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.