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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.