Page 499 - Probability and Statistical Inference

476 9. Confidence Interval Estimation

9.4.3 (Exercise 9.4.2 Continued) Suppose that X , ..., X are iid random
i1
in
2
samples from the N(µ , σ ) population, i = 1, ..., 5. The observations X , ...,
i
i1
th
X refer to the i treatment, i = 1, ..., 5. Let us assume that the treatments are
in
independent too and that all the parameters are unknown. With fixed α ∈ (0,
1), derive the (1 – α) joint confidence intervals for estimating the parameters
θ = µ – µ , θ = µ – µ and θ = µ + µ – 2µ .
1 1 2 2 2 3 3 3 4 5
9.4.4 Use the equation (9.4.20) to propose the simultaneous (1 – α) two-
sided confidence intervals for the variance ratios.
9.4.5 Let X , ..., X be iid having the common pdf f(x; µ , σ ), n ≥ 2,
i
in
i1
i
where f(x; µ, σ) = σ exp{–(x – µ)/σ}I(x > µ), (µ , σ ) ∈ ℜ × ℜ , i = 0, 1,
–1
+
i
i
..., p. The observations X , ..., X refer to a control whereas X , ..., X in
i1
0n
01
refer to the i treatment, i = 1, ..., p. Suppose that all the observations are
th
independent and all the parameters are unknown. Start with an appropriate p–
dimensional pivot depending on the sufficient statistics and then, with fixed α
∈ (0, 1), derive the simultaneous (1 – α) two-sided confidence intervals for
the ratios σ /σ , i = 1, ..., p. {Hint: Proceed along the lines of the Examples
i
0
9.4.6–9.4.7.}
9.4.6 (Exercise 9.3.10 Continued) Suppose that the random variables X ,
i1
..., X are iid having the common pdf f(x; θ ) where we denote the exponen-
i
in
tial pdf f(x; a) = a exp{–x/a}I(x > 0), θ > 0, i = 0, 1, ..., p. The observations
–1
i
th
X , ..., X refer to a control whereas X , ..., X refer to the i treatment, i
01
i1
in
0n
= 1, ..., p. Suppose that all the observations are independent and all the param-
eters are unknown. Start with an appropriate p–dimensional pivot depending
on the sufficient statistics and then, with fixed α ∈ (0, 1), derive the simulta-
neous (1 – α) two-sided confidence intervals for the ratios θ /θ , i = 1, ..., p.
i
0
{Hint: Proceed along the lines of the Examples 9.4.6–9.4.7.}