Homework 3 in STAT630: Advanced Statistical Data Analysis @ CSU
[20 pts] (Boos and Stephanski 2.20) One version of the negative binomial probability mass function is given by \[ f(y; \mu, k) = \frac{\Gamma(y + k)}{\Gamma(k)\Gamma(y + 1)} \left(\frac{k}{\mu + k}\right)^k\left(1 - \frac{k}{\mu + k}\right)^y, \quad y = 0, 1, \dots \] where \(\mu\) and \(k\) are parameters. Assume that \(k\) is known and put \(f(y; \mu, k)\) in the generalized linear model form (page 37 of the notes), identifying \(b(\theta)\), etc., and derive the mean and variance of \(Y\), \(\text{E}(Y) = \mu\), \(\text{Var}(Y) = \mu + \mu^2/k\). Additionally, find the canonical link function. Why might we not want to use this link function?
[20 pts] Synthetic data are available here. The predictor variable is
x and the response variable is y. Perform a
negative binomial regression on this data using a log link function.
That is, use the model from problem 1, with \(g(\mu) = \log(\mu)\), \(\theta = \beta_0 + \beta_1 x\), and \(k = 20\). To do this problem, create a
likelihood function, and then use a numerical optimizer in R to find the
maximum likelihood estimates for \(\beta_0\) and \(\beta_1\). Report your estimates and a 95%
confidence interval.
[10 pts] (Boos and Stephanski 2.33) Derive the Fisher information matrix for the \(\text{multinomial}(n; \boldsymbol p)\) distribution.