Johanna G. Nešlehová Mc , Winter Term 2022
Generalized Linear Models MATH 523 Practice Problems for Week 3 Q1 Lecture 3a
Consider the Negative Binomial distribution with parameters μ > 0 and θZ > 0; the corresponding probability mass function is given by
Γ(y+θz) θz θz μ y
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f (y; μ, θz ) = Γ(y + 1)Γ(θ ) μ + θ μ + θ , y = 0, 1, . . . ,
where Γ(·) denotes the Gamma function. Assume throughout that θZ, the “number of
successes until the experiment is stopped”, is known.
(1) Show that the Negative Binomial family with known θZ is an exponential disper- sion family. Identify the functions a, b, and c, and the canonical and dispersion parameters.
(2) Using the formulas derived in class, calculate E(Y ) and var(Y ) of a Negative Binomial random variable Y .
(3) Find the mean-variance relationship.
(4) Find the canonical link for a Negative Binomial GLM and discuss its pros and cons.
(5) Can you think of another link function that might be more appropriate than the canonical link?
This problem is part of Assignment 1, due on January 28 at noon.
Q2 Lecture 3b
Complete the file RClass2-ExploringLinkFunctions-DoItYourself.Rmd, knit it to a PDF and submit the PDF via Crowdmark.
Due January 24 at noon.
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