NB = Number of trials(e.g. say coin toss) "n" required for k success(heads of coin)
p/probability = fairness of coin is to be estimated
{
pmf Derivation
The trial ends on observing kth success. So nth trial has the kth success.
So n-1 trial have k-1 success and those success can be distributed in all possible ways among n-1 location i.e.
n-1Ck-1 ways, and each way/pattern having probability of p(k-1)(1-p)(n-k)
And we know that probability of success at nth position is "p".
Probability of n-1 trials and nth trial are independent so we can multiply them to get join distribution
so
PMF = p* n-1Ck-1 p(k-1)(1-p)(n-k)
= n-1Ck-1 p(k)(1-p)(n-k)
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Compounding probability distribution
Lets say p = Beta(α, β) i.e. parameter
P(no of trials=n, prob of success = p)
= NBpmf(n, k, p) * Betapmf(p|α, β)
For each value of "p" between 0 and 1, we calculate above probability and sum it up and thus get rid of variable p. [Posterior predictive distribution]
Beta Compounded Negative Binomial PMF = ∫ NBpmf(n, k, p) * Betapmf(p|α, β)
BNBpmf = ∫ n-1Ck-1 p(k)(1-p)(n-k) * p(α-1)(1-p)(β-1)/B(α, β) dp
= n-1Ck-1/B(α, β) ∫ p(k+α-1)(1-p)(n-k+β-1) dp
= n-1Ck-1 B(k+α, β+n-k)/B(α, β)
}
Now assume following sample is observed for "n" i.e number of trials for achieving k success and we want to estimate p - probability of success
[n1 , n2 , n3 , n4...….nm]
pmf1 = n1-1Ck-1 p(k)(1-p)(n1-k)
Joint distribution of all the observed sample that are independent will product of each pmf
L = i=1∏m pmfi
Take log to make taking derivative simple,
LL = i=1Σm log(pmfi
= i=1Σm log(ni-1Ck-1 p(k)(1-p)(ni-k))
For maximizing Log likelihood take derivative and equate to zero
⛛LL =
p = mk/Σni
