Page 587 - Probability and Statistical Inference

564 12. Large-Sample Inference

12.2.2 Suppose that is the MLE of θ(∈ ℜ − {0} which satis-
fies both (12.2.3)-(12.2.4). Then, find the asymptotic distribution of

12.2.3 Suppose that a random variable X has the Logistic distribution with
its pdf f(x; θ) = e −(x−y) {1 + e −(x−y) } I (x ∈ ℜ) where θ(∈ ℜ) is the unknown
−2
parameter. Show that the Fisher information, I (θ) = 1/3. {Hint: With y = x −
θ, express I (θ) as

Evaluate this integral by substituting u = {1 + e } .}
−y −1
12.2.4 (Exercise 12.3.3 Continued) Suppose that X , ..,X are iid random
n
1
variables having the common Logistic pdf f(x; θ) = e −(x−θ) {1 + e −(x−θ) } I(x ∈
−2
ℜ) where θ(∈ ℜ) is the unknown parameter.
(i) Write down the likelihood equation (12.2.2) in this situation. Is it
possible to obtain an analytical expression of the MLE, ≡
(X) where X = (X , ...,X )? If not, how could one find the esti
1 n
mate (X) for θ having obtained the data X?
(ii) Show that the MLE, is consistent for θ;
(iii) Show that asymptotically (as n → ∞), is dis
tributed as N(0, 3). {Hint: Use Exercise 12.2.3.}
12.3.1 A coffee dispensing machine automatically fills the cups placed
underneath. The average (µ) amount of fill must not vary too much from the
target (4 fl. ounces) because the overfill will add extra cost to the manufac-
turer while the underfill will generate complaints from the customers. A ran-
dom sample of fills for 35 cups gave = 3.98 ounces and S = .008 ounces.
Test a null hypothesis H : µ = 4 versus an alternative hypothesis H : µ ≠ 4 at
0 1
an approximate 1% level.
12.3.2 (Exercise 12.3.1 Continued) Obtain an approximate 95% confi-
dence interval for the average (µ) amount of fill per cup.
12.3.3 Sixty automobiles of the same make and model were tested by
drivers with similar road habits, and the gas mileage for each was recorded
over a week. The summary results were = 19.28 miles per gallon and s =
2.53 miles per gallon. Construct an approximate 90% confidence interval for
the true average (µ) gas mileage per gallon.
12.3.4 A company has been experimenting with a new liquid diet. The
investigator wants to test whether the average (µ) weight loss for individu-
als on this diet is more that five pounds over the initial two-week period.
Fifty individuals with similar age, height, weight, and metabolic structure