**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

Y

_{i}= 2sinh^{-1}√ (X_{i}/K)

It can be shown that Yi is normally distributed with mean
2sinh ^{-1}√Ph_{i} and variance K-1.

Using a series expansion, one has:

sinh ^{-1}√Ph_{i} =log2√Ph_{i}
+_ Ph_{i}– (3/32) (Phi)2 + …………,
Ph_{i}>1

Hence, if Ph_{i} is large sinh ^{-1}√Phi
can be approximated by log2√Ph_{i}

It follows that

W

_{i}= 2sinh^{-1}√(X_{i}/K) - log√h_{i}

Is assymptotically normal with mean log2√P and variance
K^{-1}

This result suggests the following iterative method for estimating K.

- Step 1: Choose a start value of K, say K
_{0}. - Step 2: Calculate the Y
_{i}’s from observed X_{i}and K_{0}. - Step 3: Calculate sample variance of W
_{i}’s denoted by S12(W) and define K1 by

K_{1}^{-1}= S_{1}^{2}(W) - Step 4: Repeat the procedure using K1 to calculate Wi’s and
define K2 by

K_{2}^{-1}= S_{1}^{2}(W) - Step 5: Stop the procedure when the estimates of K become approximately
equal,

that is, | K_{i+1}- K_{i}|±ε

μ = Σ X

_{i}/ Σ h_{i}

i=1 i=1

and P is estimated by

P = μ / K