Statistics 8701 (Geyer, Spring 2003) Homework No. 4

Policy

The policy is the same as the policy for homework number 2.

This assignment is due (initially) ?????

Problem 1

(Continues Problem 3 of Homework 3)

Use your sampler for the posterior distribution of the canonical parameter θ = logit(p) = log(p / (1 − p)) for binomial data with sample size 10 and are 3 successes to calculate

the posterior mean E(θ | x),
the posterior variance var(θ | x),
and the posterior median med(θ | x).

Calculate Monte Carlo standard errors (MCSE) for each. Use a large enough sample size so that the half-width of a 95% confidence interval (that is, 1.96 MCSE) for each is less than 1% of the quantity being calculated.

Calculating MCSE for the median directly is difficult, because the asymptotic standard error for a median depends on the density at the median, which is unknown here because of the unknown normalizing constant of the posterior density. It might just be simpler to use the exact confidence interval for the median associated with the sign test.

Problem 2

A problem about control variates. Estimate the probability that a bivariate standard normal random vector (mean zero and variance matrix the identity) falls in a regular hexagon inscribed in the unit circle (by the circular symmetry of the normal distribution it doesn't matter which inscribed regular hexagon you use).

(a)

Consider using as a control variate the empirical probability that the random vector falls in the unit circle, the theoretical probability of which is easily determined (explain how). Use a preliminary sample to estimate the regression coefficient.

(b)

Using a second, independent sample to estimate the probability of falling in the hexagon with and without control variate and provide Monte Carlo standard errors for both. What is the asymptotic relative efficiency of the two methods of estimation (ratio of asymptotic variances)?

Problem 3

Problem 5.7 in Gentle.

Problem 4

(Alternative Method for Problem 3 of Homework 3)

For the same model and data as Problem 3 of Homework 3 sample the distribution using importance sampling from a double exponential distribution. (The word sample is in quotes, because the samples aren't like ordinary samples. They need importance weights.)

If you like, you can use a generalized double exponential distribution with density of the form

f(x) = p α e^{− &alpha x} I_{(0, ∞)}(x) + (1 - p) β e^{&beta x} I_{(− ∞, 0)}(x)

as the importance sampling distribution.

As always, with importance sampling, an essential part of the job is the theoretical task of proving the Monte Carlo error has finite variance. This will depend on which double exponential or generalized double exponential distribution you use. You can't know to use a good one unless you've done the theory!

Calculate by Monte Carlo all three quantities described in Task 1 above. Use normalized importance weights in all calculations. Also estimate Monte Carlo standard errors (MCSE). The last may be hard for the median (I'm not sure right now how to do that myself).