Let y binomial(n, ). We consider the maximum likelihood based inference on the parameter . (a)

Let y ∼ binomial(n, π). We consider the maximum likelihood based inference on the parameter π.

(a) Write down the log-likelihood function L(π).

(b) Find the score function u(π) = y π − n − y 1 − π and the Fisher’s information I(π) = n π · (1 − π) .

(c) Find the MLE of π, ˆπ. Show that E(ˆπ) = π and Var(ˆπ) = π(1 − π) n = 1 I(π) .

(d) Consider the test of H0 : π = π0. Show that the Wald test statistic has the form of G = (ˆπ − π0)2 πˆ(1 − πˆ)/n; the score test is given by S = (ˆπ − π0)2 π0(1 − π0)/n; and the LRT can be simplified as LRT = 2  y log y nπ0 + (n − y) log n − y n − nπ0  .