#StackBounty: #probability #hypothesis-testing Checking technical credit default cases with hypothesis testing

Bounty: 50

My question comes from the field of banking but turns out to be a general statistical one.
Let us assume a model might exist that tells us the default probability $PD_i$ of case $i$.

However, there are circumstances when a technical deficiency results in a big chunk of cases being marked as default while these are actually performing credits. Such chunks of technical defaults can usually be recognized (by time stamp e.g.) and then analyzed together.

The question is: in statistical terms what is a good model to determine the number of cases that I need to check for being a real default or just a technical one to get confidence for the whole chunk?

I thought about a hypothesis of the following kind:
$$
H_0: {text{the fraction of defaults in the chunk is } p }
$$

Then if I select $k$ cases and can confirm that none of them was a genuine default the probability (assuming a binomial distribution, a hypergeometric one with some extra care is possible too) is
$$
P[text{no genuine default among } k text{ cases}] = (1-p)^k
$$

and I would choose $k$ high enough to get e.g.
$$
(1-p)^k < 0.05.
$$

The question is how to determine $p$. One could derive $p$ from the PD model of the portfolio.
If I can reject $H_0$ at the 5% significance level can I accept the alternative hypothesis of
$$
H_1: {text{the fraction of defaults in the chunk is less than } p }
$$

or even
$$
H_1′: {text{the fraction of defaults in the chunk is } 0 }?
$$

I think this has something to do with sample size choice for survey design. Could anyone please point me in the right direction? Thanks in advance.


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