Negative Binomial GLM: The negative binomial probability mass function is often

Question

Negative Binomial GLM: The negative binomial probability mass function is often written as: f(Y =y) = ,y=0,1,2, where is a dispersion parameter (a) Arrange this distribution in an exponential family form (Equation 4.17 in Agresti) anod identify all the relevant components. Assume the dispersion parameter is known. (b) What is the canonical link and variance function for the negative binomial GLM with known ? (c) Using the components of the exponential family form, find the expectation and variance of Y. (d) Using the components of the exponential family form, find the likelihood equations. (e) Instead of the canonical lnk, now assume a log link. Find the likelihood equations. (f) Why do you think the log link is commonly used instead of the canonical link for the negative binomial GLM?

Explanation / Answer

Negative Binomial GLM: The negative binomial probability mass function is often written as: f(Y =y) = ,y=0,1,2, where is a dispersion parameter (a) Arrange this distribution in an exponential family form (Equation 4.17 in Agresti) anod identify all the relevant components. Assume the dispersion parameter is known. (b) What is the canonical link and variance function for the negative binomial GLM with known ? (c) Using the components of the exponential family form, find the expectation and variance of Y. (d) Using the components of the exponential family form, find the likelihood equations. (e) Instead of the canonical lnk, now assume a log link. Find the likelihood equations. (f) Why do you think the log link is commonly used instead of the canonical link for the negative binomial GLM?

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