Censored Poisson regression

I am working on a data that provides the number of failures within a period of time for multiple machiones, so I am considering Poisson regression. The problem with the data is that we don’t believe there can be over 5 failures within the period. For example, there are over 70 failures for some machines, and that can’t be true. The censoring here means whenever there are over 5 failures, we believe that is greater or equal to 5 failures, not the obvious wrong number.

I read through the censoring chapter in the Stan Manual. There are descriptions on upper and lower censoring for normal distribution with normal_lccdf() and normal_lcdf function, but not for Poisson distribution.

I guess we can customize a censored log of complementary cdf for a Poisson regression just like the normal distribution. But since I am quite a newbie in Stan, I wonder if there is anybody who wrote this censored Poisson regression code before and share with me.

I don’t know what this means—the last sentence seems to contradict the first about the possible number of failiures. Are you saying that in practice that you treat observations > 5 as upper censored at 5?

In general, we tend to not use hard bounds, even if the data is bounded, because new data might not have the same bounds.

Is the 5 upper limit some kind of physical constraint, like failures take 2 hours and the time period is 10 hours, so there can’t be more than 5?

It works exactly the same way, except the bounds have to be non-negative integers.