Page 496 - Probability and Statistical Inference

9. Confidence Interval Estimation 473

(1 – α) two–sided confidence interval for µ – µ based on sufficient statis-
2
1
tics for (µ , µ ). Is the interval shortest among all (1 – α) two–sided confi-
2
1
dence intervals for µ – µ depending only on the sufficient statistics for
1
2
(µ , µ )?
1 2
9.3.2 (Example 9.3.1 Continued) Suppose that the random variables X ,
i1
..., X are iid N(µ , k σ ), n ≥ 2, i = 1, 2, and that the X ’s are independent
2
i
1j
i
ini
i
of the X ’s. Here we assume that all three parameters µ , µ , σ are unknown
2j
1
2
but k , k are positive and known, (µ , µ , σ) ∈ ℜ × ℜ × ℜ . With fixed α ∈
+
1
2
2
1
(0, 1), construct a (1 – α) two–sided confidence interval for µ – µ based on
1
2
sufficient statistics for (µ , µ , σ). Is the interval shortest on the average
1
2
among all (1 – α) two–sided confidence intervals for µ – µ depending on
1
2
sufficient statistics for (µ , µ , σ)? {Hint: Show that the sufficient statistic is
2
1
Start creating a pivot by
standardizing with the help of an analog of the pooled sample vari-
ance. For the second part, refer to the Example 9.2.13.}
9.3.3 (Example 9.3.1 Continued) Suppose that the random variables X ,
i1
2
..., X are iid N(µ , σ ), n ≥ 2, i = 1, 2, and that the X ’s are independent of
1j
i
i
ini
the X ’s. Here we assume that all three parameters µ , µ , σ are unknown,
2j
1
2
+
(µ , µ , σ) ∈ ℜ × ℜ × ℜ . With fixed α ∈ (0, 1), construct a (1 – α) two-sided
2
1
confidence interval for (µ – µ )/σ based on sufficient statistics for (µ , µ ,
1
2
2
1
σ). {Hint: Combine the separate estimation problems for (µ – µ ) and σ via
2
1
the Bonferroni inequality.}
9.3.4 (Example 9.3.2 Continued) Suppose that the random variables X ,
i1
..., X are iid having the common pdf f(x; µ , σ ), i = 1, 2, where we denote
in
i
i
–1
f(x; µ, s) = σ exp{–(x – µ)/σ}I(x > µ). Also let the X ’s be independent of
1j
the X ’s. Here we assume that the location parameters µ , µ are unknown
2
1
2j
but σ , σ are known, (µ , σ ) ∈ ℜ × ℜ , i = 1, 2. With fixed α ∈ (0, 1),
+
1
i
i
2
construct a (1 – α) two–sided confidence interval for µ – µ based on suffi-
1
2
cient statistics for (µ , µ ).
1
2
9.3.5 (Example 9.3.2 Continued) Suppose that the random variables X ,
i1
..., X are iid having the common pdf f(x; µ , σ), i = 1, 2 and n ≥ 2, where we
i
in
–1
denote f(x; µ, σ) = σ exp{–(x – µ)/σ}I(x > µ). Also let the X ’s be indepen-
1j
dent of the X ’s. Here we assume that all three parameters are unknown, (µ ,
1
2j
+
µ , σ) ∈ ℜ × ℜ × ℜ . With fixed α ∈ (0, 1), construct a (1 – α) two-sided
2
confidence interval for (µ – µ )/σ based on sufficient statistics for (µ , µ ,
1
2
2
1
σ). {Hint: Combine the separate estimation problems for (µ – µ ) and σ via
2
1
the Bonferroni inequality.}
9.3.6 Two types of cars were compared for their braking distances. Test
runs were made for each car in a driving range. Once a car reached the
stable speed of 60 miles per hour, the brakes were applied. The distance