Page 408 - Probability and Statistical Inference

7. Point Estimation 385

where θ(> 0) is the unknown parameter. Derive the MLE for θ. Compare this
MLE with the method of moments estimator obtained earlier. Is the MLE
sufficient for θ?
7.2.18 (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 and σ are both unknown, −∞
-1
< µ < ∞, 0 < σ < ∞, n ≥ 2. Derive the method of moment estimators for µ
and σ.
7.2.19 Suppose that Y , ..., Y are independent random variables where Y i
n
1
is distributed as N(β + β x , σ ) with unknown parameters β , β . Here, σ(>
2
0
1 i
0
1
2
0) is assumed known and x ’s are fixed real numbers with θ = (β , β ) ∈ ℜ ,
i 0 1
i = 1, ..., n, n ≥ 2. Denote with and assume that
a > 0. Suppose that the MLE’s for β and β are respectively denoted by
0 1
Now, consider the following linear regression problems.
(i) Write down the likelihood function of Y , ..., Y ;
1 n
(ii) Show that and is normally distributed
2
with mean β and variance σ /a;
1
(iii) Show that and is normally distributed with
2
mean β and variance σ {1/n + /a}.
2
0
7.2.20 Suppose that X , ..., X are iid with the common pdf f(x; θ) =
n
1
θ xexp{−x /(2θ)}I(x > 0) where 0 < θ < ∞ is the unknown parameter. Esti-
2
−1
mate θ by the method of moments separately using the first and second popu-
lation moments respectively. Between these two method of moments estima-
tors of the parameter θ, which one should one prefer and why?
2
7.3.1 Suppose that X , ..., X are iid N(0, σ ) where 0 < σ < ∞ is the
4
1
unknown parameter. Consider the following estimators:
(i) Is T unbiased for σ , i = 1, ..., 4?
2
i
(ii) Among the estimators T , T , T , T for σ , which one has the
2
4
2
1
3
smallest MSE?
(iii) Is T unbiased for σ? If not, find a suitable multiple of T which
5
5
is unbiased for σ. Evaluate the MSE of T .
5
7.3.2 (Example 7.3.1 Continued) Let X , ..., X be iid N(µ, σ ) where µ, σ
2
n
1
are both unknown with −∞ < µ < ∞, 0 < σ < ∞, n ≥ 2. Denote
Let V = cU be an estimator of σ where c(> 0) is a
2
constant.