Suppose y is a discrete random variable that potentially take values on the following set of integers,
y = {0, 1, 2},
with the following probability mass function parametrized by θ € [0, 1]
P(y=0) = 1-θ
P(y = 1) = 1/2 θ
P(y = 2) = 1/2 θ
Suppose we observed a i.i.d. sample of n= 10 observations taken from the distribution above,
y = (y_1, y_2, ..., y_n) = (0,1,2,1,0,1,2,2,1,0).
We are interested in estimating θ based on data sample y. Answer the following questions.
1. MLE
(a) Derive the log-likelihood function of the data sample log L (θ; y) = P(y|θ). (b) Take the first derivative of log L (θ;y) with respect to θ, and derive the analytical expression for θ_MLE. Based on your data sample y, compute θ_MLE numerically.
(c) Take the second derivative of log L - (θ;y) with respect to θ and obtain the analytical expression for the Hessian H (θ) of log-likelihood. Verify that H (θ_MLE) < 0.
(d) Compute the standard error of θ_MLE, denoted as se (θ_MLE).
