# Test if a sample comes from a theoretical distribution (with a specific type of censoring)

My situation is the following. I am estimating certain numbers from a dataset split in $$n$$ parts, one number for each part. How and why exactly is not relevant to the current question, but as a toy example which is not too far from the truth let's say that each part $$i$$ of the dataset corresponds to the results of $$k_i$$ tosses from a biased coin with probability $$p_i$$ of giving head and the number is the fraction of heads, approximating $$p_i$$. This way I get a sample of $$n$$ values $$(x_1,\ldots,x_n)$$ and I would like to test whether they can come from a given theoretical distribution (in my case, I want to check for normal in certain cases, gamma in others - different from the toy example where the values are between $$0$$ and $$1$$).

I have the additional problem that if the correct value for $$x_i$$ becomes too high or too low, then my estimation becomes wrong, and I can easily see this from the sample. In the toy example of the coins, this would correspond to the cases where I get $$0$$ heads (or $$0$$ tails). Thus, I want to drop these observations from my sample, which censors the data.

I am very new to this type of problems. What are some good tests/procedures I can use to test if my sample comes from a given theoretical distribution in this case? Both explanations and references are very welcome.

My thoughts so far

How I could proceed is as follows. Under the hypothesis of the sample being drawn from the given distribution, if I drop $$m$$ observations because the $$x_i$$ have too low value ($$0$$ heads in the toy example), then I would approximately be drawing from the given distribution clipped at the $$\tfrac{m}{n}$$ quantile, so I could simply apply a goodness of fit test for the clipped. I am not sure how good of a test this would be though.