Nandini Das, Indian Statistical Institute
Appendix 1 for Study on Implementing Control Charts Assuming Negative Binomial Distribution with Varying Sample Size in a Software Industry
For Xi defined previously, the usual moment estimators of K and P are not appropriate since the variance of Xi is the quadratic function of hi. The following transformation is used for this purpose. Let
Yi = 2sinh -1√ (Xi /K)
It can be shown that Yi is normally distributed with mean 2sinh -1√Phi and variance K-1.
Using a series expansion, one has:
sinh -1√Phi =log2√Phi +_ Phi– (3/32) (Phi)2 + …………, Phi>1
Hence, if Phi is large sinh -1√Phi can be approximated by log2√Phi
It follows that
Wi = 2sinh -1√(Xi /K) - log√hi
Is assymptotically normal with mean log2√P and variance K-1
This result suggests the following iterative method for estimating K.
μ = Σ Xi / Σ hi
and P is estimated by
P = μ / K