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Maybe let me first describe the real world application of my problem before going to a more mathematical model.

Suppose I have a system that measures my current position up to an unknown error (let's say this would be a GPS-device). At every point that is recorded I also save the accurate position obtained by other means. Now I want to estimate an prediction interval of the recorded error, thus I want a reliable value that say 95 % of all data points are closer than the estimated value to the real position.

Which means I basically would like to estimate the probability distribution function of the values $$e_i:=|X_i-P_i|,$$ where $X_i$ is my recorded (random) position and $P_i$ the known reference value. Thus far everything is good. Problems do now arise looking at the sample values of the $e_i$. I see highly correlated data points and a distribution function that is far away from all parametric distributions I know of.

Do you know of a robust, non-parametric method of estimating the distribution (or at least one predefined quantile of that distribution) with correlated data? Sample sizes shall not be a problem, I have about 50000 data points available.

Of course one way would be to compute the empiric distribution function, but does this still work with correlated data? The limit theorem I know of assumes independent samples to guarantee correct asymptotical behavior.

Thanks for your input.

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    $\begingroup$ What makes the data points correlated? $\endgroup$ – Aniko Sep 9 '11 at 21:02
  • $\begingroup$ In practice: The system starts estimating the position whenever no position can be obtained through the given sensors. This usually works fairly well for the first point estimated and gets worse for every point that has to be estimated before a measured position can be received. Thus, bigger errors often do not come alone but in groups. Furthermore, as accuracy probably is a function of the position, the accuracy might change depending on the position (which, writing it this way, brings up the question if there could be done regression of some kind...). $\endgroup$ – Thilo Sep 10 '11 at 5:55

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