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.


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