Reference no: EM132279004
Assignment - Questions on Fundamental Theory of Statistical Inference
Q1. Let θ^ be an unbiased estimator of a scalar parameter θ ∈ Ωθ ≡ R satisfying Eθ(θ2^) < ∞ for all θ ∈ Ωθ. We say that θ^ is a uniform minimum variance unbiased (UMVU) estimator if Varθ(θ^) < Varθ(θ∼) for all θ ∈ Ωθ and any other unbiased estimator θ∼. By considering the estimator θ^λ = θ^ + λU, or otherwise, show that a necessary condition for θ^ to be a UMVU estimator is that Eθ(θ^U) = 0 for all θ ∈ Ωθ and all estimators U with Eθ(U) = 0 and Eθ(U2) < ∞. [So, θ^ has to be 'uncorrelated with every unbiased estimator of 0'].
For n ≥ 2, let X1, . . . , Xn be independent, identically distributed uniform U(θ, 2θ), for some θ > 0. Show that θ∼ = 2/3 X1 is an unbiased estimator of θ. Find an unbiased estimator θ^ which is a function of a minimal sufficient statistic and satisfies Varθ(θ^) < Varθ(θ∼) for all θ > 0. Is θ^ a UMVU?
Q2. What are the main similarities and differences between Fisherian and Frequentist approaches to statistical inference?
Q3. Let X = (X1, . . . , Xp)T, p ≥ 3, have a p-dimensional normal distribution with mean θ = (θ1, . . . , θp)T and identity covariance matrix [so that X1, . . . , Xp are independent and Xi is distributed as N(θi, 1)].
Given an estimator δ(X) of θ, define its risk as
R(θ, δ(X)) = Eθ||θ - δ(X)||2,
where || · || is the Euclidean norm. The James-Stein estimator is
δJS(X) = (1 - ((p-2)/||X||2))X.
We then have that W = ||X||2 is distributed as 'non-central chi-squared with p degrees of freedom and non-centrality parameter λ = ||θ||2, χ2p(λ), and it may be shown that the density function of W is
fW(w) = i=1∑∞((e-λ/2(λ/2)i)/i!) fp+2i(w), w > 0,
where fq denotes the density function of χ2q, the chi-squared distribution on q degrees of freedom.
It may also be shown that for real valued function h(·),
E{h[χ2p,(λ)] χ2p'(λ)} = p'E{h[χ2p'+2(λ)]} + λE{h[χ2p'+4(λ)]}.
Using these results together with Jensen's inequality [E(1/W) ≥ 1/E(W)], or otherwise, show that
1/(p-2+||θ||2) ≤ E(1/W) ≤ 1/(p-2) (p/(p+||θ||2)).
and that
E(1/W) = ½0∫1e-λt/2(1-t)(p/2)-2 dt.
Deduce that the risk of the James-Stein estimator depends on θ only through the value of λ and is an increasing function of λ. What is the limiting value of the risk as λ → ∞?
Attachment:- Assignment File.rar