﻿ Appendix - Estimation of K and P for Negative Binomial Control Chart - ASQ

# Appendix - Estimation of K and P for Negative Binomial Control Chart

Contents

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.

• Step 1: Choose a start value of K, say K0.
• Step 2: Calculate the Yi’s from observed Xi and K0.
• Step 3: Calculate sample variance of Wi’s denoted by S12(W) and define K1 by
K1-1 = S12(W)
• Step 4: Repeat the procedure using K1 to calculate Wi’s and define K2 by
K2-1 = S12(W)
• Step 5: Stop the procedure when the estimates of K become approximately equal,
that is, | Ki+1- Ki |±ε

μ = Σ Xi / Σ hi
i=1 i=1

and P is estimated by

P = μ / K