Examples and Problems in Mathematical Statistics

PART III: PROBLEMS

Section 5.2

5.2.1 Let X1, …, Xn be i.i.d. random variables having a rectangular distribution R(θ1, θ2), -∞ < θ1 < θ2 < ∞.

(i) Determine the UMVU estimators of θ1 and θ2.

(ii) Determine the covariance matrix of these UMVU estimators.

5.2.2 Let X1, …, Xn be i.i.d. random variables having an exponential distribution, E(λ), 0 < λ < ∞.

(i) Derive the UMVU estimator of λ and its variance.

(ii) Show that the UMVU estimator of ρ = e– λ is

where T = and a+ = max (a, 0).

(iii) Prove that the variance of is

where P(j; λ) is the c.d.f. of P(λ) and H(k| x) = . [H(k| x) can be determined recursively by the relation

and H(1|x) is the exponential integral (Abramowitz and Stegun, 1968).

5.2.3 Let X1, …, Xn be i.i.d. random variables having a two–parameter exponential distribution, X1 ∼ μ + G(λ, 1). Derive the UMVU estimators of μ and λ and their covariance matrix.

5.2.4 Let ...