In a recent task at my job I have come with a situation I cannot find references to research made about it. Basically, I have two procedures and I want to test if they behave exactly the same way (i.e., they will always give the same response when fed with the same input).
More precisely, say I have procedure 1 which, when fed with an input, will give a sequence as a response, say ABAAB. This procedure is the industry standard, but it is expensive.
Now, there is a (cheaper) proposed procedure 2 which should behave exactly the same way. So, if for three consecutive and (maybe) dependent inputs procedure 1 outputs ABCAA, ACAAB and CABBA; I should see the same three sequences as output from procedure 2.
Lets call $X_n$ the Bernoulli variable which equals 1 if both procedures give the same answer and 0 otherwise. So $X_n \sim Ber(p)$ and my null hypothesis is $p = 1$.
I cannot figure how to test this hypothesis with a finite sample. I can easily propose tests for the hypothesis $p > 1 - \epsilon$, but I would like to check if there is some work done regarding the original problem. Thank you in advance.