HW 17 Stat 705  Fall 2015

Assigned Monday 11/30/15 DUE Wednesday 12/9/15

In HW12, you previously analyzed the dataset "Insurance" from the MASS 
package using linear regression techniques.Now we will consider the data 
as being subject to a Poisson model, 

         Claims ~ Poisson(Holders * lambda)               (*)

(A) Find the MLE for lambda, and also note that if  lambda  has a prior 
density Gamma(1,0.1), then the posterior density is proportional to
lambda^(sum(Y_i)) * exp(-lambda*(0.1+sum(X_i)))  and is therefore again 
Gamma, where Y_i = Claims[i] and X_i = Holders[i].

NOTE that the MLE for lambda in this problem has a closed-form formula, 
which you will want to find for part (C).

------------

(B) For this dataset, implement 4 methods to simulate (arbitrarily many) 
"pseudo"-datasets  (Y_1,...,Y_{64}), as follows.For methods (1) and (2), 
the vector of numbers  (X_1,...,X_{64}) of Holders is regarded  as fixed (nonrandom).
(1) Generate the Y pseudo-sample via parametric bootstrap.
(2) Generate the Y sample by drawing a random parameter  lambda  from 
the posterior density of lambda given the observed Y-data in "Insurance",
and then sample new batches of 64 Y's (for the same fixed X's) using the 
Poisson model.
(3) Generate a nonparametric bootstrap sample of X's (by iid 
sampling with replacement fom the given numbers { Holders[1], ..., Holders[64] }, 
and then a parametric bootstrap of Y's conditionally given X's using the 
model (*) with the MLE-estimated parameter  lambda . 
(4) Generate a nonparametric bootstrap sample of (X_i,Y_i)  pairs (iid 
sampled with replacement from the set of 64 pairs).

------------

(C) Apply the functions you wrote in generating pseudo-samples 4 for 
methods (1)-(4), with 1000 replications each, to find a 90% two-sided 
confidence interval [called "credible interval" in the Bayesian setting] 
for  lambda  based on the original dataset. The interpretations of these 
intervals are slightly different when generated by the 4 methods: give the 
best verbal description you can for how to interpret each one.

NOTE: your intervals should ideally allow probability 0.05 of falling below 
the true value and 0.05 of falling above it.