p> <p>ii) For more detail on polyaches, see section 7.2B of Chapter 7 of Rose and Smith, <em>Mathematical Statistics with Mathematica</em> (am one of the authors), a free download of which is available <a href="http://www.mathstatica.com/book/Rose_and_Smith_2002edition_Chapter7.pdf" rel="nofollow noreferrer" title="here">here</a>.</p> <p>While <em>mathStatica</em> does not have an automated converter to express <code>PolyH</code> in terms of sample central moments $m_i$ (nice idea), doing that conversion yields:</p> <p><a href="https://i.stack.imgur.com/gxhYL.png" rel="nofollow noreferrer"><img src="https://i.stack.imgur.com/gxhYL.png" alt="enter image description here"></a> </p> <hr> <p><strong>Putting it all together:</strong> </p> <p>An unbiased estimator of $\frac{\mu_4}{n}-\frac{(n-3)}{n(n-1)} {\mu_2^2}$ is thus:</p> <p><a href="https://i.stack.imgur.com/Xhp0J.png" rel="nofollow noreferrer"><img src="https://i.stack.imgur.com/Xhp0J.png" alt="enter image description here"></a></p> <p>or, more compactly, in terms of sample central moments $m_i$:</p> <p>........... <a href="https://i.stack.imgur.com/j4qkd.png" rel="nofollow noreferrer"><img src="https://i.stack.imgur.com/j4qkd.png" alt="enter image description here"></a></p> <p>And as a check, we can run the expectations operator over the above (the $1^\text{st}$ <code>RawMoment</code> of <code>sol</code>), expressing the solution in terms of <code>Central</code> moments of the population:</p> <p><a href="https://i.stack.imgur.com/r9CcN.png" rel="nofollow noreferrer"><img src="https://i.stack.imgur.com/r9CcN.png" alt="enter image description here"></a></p> <p>... and all is good.</p>