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I have $m$ weakly stationary observations $X_1,X_2,\cdots,X_m$ from a Markov chain. I want to estimate the variance of the mean. My idea was to use nonparametric bootstrapping to make $n$ bootstrap samples from the $m$ observations and compute $n$ bootstrap means $\mu^*_i$. Then estimate the variance using $$ \widehat{\mathrm{Var}(\overline X)} = \frac{1}{n-1}\sum_{i=1}^n ( \mu^*_i - \overline \mu^*)^2 $$ Is it a problem that the observations are correlated, or is this fine?

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  • $\begingroup$ By "unparameterised bootstrap" do you mean nonparametric bootstrap or are you referring to something else? $\endgroup$ – Glen_b Jul 9 '17 at 13:58
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No you can't use standard bootstrap. There are many dependent bootstraps.

Jens-Peter Kreiss, Efstathios Paparoditis, Bootstrap methods for dependent data: A review, Journal of the Korean Statistical Society, Volume 40, Issue 4, 2011, Pages 357-378, ISSN 1226-3192, Keywords: Stochastic processes; Bootstrap methods; Time series

Here is a review , Here it is not behind paywall.

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