Page 492 - Probability and Statistical Inference

9. Confidence Interval Estimation 469

But, the numerical determination of the numbers a and b is not so simple.
The following result of Hewett and Bulgren (1971) may help:

1/p
Now, equating P{a < F n–1,n–1 < b} with (1 – α) , we may determine ap-
proximate choices of a and b corresponding to equal tails
1/p
(= ½(1 – (1 – α) )) of the F n–1,n–1 distribution. This approximation works
well when n ≤ 21.

9.5 Exercises and Complements

9.1.1 Suppose that X , X are iid with the common exponential pdf f(x; θ) =
1
2
–1
θ exp{–x/θ}I(x > 0) where θ(> 0) is the unknown parameter. We are given
α ∈ (0, 1).
(i) Based on X alone, find an appropriate upper (lower) (1 – α) confi-
1
dence interval for α;
(ii) Based on X , X , find an appropriate upper (lower) and two–sided
2
1
(1 – α) confidence interval for θ.
Give comments and compare the different confidence intervals.
9.1.2 Let X have the Laplace pdf f(x; θ) = ½exp{– |x – θ |}I(x ∈ ℜ)
where θ(∈ ℜ) is the unknown parameter. We are given α ∈ (0, 1). Based on
X, find an appropriate upper (lower) and two–sided (1 – α) confidence
interval for θ.
2 –1
9.1.3 Let X have the Cauchy pdf f(x; θ) = 1/π{1 + (x – θ) } I(x ∈ ℜ)
where θ(∈ ℜ) is the unknown parameter. We are given α ∈ (0, 1). Based on
X, find an appropriate upper (lower) and two–sided (1 – α) confidence inter-
val for θ.
1
9.1.4 Suppose that X has the Laplace pdf f(x; θ) = - exp{– |x|/θ}I(x ∈ ℜ)
2θ
where θ(∈ ℜ ) is the unknown parameter. We are given α ∈ (0, 1). Based on
+
X, find an appropriate upper (lower) and two–sided (1 – α) confidence inter-
val for θ.
+
2
9.2.1 Suppose that X has N(0, σ ) distribution where σ(∈ ℜ ) is the un-
known parameter. Consider the confidence interval J = (|X|, 10 |X|) for the
parameter σ.
(i) Find the confidence coefficient associated with the interval J;
(ii) What is the expected length of the interval J?