Homework2:




1. Given an iid sample of observations: y1, y2, ... yn with
density function f(y, \theta). Assume \theta is a one dimensional parameter.

Show that the score test and the Wald test are asymptotically equivalent.
(Please list clearly the conditions needed and also clearly explain
what is "equivalent" means)


2. (Logistic regression model) Suppose observations Y1, Y2, .... Yn are
independent Bernoulli random variables with success probability

p_i =  exp( \beta x_i ) / [1+ exp( \beta X_i )]

where \beta is a parameter and x_i are observed constants.

Compute the Fisher information about \beta based on the sample (Y_i, x_i); i=1, ... n.


3.  (Proportional hazards regression model) Suppose observations Y1, Y2, ..., Yn
are independent random variables with cumulative hazard function

  \Lambda_i(t) =  t^2 exp( \beta x_i ) 


Compute the Fisher information about \beta based on the sample (Y_i, x_i) i=1, ... n.

Now suppose the cumulative hazard function is

   t^{\alpha} exp( \beta x_i )
  
where \alpha is a nuisance parameter. What is the (effective) Fisher information
for \beta in this model?