Question

Negative binomial probability function:

T(kx fx(x) P(X1) = T(2 +1)F(k) k k+j

$\mu>0$ is the mean

is the dispersion parameter

Let there be two groups with numbers and means of $\mu_1, \mu_2$

1) Write down the log-likelihood for the full model

2) Calculate the likelihood equations and find the general form of the MLE for $\mu_1$ and

3) Write down the likelihood function in the reduced model (ie. assuming ) and derive the MLE for $\mu$ in general terms

4) Using the above answers only, give the MLE and standard error for $\delta$ where $\delta=\mu_1-\mu_2$

5) Compute the expected information and the asymptotic variance-covariance matrix of the MLEs in the full model

6) Give the formula for an approximate Wald test of $\delta=0$

Answer 1

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