# 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!

## 1 Answer

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