代写MATH0099 Statistical Methods and Data Analytics 2018 Problem Sheet 6代做迭代

日期：2024-08-22 06:26

MSc Financial Mathematics

Statistical Methods and Data Analytics 2018

MATH0099

Problem Sheet 6

An even stronger criterion than equivariance of an estimator is invariance: an estimator δ is invariant, if

δ(x1 + a, . . . , xn + a) = δ(x1, . . . , xn),   ∀a ∈ R.

Problem 1.   Let X1, . . . , Xn be iid copies of a random variable X with pdf (1/σ)p((x − θ)/σ). Suppose we wish to estimate σ2.

Show that estimators of the form. kS2 , where k is a positive constant and S2 the sample variance, are invariant with respect to the transformation

ga,1(x1, . . . , xn) = (x1 + a, . . . , xn + a),

but not with respect to the transformations

            ga,c(x1, . . . , xn) = (cx1 + a, . . . , cxn + a),

g0,c(x1, . . . , xn) = (cx1, . . . , cxn).

Problem 2.   Let X1, . . . , Xn be iid copies of a random variable X with pdf p(x − θ). The Pitman estimator (cf. Lecture 7) is given by

1. Show by direct verification of the definition that δ(x) is equivariant with respect to the transformation

g(x1, . . . , xn) = (x1 + a, . . . , xn + a), a ∈ R.

2. Show that if p(x − θ) is N(θ, 1) then δ(X) = X.

3. Show that if p(x−θ) is uniform. on (θ−1/2, θ+1/2) then δ(X) = (1/2)(X(1)+X(n)).

Problem 3.   Let X = (X1, . . . , Xn) have joint distribution with density

p(x − θ) = p(x1 − θ, . . . , xn − θ).

Let δ be equivariant for estimating θ with invariant loss function L(θ, a). Prove that the bias of δ is constant.