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I am interested in how to validly bootstrap data with an unknown correlation structure. Let's say I am bootstrapping in order to obtain inference for some smooth function of the data similar to a sample mean.

I found an authoritative review paper (1) and several papers discussing bootstrap approximations to GEE (2, 3), and of course there is a large literature on bootstrapping time series data, etc. However, these approaches seem to assume a known correlation structure or at least that there are known clusters (e.g., subjects each with repeated observations), even though GEE does not require this assumption.

How would I bootstrap if I knew nothing about the correlation structure? Basically I want a bootstrap equivalent to, for example, fitting GEE with the assurance that I still have asymptotic consistency even if I misspecify the covariance structure.

And what would be the consequences of using the plain-vanilla nonparametric bootstrap (i.e., resampling individual observations with replacement)? Would this approach be conservative, anticonservative, or indeterminate?

References

1. Field, Christopher A., and Alan H. Welsh. Bootstrapping clustered data. Journal of the Royal Statistical Society: Series B (Statistical Methodology) 69.3 (2007): 369-390.

2. Cheng, Guang, Zhuqing Yu, and Jianhua Z. Huang. The cluster bootstrap consistency in generalized estimating equations. Journal of Multivariate Analysis 115 (2013): 33-47.

3. D'Angelo, Gina M., et al. Bootstrapping GEE models for fMRI regional connectivity. Neuroimage 63.4 (2012): 1890-1900.

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