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 $$NB({\mu }_i=m_i\lambda ,\varphi )$$, 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:

$$\begin{array}{}\text{(1)}& f(y_i;\mu ,\varphi )=P(Y=y_i)=\frac{\mathrm{\Gamma }({\varphi }^{-1}+y_i)}{\mathrm{\Gamma }({\varphi }^{-1})\mathrm{\Gamma }(y_i+1)}{\left(\frac{1}{{\varphi }^{-1}{\mu }^{-1}+1}\right)}^{y_i}{\left(\frac{1}{\varphi \mu +1}\right)}^{{\varphi }^{-1}}\end{array}$$

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

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

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

$$l_{Y|Z=z}(\varphi )=[\sum _{i=1}^n\log \mathrm{\Gamma }(y_i+{\varphi }^{-1})]+\log \mathrm{\Gamma }(n{\varphi }^{-1})-\log \mathrm{\Gamma }(z+n{\varphi }^{-1})-n\log \mathrm{\Gamma }({\varphi }^{-1})$$

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: $$l_{Y|Z=z}(\mathbf{y};\varphi )=\log f(\mathbf{y};\mu ,\varphi )-\log f_z(z;n\mu ,n^{-1}\varphi )$$

$$=\sum \log \mathrm{\Gamma }(y_i+{\varphi }^{-1})-n\log \mathrm{\Gamma }({\varphi }^{-1})+\sum y_i\log \left(\frac{\varphi \mu }{1+\varphi \mu }\right)+n{\varphi }^{-1}\log \left(\frac{1}{1+\varphi \mu }\right)$$

$$-\log \mathrm{\Gamma }(z+n{\varphi }^{-1})+\log \mathrm{\Gamma }(n{\varphi }^{-1})-z\log \left(\frac{\varphi \mu }{1+\varphi \mu }\right)-n{\varphi }^{-1}\log \left(\frac{1}{1+\varphi \mu }\right)$$

$$=[\sum _{i=1}^n\log \mathrm{\Gamma }(y_i+{\varphi }^{-1})]+\log \mathrm{\Gamma }(n{\varphi }^{-1})-\log \mathrm{\Gamma }(z+n{\varphi }^{-1})-n\log \mathrm{\Gamma }({\varphi }^{-1})$$