Many thanks to Xavier de Luna, Howard Grubb, David Jones and
Richard Lockhart who replied to my query:
> Under `mild conditions' on the autocorrelations, the variance
> of the average of a time series of length $n$ is $O(n^{-1})$,
> i.e. of form $a/n$ for large $n$. Is any corresponding
> result known for the median of the series?
>
> (If the data were white noise, the answer would be that
> the variance of the median is also $O(n^{-1})$. Does
> the presence of autocorrelation have a dramatic effect?)
A partial answer is given by Gastwirth and Rubin (Annals of
Statistics, 1975, page 1070), who show that for Gaussian
series with `mild dependence' the variance of a robust
estimator of location is $O(n^{-1})$ in some generality;
they treat the special case of the median.
They also give results for a few other special models, but
there seems to be no obvious reason that the results would
not hold more widely. Some discussion of this is given in
Section 8.3 of `Statistics for Long-Memory Processes' by
Jan Beran (1994), Chapman and Hall.
The asymptotic efficiency of the median relative to the mean
depends on the autocorrelations; the median can be more
efficient than the mean in some cases (unlike with white
noise where the efficiency would be $2/\pi<1$). But at
least for the Gaussian case with short-range dependence the
variances always seem to be $O(n^{-1})$ (modulo
pathologies where certain combinations of autocorrelations
equal zero).
Many thanks again.
--
Professor A. C. Davison
Department of Mathematics
Swiss Federal Institute of Technology Lausanne
CH-1015 Lausanne EPFL
Switzerland Tel: + 41 (0)21 693 5502
Sec: + 41 (0)21 693 2565
http://statwww.epfl.ch/ Fax: + 41 (0)21 693 4250
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