Homework Problem 10, Due Friday October 21, 2016.
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Assigned 10/11/2016, due 10/21 by 5pm.


Consider the model

Y_i = exp(alpha+beta*X_i) + gamma*X_i*eps_i    i.i.d. , i=1,...,n

where (X_i,Y_i)  are observed data-pairs, and X_i ~ Unif[0,1],
with eps_i ~ N(0,1)  independent of X_i  and with parameters
(alpha,beta,gamma) unknown.


(1) Show by simulation that the maximum likelihood estimators (MLEs) 
of (alpha, beta,gamma), considered as functions of the dataset

(X_i,Y_i), i=1,...,n   are approximately normally distributed when n=150.

To do this, first fix a set of three parameter values (alpha, beta, gamma) 
more or less arbitrarily, say alpha=1, beta=2, gamma = 0.5. (I suggest that
you take gamma positive, but alpha and beta can be any real numbers.

Next write code to simulate a batch of 150 X's and the use the model 
formula to write Y_i as conditionally normally distributed with mean and 
variance given X_i, using the fact that eps_i ~ N(0,1) is independent of X_i.
This conditional normal distribution should tell you how to simulate the Y_i's 
given the X_i's within your batch of data, i=1,...,150.

Simulate enough batches of this size (e.g, 500 or 1000) that you can give 
some quantitative bounds (confidence intervals) on the differences between 
the actual distributions of these MLEs  and their approximating normal 
distributions with same means and variances.


(2) Repeat this exercise for a few smaller values of n, e.g. 100, 80, 50.

At which smaller value of n would you say the normal approximation for 
the MLEs has broken down ?


I AM NOT LOOKING FOR A SPECIFIC BREAKPOINT IN n, JUST A QUANTIFICATION 
OF APPROXIMATE NORMALITY THAT SHOWS SOME DEFECTS WHEN n > 20  GETS 
SMALL ENOUGH.

ALSO: IT IS ENOUGH IN THIS PROBLEM TO LOOK ONLY AT THE MARGINAL 
DISTRIBUTIONS OF EACH OF THE PARAMETER MLE's for alpha, beta, and gamma, 
NOT FOR ANY ASPECT OF THEIR JOINT DISTRIBUTION.