Assignment -
Solve the following using R software. Submit your R codes with the appropriate graphs.
(1) Firs Part:
(a) Give R code to generate (and then test the distribution in order to check your work) 5,000 pseudo-random variates from the density on [0, 7] proportional to:
1/(1+0.1*x^2)^2 + x^3 *exp(-x) + 0.2*(abs(x-3)<2) , 0 < x < 7
(b) Show by simulation of 10000 independent data batches of size 50 that the maximum likelihood estimators (MLEs) of each of the two parameters a,b from a dlogis(x, a, b) density (based on a sample of size 50) is approximately normal, with the mean and variance predicted from the Fisher Information for this estimation problem.
Can you detect any evidence of non-normality in the distributions of these MLEs?
(c) Generate 10000 independent random variables with density proportional to (1+2*x)*x^2*(1-x)^3 on (0,1). Do this by an efficient Accept-Reject method.
(2) Consider the problem of estimating/calculating the probability
P(X_(3) + X_(7) + X_(10) > -1.5 ) (*)
where X_i , I = 1,...,25 are independent random variables with standard normal (N(0, 1)) density and these variables are ordered to form the order-statistic sequence X_(1) < X_(2) < ... < X_{25)
Calculate (*) above in three ways, and compare their accuracy and speed:
(A) Direct Monte Carlo simulation -- you choose an appropriate number of batches of 25 iid variates to simulate to achieve accuracy comparable to the other methods.
(B) Simulation, but making use of antithetic variables.
(C) Simulation using the method of control variates, i.e. via linear regression of I[X_(3)+X_(7)+X_(10) > 4] either on X_(3)+X_(7)+X_(10) or on X_(3), X_(7), and X_(10).
(3) IMPORTANCE SAMPLING SIMULATION PROBLEM, adapted from the Robert and Casella (1999) book on Monte-Carlo, and was originally taken from a journal paper.
PROBLEM: Estimate by a Monte Carlo experiment, with and without importance sampling, the probability
P(W_j > 0 for j = 1,...6) (**)
where the (dependent) 6-vector random variables W are defined from a vector Z of 6 iid N(0, 1) random variables, by W = c( (inverse of (diag(0:5) + 0.5) ) %*% Z )
In your importance sampling approach, sample from a distribution in R^6 (6-dimensional Euclidean vector space) supported completely in the positive orthant (i.e., in the set of vectors with all positive coordinates.
Try to come up with an importance-sampling algorithm for a sampling distribution supported completely within the positive orthant in R^6; but if that requirement is too hard to meet, you may relax it.