Suppose that \(Y_{1}, \ldots, Y_{n}\) are independent and identically distributed with density

\[
\frac{1}{2 \pi}(1+\psi \cos y)(1+\sin y / 2)
\]

on the interval \(-\pi
Show that the vector statistic \(R=\cos Y\) is sufficient and that \(S=\sin Y\) is ancillary, ie., that \(S\) is distributed independently of the parameter. Show that the conditional distribution of \(R\) given \(S\) is discrete, a Bernoulli multiple with independent components. Find the conditional likelihood, and compare it with the unconditional likelihood in Exercise 12.8.

Data From Exercise 12.8

Suppose that \(Y_{1}, \ldots, Y_{n}\) are independent and identically distributed with density

\[
\frac{1}{2 \pi}(1+\psi \cos y)
\]

on the interval \(-\pi
Show that the log likelihood is concave. What does this imply about maximumlikelihood estimation?