Homework 8.2: MLE of microtubule catastrophe data (35 pts)

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Refresh yourself about the microtubule catastrophe data we explored in homeworks 3.3, 6.2, and 7.1. We will again work with this data set here.

a) In their paper, Garnder, Zanic, and coworkers modeled microtubule catastrophe times as Gamma distributed. Perform a maximum likelihood estimate for the parameters of the Gamma distribution. Because you showed in homework 7.1 that there is little difference between labeled and unlabeled tubulin, you only need to work this out for the labeled tubulin now and in part (b). Be sure to include confidence intervals for your MLEs and discuss the method you used to get the confidence interval.

b) Obtain a MLE estimate for the parameter \(\beta_1\) and \(\beta_2\) from the model you derived in homework 6.2c. As a reminder, you derived that the PDF for microtubule catastrophe times is

\begin{align} f(t;\beta_1, \beta_2) = \frac{\beta_1\beta_2}{\beta_2 - \beta_1}\left(\mathrm{e}^{-\beta_1 t} - \mathrm{e}^{-\beta_2 t}\right). \end{align}

Again, include confidence intervals. Be careful; this is a very tricky calculation. It is possible to analytically compute the MLE. If you choose to do it numerically, you need to think about what happens when \(\beta_1 \approx \beta_2\). You also need to think about how you will handle the log of sums of exponentials.