Page 493 - Probability and Statistical Inference

470 9. Confidence Interval Estimation

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9.2.2 Suppose that X has its pdf f(x; θ) = 2(θ – x)θ I(0 < x < θ) where
+
θ(∈ ℜ ) is the unknown parameter. We are given α ∈ (0, 1). Consider the
pivot U = X/θ and derive a (1 – α) two–sided confidence interval for θ.
9.2.3 Suppose that X , ..., X are iid Gamma(a, b) where a(> 0) is known
1
n
but b(> 0) is assumed unknown. We are given α ∈ (0, 1). Find an appropriate
pivot based on the minimal sufficient statistic and derive a two-sided (1 – α)
confidence interval for b.
2
9.2.4 Suppose that X , ..., X are iid N(µ, σ ) where µ(∈ ℜ) is known but
1
n
+
σ(∈ ℜ ) is assumed unknown. We are given α ∈ (0, 1). We wish to obtain a
(1 – α) confidence interval for σ.
(i) Find both upper and lower confidence intervals by inverting appro-
priate UMP level θ tests;
(ii) Find a two–sided confidence interval by considering an appropriate
pivot based only on the minimal sufficient statistic.
9.2.5 Let X , ..., X be iid with the common negative exponential pdf
1
n
–1
f(x; σ) = σ exp{– (x – θ)/σ} I(x > θ). We suppose that θ(∈ ℜ) is known but
+
σ(∈ ℜ ) is unknown. We are given α ∈ (0, 1). We wish to obtain a (1 – θ)
confidence interval for σ.
(i) Find both upper and lower confidence intervals by inverting appro-
priate UMP level θ tests;
(ii) Find a two–sided confidence interval by considering an appropriate
pivot based only on the minimal sufficient statistic.
{Hint: The statistic is minimal sufficient for σ.}
9.2.6 Let X , ..., X be iid with the common negative exponential pdf f(x;
1
n
θ, σ) = σ exp{– (x – θ)/σ} I(x > θ). We suppose that both the parameters
–1
θ(∈ ℜ) and σ(∈ ℜ + ) are unknown. We are given α ∈ (0, 1). Find a (1 – θ)
two-sided confidence interval for σ by considering an appropriate pivot based
only on the minimal sufficient statistics. {Hint: Can a pivot be constructed
from the statistic where the smallest order
statistic?}
9.2.7 Let X , ..., X be iid with the common negative exponential pdf f(x;
n
1
–1
θ, σ) = σ exp{ – (x – θ)/σ} I(x > θ). We suppose that both the parameters
θ(∈ ℜ) and σ(∈ ℜ + ) are unknown. We are given α ∈ (0, 1). Derive the
joint (1 – α) two–sided confidence intervals for θ and σ based only on the
minimal sufficient statistics. {Hint: Proceed along the lines of the Example
9.2.10.}
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9.2.8 Let X , ..., X be iid Uniform(–θ, θ) where θ(∈ ℜ ) is assumed
n
1
unknown. We are given α ∈ (0, 1). Derive a (1 – α) two-sided confidence
interval for θ based only on the minimal sufficient statistics. {Hint: Can