Suppose that x ~ C(0, 1) has a Cauchy distribution. It is easily shown that =
Question:
Suppose that x ~ C(0, 1) has a Cauchy distribution. It is easily shown that η = P(x > 2) = tan-1(½)/π = 0.147 583 6, but we will consider Monte Carlo methods of evaluating this probability.
(a) Show that if k is the number of values taken from a random sample of size n with a Cauchy distribution, then k / n is an estimate with variance 0.125 802 7 .
(b) Let p(x) = 2/x2, so that ∫x∞ p (ξ) dξ 2/x. Show that if x ~ U(0, 1) is uniformly distributed over the unit interval then y = 2/x has the density p(x) and that all values of y satisfy y ≥ 2 and hence that
gives an estimate of η by importance sampling.
(c) Deduce that if x1, x2 , ... , Xn are independent U(0, 1) variates then
gives an estimate of η.
( d) Check that η̂ is an unbiased estimate of η and show that
and deduce that
so that this estimator has a notably smaller variance than the estimate considered in (a).
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