How to derive conditional maximum likelihood of the sum of independent negative binomial variables?

Question

[Dan]

In the paper Small-sample estimation of negative binomial dispersion, with applications to SAGE data, section 4.1 Conditional maximum likelihood:

For RNA sequencing data, assume the counts for a single tag across n libraries is a negative binomial random variable. Consider Y1,⋯,Yn as independent and $$NB(\mu_i = m_i \lambda, \phi)$$, where mi is the library size (i.e. total number of tags sequenced for library i) and λ represents the proportion of the library that is a particular tag. The probability mass function is:

$$f(y_i;\mu,\phi)=P(Y=y_i)=\frac{\Gamma(\phi^{-1}+y_i)}{\Gamma(\phi^{-1})\Gamma(y_i+1)}\left(\frac{1}{\phi^{-1}\mu^{-1}+1}\right)^{y_i}\left(\frac{1}{\phi\mu+1}\right)^{\phi^{-1}} \tag 1$$

$$\text{E}(Y)=\mu$$ and $$\text{Var}(Y)=\mu+\phi\mu^2$$

If all libraries are the same size (i.e. mi≡m), the sum $$Z = Y_1 + \cdots + Y_n \sim NB(nm\lambda, \phi n^{−1})$$

How to derive conditional maximum likelihood for ϕ that is independent of λ, and how to calculate the maximum of the likelihood?

$$l_{Y|Z=z}(\phi)=\left[\sum_{i=1}^{n}\log\Gamma(y_i+\phi^{-1})\right]+\log\Gamma(n\phi^{-1})-\log\Gamma(z+n\phi^{-1})-n\log\Gamma(\phi^{-1})$$

Answer 1

[Dan]

I try to answer this question. Based on the Conditional Likelihood defined in https://rss.onlinelibrary.wiley.com/doi/epdf/10.1111/j.2517-6161.1996.tb02101.x, the conditional log-likelihood for ϕ of Y conditioned on Z, dropping terms that don't involve ϕ, is: $$l_{Y|Z=z}(\mathbf y; \phi)=\log f(\mathbf y;\mu,\phi)-\log f_{z}(z;n\mu,n^{-1}\phi)$$

$$=\sum\log \Gamma(y_i+\phi^{-1}) - n \log\Gamma(\phi^{-1}) + \sum y_i\log\left({\phi\mu \over 1 + \phi\mu}\right) + n\phi^{-1}\log\left({1 \over 1 + \phi\mu}\right) $$

$$-\log\Gamma(z+n\phi^{-1})+\log\Gamma(n\phi^{-1})-z\log\left({\phi\mu \over 1 + \phi\mu}\right) - n\phi^{-1}\log\left({1 \over 1 + \phi\mu}\right)$$

$$=\left[\sum_{i=1}^{n}\log\Gamma(y_i+\phi^{-1})\right]+\log\Gamma(n\phi^{-1})-\log\Gamma(z+n\phi^{-1})-n\log\Gamma(\phi^{-1})$$