Page 461 - Probability and Statistical Inference

438 8. Tests of Hypotheses

(i) Find the minimal sufficient statistic for µ;
(ii) Given α ∈ (0, 1), find the MP level α test to choose between the
null hypothesis H : µ = µ and the alternative hypothesis H : µ =
1
0
0
µ (> µ ) where µ , µ are fixed real numbers;
1 0 0 1
(iii) Given α ∈ (0, 1), find the UMP level α test to choose between
the null hypothesis H : µ = 1 and the alternative hypothesis
0
H : µ > 1.
a
{Hints: In part (i), use the Lehmann-Scheffé Theorems from Chapter 6 to
claim that is the minimal sufficient statistic. In parts
(ii)-(iii), use the likelihood functions along the lines of the Examples 8.3.11-
8.3.12 and show that one will reject H if and only if T > k.}
0
8.4.23 Let X , X be iid with the common pdf θ exp(-x/θ)I(x > 0) where θ
-1
1
2
> 0. In order to test H : θ = 2 against H : θ > 2, we propose to use the critical
0
1
+2
region R = {X ∈ ℜ : X + X > 9.5}. Evaluate the level α and find the power
1
2
function. Find the Type II error probabilities when θ = 4, 4.5.
8.4.24 Let X , ..., X be iid random variables having the common pdf f(x;
1
n
-1 (1-θ) θ
θ) = θ x / I(0 < x < 1) with 0 < θ < ∞. Here, θ is the unknown parameter.
Given the value α ∈ (0, 1), derive the UMP level a test in its simplest
implementable form, if it exists, for deciding between the null hypothesis H :
0
θ ≤ θ and the alternative hypothesis H : θ > θ where θ is a fixed positive
0
0
1
0
number.
8.4.25 (Sample Size Determination) Let X , ..., X be iid N(µ, σ ) where
2
n
1
µ is assumed unknown but σ is known, µ ∈ ℜ, σ ∈ ℜ . We have shown that
+
given α ∈ (0, 1), the UMP level α test for H : µ = µ versus H : µ > µ rejects
0
0
0
1
H if and only if The UMP test makes sure that it has
0
the minimum possible Type II error probability at µ = µ (> µ ) among all level
0
1
α tests, but there is no guarantee that this minimum Type II error probability
at µ = µ will be “small” unless n is appropriately determined.
1
Suppose that we require the UMP test to have Type II error probability ≤
β ∈ (0, 1) for some specified value µ = µ (> µ ). Show that the sample size n
1
0
must be the smallest integer ≥ {(z + z )σ/(µ – µ )} .
2
α β 1 0
8.5.1 Let X , ..., X be iid having the common negative exponential pdf f(x;
n
1
θ) = b exp{–(x – θ)/b}I(x > θ) where b(> 0) is assumed known and θ(∈ ℜ)
-1
is the unknown parameter. With preassigned α ∈ (0, 1), derive the UMP level
α test for H : θ = θ versus H : θ ≠ θ where θ is a fixed number, in the
1
0
0
0
0
simplest implementable form. {Hint: Observe that Y = exp{-X /b}, i = 1, ...,
i
i
n, are iid Uniform(0, δ) where the parameter δ = exp{–θ/b}. Now, exploit the
UMP test from the Section 8.5.2.}