Page 416 - Probability and Statistical Inference

7. Point Estimation 393

7.7.6 (Exercises 7.2.10 and 7.5.4) Suppose that X , ..., X are iid with the
1
n
Rayleigh distribution, that is the common pdf is

where θ(> 0) is the unknown parameter. Show that the MLE’s and the
UMVUE’s for θ, θ and θ are all consistent.
-1
2
7.7.7 (Exercises 7.2.11 and 7.5.6) Suppose that X , ..., X are iid with the
1 n
Weibull distribution, that is the common pdf is

where α(> 0) is the unknown parameter but β(> 0) is assumed known. Show
2
-1
that the MLE’s and the UMVUE’s for α, α and α are all consistent.
7.7.8 (Exercises 7.2.4 and 7.2.17 Continued) Suppose that X , ..., X are
n
1
iid whose common pdf is given by

where θ(> 0) is the unknown parameter. Show that the method of moment
estimator and the MLE for θ are both consistent.
7.7.9 Suppose that X , ..., X are iid whose common pdf is given by
1 n

where µ, α are both assumed unknown with −∞ < µ < ∞ 0 < α < ∞, θθ θθ θ = (µ,
σ). One will recall from (1.7.27) that this pdf is known as the lognormal
density.
(i) Evaluate the expression for denoted by the parametric
function T(θ), for any fixed k(> 0);
(ii) Derive the MLE, denoted by T , for T(θ);
n
(iii) Show that T is consistent for T(θ).
n
7.7.10 Suppose that X , ..., X are iid N(µ, θ ) where µ and σ are both
2
1
n
assumed unknown with µ ∈ ℜ, σ ∈ ℜ , θθ θθ θ = (µ, σ), n ≥ 2. First find the
+
2
UMVUE T = T for the parametric function T(µ) = µ + µ . Show that T is
n
n
consistent for T(µ).
7.7.11 (Exercise 7.5.4) Suppose that X , ..., X are iid with the Rayleigh
1
n
distribution, that is the common pdf is