# Hypothesis testing over probability density function

I have PDF of time variable, and I'd like to create statistical test in order to decide if new data point derived from my distribution.

Say I'm setting a p-value of alpha. My question is how should I find my "rejection area". Should I search for a c (cutoff) value which all points' (e.g., minutes) densities above this cutoff summarised into (1-alpha) or should I do the search of this cutoff over the points' probabilities (in that case I'll compute the probability over minutes' interval)?

Hope my question is clear. Thank you in advance!

It's hard to test if a single point is from any distribution, unless that is clearly outside the domain of your PDF (e.g., if you're using a Beta or Uniform distribution).

For unbounded distributions, you generally want to have a lot of points and then use one of a number of goodness of fit tests to get some idea of the likelihood your data came from this distribution.

I am assuming your PDF is unimodal. In that case, one sensible way to test the point is to determine your acceptable long-run error rate (e.g., 5%) then find the shortest interval that contains 95% of the probability. There is a 5% chance your point will fall outside of this interval even if it comes from your distribution.

If you are dealing with a multi-modal distribution, you can define a set of intervals that collectively contain 95% of the probability (e.g., the set of highest density intervals that collectively contain 95% of the probability).

• Thank you very much. It is a very good answer. I'd like to ask you one more thing. My data is actually bounded (time of the day - continuous but bounded) and multi-modal.. so, prectically, I should: 1. caculate the probability of (for examle) each minute during the day from my pdf; 2. Sorting those probabilities in descending order; 3. Summarised the probabilities until I get 95% (assume the acceptable long-run error rate is 5%); 4. Set my cutoff as the probability of the "first point left out".. am I right? – staove7 Apr 26 '17 at 17:09
• @staove7 this is an open-ended problem you have. You've outlined one approach. Basically, your approach will define some intervals (plural) of time. The times not in these intervals constitute your "rejection region" – user145807 Apr 26 '17 at 17:11