Reference no: EM132279004

Assignment - Questions on Fundamental Theory of Statistical Inference

Q1. Let θ^{^} be an unbiased estimator of a scalar parameter θ ∈ Ω_θ ≡ R satisfying E_θ(θ^{2^}) < ∞ for all θ ∈ Ω_θ. We say that θ^{^} is a uniform minimum variance unbiased (UMVU) estimator if Var_θ(θ^{^}) < Var_θ(θ^∼) for all θ ∈ Ω_θ 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 E_θ(θ^{^}U) = 0 for all θ ∈ Ω_θ and all estimators U with E_θ(U) = 0 and E_θ(U²) < ∞. [So, θ^{^} has to be 'uncorrelated with every unbiased estimator of 0'].

For n ≥ 2, let X₁, . . . , X_n 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 Var_θ(θ^{^}) < Var_θ(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, . . . , X_p)^T, p ≥ 3, have a p-dimensional normal distribution with mean θ = (θ₁, . . . , θ_p)^T and identity covariance matrix [so that X₁, . . . , X_p are independent and X_i is distributed as N(θ_i, 1)].

Given an estimator δ(X) of θ, define its risk as

R(θ, δ(X)) = E_θ||θ - δ(X)||²,

where || · || is the Euclidean norm. The James-Stein estimator is

δ^JS(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, χ²_p(λ), and it may be shown that the density function of W is

f_W(w) = _i=1∑^∞((e^-λ/2(λ/2)ⁱ)/i!) f_p+2i(w), w > 0,

where f_q denotes the density function of χ²_q, the chi-squared distribution on q degrees of freedom.

It may also be shown that for real valued function h(·),

E{h[χ²_p,(λ)] χ²_p'(λ)} = p'E{h[χ²_p'+2(λ)]} + λE{h[χ²_p'+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