Estimate parameters of Negative Binomial Distribution - NB?

 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  k_th success. So n_th trial has the k_th 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-1C_k-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 n_th trial are independent so we can multiply them to get join distribution

so

PMF = p* ^n-1C_k-1 p^(k-1)(1-p)^(n-k) 

^= n-1C_k-1 p^(k)(1-p)^(n-k)

----------------------------------------------------------------------

Compounding probability distribution

Lets say p = Beta(α, β) i.e. parameter

P(no of trials=n,  prob of success = p)

 = NB_pmf(n, k, p) * Beta_pmf(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 = ∫ NB_pmf(n, k, p) * Beta_pmf(p|α, β)

BNB_pmf = ∫ ^n-1C_k-1 p^(k)(1-p)^(n-k) * p^(α-1)(1-p)^(β-1)/B(α, β) dp

= ^n-1C_k-1/B(α, β)  ∫ p^(k+α-1)(1-p)^(n-k+β-1) dp

= ^n-1C_k-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...….n_m]

 pmf₁ = ^n1-1C_k-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 pmf_i

Take log to make taking derivative simple, 

LL = _i=1Σ^m log(pmf_i

=  _i=1Σ^m log(^ni-1C_k-1 p^(k)(1-p)^(ni-k)) 

For maximizing Log likelihood take derivative and equate to zero

⛛LL = 

p = mk/Σn_i