final report - probability.ca

+ # plot(ell, dim^ep*sx1/it)
+ result = list(ell, accr, dim^ep * sx1/it)
+ return(result)
+ }
Subsequently, one can scan through the returned vectors easily using the following code, which also yields
theoretical curves for the same example:
> ScanGam = function(it = 5e+05, dim = 50, ep = 0, n = 50) {
+ ell = 10 + (1:n) * 3
+ ER = 1/75
+ vsd = ell/dim^ep
+ a = 5
+ # initialize the speed measure
+ esp = 0
+ for (i in (1:it)) {
+ x = rgamma(2 * n, a, 1)
+ y = rnorm(2 * n, x, rep(vsd^0.5, each = 2))
+ eps = matrix(log(dgamma(y, a, 1)/dgamma(x, a, 1)), 2,
+ n)
+ # calculate the ratio
+ eps = (eps[1, ] + eps[2, ] - ell * ER/2)/sqrt(ell * ER)
+ esp = esp + pnorm(eps)
+ }
+ # scan the vectors
+ ell[which.max(2 * ell * esp/it)]
+ OAR = 2 * esp[which.max(2 * ell * esp/it)]/it
+ plot(2 * esp/it, 2 * ell * esp/it, type = "l")
+ result = list(ell, 2 * esp/it, 2 * ell * esp/it, which.max(2 *
+ ell * esp/it), ellhat, OAR)
+ return(result)
+ }
Acknowledgement. Words cannot express the gratitude I have for my supervisor, Professor Jeffrey S. Rosenthal. He
graciously lent me this time and introduced me to the exciting field of optimal scaling of Monte Carlo algorithms,
providing positive encouragement throughout the process. Without his guidance, none of this would have been
possible.
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