Estimates of the variance of the variance component of a mixed effects model Say $y=X\beta+ Zu +\epsilon$ is our mixed effects model where $u=(u_1,..,u_r)$ and $u_{j} \stackrel{i.i.d.}{\sim} N(0, \sigma^2_{a})$ for $j=1,...,r$ and $\epsilon=(\epsilon_1,...,\epsilon_n)$ are i.i.d. $N(0, \sigma^2_{b})$, furthermore $\epsilon_j$ and $u_i$ are also assumed to be independent for all $j$'s and all $i$'s. 
I am interested in Var($\widehat{\sigma_{a}^2}$)  i.e., the variance of the estimate of $\sigma^2_{a}$. In the R package ${\bf lme4}$ they do provide the M.L.Es but they do not provide the estimate of the Var($\widehat{\sigma_{a}^2}$). I don't think there is any closed form expression for the estimate of Var($\widehat{\sigma_{a}^2}$) and I was wondering if anyone knew of a R implementation (or any easily implementable algorithm) of how to calculate this quantity.  
 A: (Leaving my previous answer to the wrong question in tact for posterity, hopefully this time I'm answering the question actually being asked...)
A question about the variance of the variance estimates was recently posted on R-SIG-MIXED-MODELS. Ben Bolker, one of the lme4 authors, has already worked out how to do this for ML estimates, for REML the problem is apparently a bit harder due to the internal parameterization (links below). 
The full answer is a bit long, but the basic idea is to use confidence intervals, as I suggested in my comment. Modern lme4 provides only profile and bootstrap confidence intervals for the random-effect components, which aren't as straightforwardly related to the variance/standard error of those estimates as the Wald confidence intervals are, but perhaps provide the better measure of the estimate's variability. If you do want to go the Wald confidence-interval route, from which you can rapidly compute the standard error and hence the variance on those estimates, then check out Ben Bolker's longer explication (with code). There is also an older version that not completely identically in methodology and focus (much in the same way that nlme differs from lme4, that might be worth taking a look at.
A: If you are willing to fit the mixed model using ANOVA Type-1 estimation you can use R-package VCA which has two approaches to estimation of the variance of variance components implemented following Searle et al. (1992) "Variance Components" and alternatively an approximation of Giesbrecht and Burns (1985) Two-Stage Analysis Based on a Mixed Model: Large-Sample Asymptotic Theory and Small-Sample Simulation Results, Biometrics 41, p. 477-486.
