# Books on non-parametric theory with correlated data

Short question: I am looking for books that deal with correlated data in a systematic and theoretical way.

Long version:

Right now I am developing approximation algorithms for time series data (more concrete I want to approximate a function $f:[a,b]\rightarrow\mathbb{R}^3$ where $[a,b]$ is a time interval). The raw approximation of that data shall not be of concern for this question.

However, what has a certain relevance for my works is a proof of correctness of the developed algorithm. That means that I have to take the measurement error of the data that is to be approximated into account. Problem with that is that I do not know a lot on the distribution of the error. Furthermore, I know that high absolute errors are likely to be correlated. Thus, I need to use statistical methods that do not fail completely with correlated data for my theoretical examination of the data. (Intuitively I would guess that quantiles already might be enough for my purpose, if I can justify their use reasonably.)

My problem with that is that I do not have a strong knowledge on methods that hold with correlated data. All statistical methods I remember have that small but important iid among the list of their requirements.

Thus, the question is: Can you recommend any good books that build up statistical methods for correlated data? As I'm looking for methods and used theory, I do need a book dealing with probability theory rather than just practical methods. My background on statistics is a bit rusty but certainly should be enough for understanding theoretical books as well. For me it is important to gain an understanding on the mathematics used for that kind of problem rather than just to get to know a lot of different tests.

You could try:

Kohn R., Schimek M.G., Smith M. (2000) Spline and kernel regression for dependent data. In Schimekk M.G. (Ed) (2000) Smoothing and Regression: approaches, computation and application. John Wiley & Sons, Inc.

I used an example from that chapter to illustrate one of the problems of smoothing correlated data on my blog a while back.

• No worries @Thilo – Gavin Simpson Nov 12 '12 at 21:21

I know P K. Sen's book "Sequential Nonparametrics" deals with data correlated in time. I am not sure how much detail it goes into time series. I suspect it is very theoretical and does not deal with specific time series models like ARMA. Some frequency domain tests for white noise are nonparametric. I am thinking of Fisher's test in particular.

If you want to do this using the bootstrap approach you can use Lahiri's book (which is heavily mathematical). The title is "Resampling Methods for Dependent Data" published by Springer-Verlag in 2003.

• @Chl Why did you change my answer to a comment?? The question was to cite some sources for nonparametric methods on dependent data. The citation was an answer. It is different from my first asnwer because he changed the question from being about time series to any type of dependent data. – Michael R. Chernick May 24 '12 at 20:14
• I can accept combining the answers. That would be right. – Michael R. Chernick May 24 '12 at 20:20
• Sorry, @Michael, if my action was misunderstood there. I've just merged your two replies into a single one. (I noticed you often leave separate replies, and I was, in this particular case, responding to a flag raised by one user.) Multiple replies are ok when these are different replies to the original post; otherwise, it's better to update your post. If the question has been updated, you can change your answer as well. We keep an history of every change there, as you may know. Thanks for your understanding. – chl May 24 '12 at 21:05
• That is fine Chl. – Michael R. Chernick May 24 '12 at 21:53