I am looking at an estimator that counts the ratio of successes in a situation where $n$ people do $m$ Bernoulli trials each, except all the trials after the last success are eliminated. The probability $p$ of success is the same for all people and trials. To give you an example, here is one possible sequence with $n =5, m = 5$, where $0$ denotes failure and $1$ denotes success. $$ 101 \\ 1101 \\ 01\\ 00101\\ 0001\\ $$ so with the failures put in $$ 10100 \\ 11010 \\ 01000\\ 00101\\ 00010\\ $$ There are 9 successful trials out of 18 trials. That gives us $ \hat{\theta} = \frac{9}{18}=\frac{1}{2} $.
I want to calculate the bias of the estimator and want to see if it's consistent.
Now here's what I have done and tried so far.
Normally, in the case where all the trials are counted in the ratio, we would arrive at $ \hat{\theta} = \frac{X}{nm}=\frac{9}{25}$. That would be the MLE estimator $$ \hat{\theta}=\frac{X}{n'}\\n' = nm$$ for a Binomial distribution with probability $p$ and $nm$ trials; its bias would be zero and it would also be consistent. I am not sure how to take into account those "missed" trials in the denominator. I think that changes the estimator completely.
Because the "missed" trials are all failures I tried to come up with the probability of there being $y$ failures all starting at place $k$ in a sequence of $n$ Bernoulli trials (my thinking here was that, if we have a sequence of length 3, say 101, we know that there is a successful trial at place 3 and $m-3$ failures right after). By taking those probabilities into account I could calculate the average number of those failures and take them into account in the denominator. I've not got far though. Any suggestions?