Ahandy rule of thumb in statistics and life is as follows:

conditioning often makes things better.

this problem explores how the above rule of thumb applies to estimating unknown parameters. let θ be an unknown parameter that we wish to estimate based on data x1, x2, xn (these are r. v.s before being observed, and then after the experiment they "crystallize" into data). in this problem, θ is viewed as an unknown constant, and is not treated as an r. v. as in the bayesian approach. let t be an estimator for θ (this means that t1 is a function of xi, ,x, which is being used to estimate θ)

a strategy for improving ti (in some problems) is as follows. suppose that we have an r. v. r such that t2 = e(tir) is a function of xi, , xn (in general. e(tir) might involve unknowns such as θ but then it couldn't be used as an estimator). also suppose that p(t1-t2) < 1, and that e(z? ) įs finite.

(a) use jensen's inequality to show that t2 is better than ti in the sense that the meain squared error is less, i. e

hint: use adam's law on the right-hand side

(b) the bias of an estimator for θ is defined to be b(t)-e(t)-θ. an important identity in statistics, a form of the bias-variance tradeoff, is that mean squared error is variance plus squared bias e(t-0). var(t) + (b(t))2

use this identity and eve's law to give an alternative proof of the result from (a)

(c) now suppose that x1,x2, are 1.1.d. with mean θ. and consider the special case t-xi, r → squared error than t1 for n-2. also, say what happens to t1 and t2 as n → oo. 1x. find t2 in simplified form, and check that it has lower mean