How to compute the standard error of an L-estimator? I am trying to compute the standard error of the sample spectral risk measure, which is used as a metric for portfolio risk. Briefly, a sample spectral risk measure is defined as 
$q = \sum_i w_i x_{(i)}$, where $x_{(i)}$ are the sample order statistics, and $w_i$ is a sequence of monotonically non-increasing non-negative weights that sum to $1$.  I would like to compute the standard error of $q$ (preferrably not via bootstrap). I don't know much about L-estimators, but it looks to me like $q$ is a kind of L-estimator (but with extra restrictions imposed on the weights $w_i$), so this should probably be an easily solved problem. 
edit: per @srikant's question, I should note that the weights $w_i$ are chosen a priori by the user, and should be considered independent from the samples $x$.
 A: As you know, from
$$Var[q] = Var[\sum_i w_i x_{(i)}] = \sum_i\sum_j w_i w_j Cov[x_{(i)}, x_{(j)}]$$
it follows you need only compute the variances and covariances of the order statistics.  To do this, diagonalize the covariance matrix!  Although this cannot be done in general, M. A. Stephens has obtained (heuristically) an asymptotic diagonalization.  (The eigenvectors are Hermite polynomials.)  In the spirit of PCA, limiting your calculations to the largest few eigenvalues can greatly reduce the computational effort and might produce a reasonable approximation, depending on the structure of the $w_i$.  In fact, if you adjust that weight vector to be a linear combination of a small number of the eigenvectors, that will assure a simple calculation of $Var[q]$ and perhaps not cost you much in terms of the accuracy of $q$ itself.  At worst, a preliminary eigendecomposition of $\vec{w}$ will then require only $O(N)$ calculations of the variance rather than $O(N^2)$.
A: It looks like I am probably stuck with a bootstrap. One interesting possibility here is to compute the 'exact bootstrap covariance', as outlined by Hutson & Ernst. Presumably the bootstrap covariance gives a good estimate of the standard error, asymptotically. However, the approach of Hutson & Ernst requires computation of the covariance of each pair of order statistics, and so this method is quadratic in the number of samples. Maybe I should just stick with the bootstrap!
