The sampling distribution of sample variances

A colleague and her students asked me the other day whether I knew of a citation that gives the covariance between the sample variances of two outcomes from a common sample. This sort of question comes up in meta-analysis problems occasionally. I didn’t know of a convenient reference that directly answers the question, but I was able to suggest some references that would help (listed below). While the students work on deriving it, I’ll provide the answer here so that they can check their work.

Suppose that we have a sample of \(n\) observations \(\mathbf{y}_1,...,\mathbf{y}_n\) from a \(p\)-dimensional multivariate normal distribution with mean \(\boldsymbol\mu\) and covariance \(\boldsymbol\Sigma = \left[\sigma_{jk}\right]_{j,k=1,...,p}\). Let \(\mathbf{\bar{y}}\) denote the (multivariate) sample mean, with entries \(\bar{y}_1,...,\bar{y}_p\). Let \(\mathbf{S}\) denote the sample covariance matrix, with entries \(\left[s_{jk}\right]_{j,k=1,...,p}\) where

\[ s_{jk} = \frac{1}{n - 1}\sum_{i=1}^n (y_{ij} - \bar{y}_j)(y_{ik} - \bar{y}_k). \]

Then \((n - 1)\mathbf{S}\) follows a Wishart distribution with \(n - 1\) degrees of freedom and scale matrix \(\boldsymbol\Sigma\) (Searle, 2006, p. 352; Muirhead, 1982, p. 86; or any textbook on multivariate analysis).

The sampling covariance between two sample covariances, say \(s_{jk}\) and \(s_{lm}\), can then be derived from the properties of the Wishart distribution. Expressions for this are available in Searle (2006) or Muirhead (1982). The former is a bit hard to parse because it uses the \(\text{vec}\) and Kronecker product operators; Muirhead (1982, p. 90) gives the following simple expression:

\[ \text{Cov}\left(s_{jk}, s_{lm}\right) = \frac{\sigma_{jl}\sigma_{km} + \sigma_{jm}\sigma_{kl}}{n - 1}. \]

For sample variances, this reduces to

\[ \text{Cov}\left(s_j^2, s_l^2\right) = \frac{2\sigma_{jl}^2}{n - 1}. \]

The formula also reduces to the well-known result that the sampling variance of the sample variance is

\[ \text{Var}\left(s_j^2\right) = \frac{2 \sigma_{jj}^2}{n - 1}. \]

One application of this bit of distribution theory is to find the sampling variance of an average of sample variances. Suppose that we have a bivariate normal distribution where both measures have the same variance \(\sigma_{11} = \sigma_{22} = \sigma^2\) and correlation \(\rho\). One estimate of this common variance is to take the simple average of the sample variances, \(s_{\bullet}^2 = \left(s_1^2 + s_2^2\right) / 2\). Then using the above:

\[\begin{aligned} \text{Var}\left(s_{\bullet}^2\right) &= \frac{1}{4}\left[\text{Var}\left(s_1^2\right) + \text{Var}\left(s_2^2\right) + 2\text{Cov}\left(s_1^2, s_2^2\right) \right] \\ &= \frac{\sigma^4 \left(1 + \rho^2\right)}{n - 1}. \end{aligned}\]

To see that this is correct, consider the extreme cases. If the two measures are perfectly correlated, then averaging the sample variances has no benefit because \(\text{Var}\left(s_{\bullet}^2\right) = \text{Var}\left(s_1^2\right) = \text{Var}\left(s_2^2\right)\). If they are exactly uncorrelated, then averaging the sample variances is equivalent to pooling the sample variance from two independent samples.

References

Muirhead, R. J. (1982). Aspects of Multivariate Statistical Theory. New York, NY: John Wiley & Sons.

Searle, S. R. (2006). Matrix Algebra Useful for Statistics. Hoboken, NJ: John Wiley & Sons.

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