Suppose to have a set of exact data $x_i, i=1,\dots,N$ and to calculate its median value $m$. Then, a sound way to estimate the error $\delta m$ on $m$ would be bootstrapping. (I think...)

But what if each $x_i$ has an associated error $\delta x_i$? In this case bootstrapping would not retain the information on the error of the single data.

Is this loss of information a common assumption when using bootstrap? Is there some other method to calculate $\delta m$ that retains the information given by the $\delta x_i$?

  • $\begingroup$ (1) You don't need bootstrapping to estimate the sample distribution of the median: see stats.stackexchange.com/questions/45124. (2) When the data are measured with error, what is the problem? True, you cannot separate the measurement error variance from the variance in the underlying values, but that's the only issue; and it points to the only effective solution, which is to find a way (through observational design and/or modeling) to estimate the measurement errors. $\endgroup$ – whuber May 25 '16 at 13:56

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