Page 407 - Probability and Statistical Inference

384 7. Point Estimation

7.2.10 Suppose that X , ..., X are iid with the Rayleigh distribution, that is the
n
1
common pdf is
f(x;θ) =

where θ(> 0) is the unknown parameter. Find the MLE for θ. Is the MLE
sufficient for θ?
7.2.11 Suppose that X , ..., X are iid with the Weibull distribution, that is
n
1
the common pdf is
f(x;α) =

where α(> 0) is the unknown parameter, but β(> 0) is assumed known. Find
the MLE for α. Is the MLE sufficient for α?
7.2.12 (Exercise 6.2.11 Continued) Let X , ..., X be iid having the com-
n
1
-1
mon pdf σ exp{−(x − µ)/σ}I(x > µ) where µ and σ are both unknown, −∞ <
µ < ∞, 0 < σ < ∞, n ≥ 2. Show that the MLE for µ and σ are respectively X ,
n:1
the smallest order statistic, and Then, derive the MLE’s
for µ/σ, µ/σ and µ + σ.
2
7.2.13 (Exercise 7.2.12 Continued) Let X , ..., X be iid having the com-
n
1
mon pdf σ exp{ −(x − µ)/σ}I(x > µ) where µ is known but σ is unknown, −
-1
∞ < µ < ∞, 0 < σ < ∞. Show that the MLE for σ is given by
7.2.14 (Exercise 6.3.12 Continued) Let X , ..., X be iid having the com-
n
1
mon Uniform distribution on the interval (−θ, θ) where 0 < θ < ∞ is the
unknown parameter. Derive the MLE for θ. Is the MLE sufficient for θ? Also,
derive the MLE’s for θ and θ .
−2
2
7.2.15 (Exercise 6.3.13 Continued) Let X , ..., X be iid N(µ , σ ), Y , ...,
2
1
1
1
m
Y be iid N(µ , σ ), and also let the X’s be independent of the Y’s where −∞ <
2
n
2
µ , µ < ∞, 0 < σ < ∞ are the unknown parameters. Derive the MLE for (µ ,
1
1
2
µ , σ ). Is the MLE sufficient for (µ , µ , σ )? Also, derive the MLE for (µ -
2
2
2
1
2
1
µ )/σ.
2
7.2.16 (Exercise 6.3.14 Continued) Let X , ..., X be iid N(µ , σ ), Y , ...,
2
1
1
m
1
Y be iid N(µ , kσ ), and also let the X’s be independent of the Y’s where −∞
2
2
n
< µ , µ < ∞, 0 < σ < ∞ are the unknown parameters. Assume that the number
2
1
k (> 0) is known. Derive the MLE for (µ , µ , σ ). Is the MLE sufficient for
2
2
1
(µ , µ , σ )? Also, derive the MLE for (µ − µ )/σ.
2
1 2 1 2
7.2.17 (Exercise 7.2.4 Continued) Suppose that X , ..., X are iid whose
n
1
common pdf is given by