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I have a random variable bounded on [0,1]. The mean can be modeled as a random process. Assume the change of the mean |mu_i - mu_{i+1}| is bounded with a known bound. Given N (N is not large) samples (X_1,..., X_N) I want to estimate the upper confidence bound on the mean.

I was unable to find anything in the literature and was hoping someone might have some pointers to relevant research :)

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You have to look for the distribution of the difference between two consecutive samples pdf(|X_i - X_{i+1}). This distribution can be compared to analytically (if the probability density functions pdf are nice) or Monte-Carlo-generated distributions (if pdf is not so nice) with given bound called the expected distribution. You can use standard methods like a fitting routine to get the best matching bound and the corresponding confidence intervals. Here the low number of samples will challenge you. Here it is important to use the correct pdfs in advance for the construction of expected distributions, otherwise the calculation might give misleading results.

I call the former described approach the first order estimate, as you only use information of directly consecutive samples. However I assume more information is hidden in higher orders, e.g. the next-to-next sample and so on. The way is equivalent, but at the and you can fit multiple distributions simultaneous, which should lead to more accuracy.

I'm sorry that I cannot give you any pointers to research

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