Exercises for Wed 15 October
1. On Wed Oct 8 Céline Cunen and Aliaksandr Hubin laudably went through various exercises from Ch 6.
2. For Wed Oct 15 I will round off Ch 6 and start on Ch 7. Exercises are as follows.
E1. For an i.i.d. sample of size n from the N(mu,sigma^2), show that the usual parameter estimates are sufficient for the two parameters. Construct an estimator \hat g(x) of the normal density f(x,mu,sigma) with the property that it is unbiased on the log-scale. Illustrate in your computer, with simulated data set of sizes 5, 10, 50, where you plot both your \hat g(x) and the more obvious plug-in version f(x,\hat mu,\hat sigma).
E2. Suppose Y_i is normal N(a + b x_i, sigma^2) for i = 1, ..., n, with these being independent. (a) For sigma known, find sufficient statistics. (b) Find sufficient statistics for the case of all three parameters being unknown. Comment on your findings.
E3. Suppose X_n and Y_n are independent, and that they both tend in distribution to the standard normal. Find the limit distributions of (a) X_n^2 + Y_n^2 and (b) X_n/Y_n.
E4. Assume X_1, ..., X_n are i.i.d. from the exponential model with density theta exp(-theta x) for x positive. (a) Show that T, the sum of observations, is sufficient. (b) Suppose n = 10 and that I've thrown away the data, apart from noting that T = 13.579. Generate a data set with the same information content as the original one.