Page 457 - Probability and Statistical Inference

434 8. Tests of Hypotheses

8.3.19 Let X , ..., X be iid positive random variables having the common
1 n
lognormal pdf 0) with
– ∞ < µ < ∞, 0 < σ < ∞. Here, µ is the only unknown parameter.

(i) Find the minimal sufficient statistic for µ;
(ii) Given α ∈ (0, 1), find the MP level a test for deciding 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
8.3.20 Suppose that X is an observable random variable with its pdf given
by f(x), x ∈ ℜ. Consider the two functions defined as follows:

Derive the MP level α test for

Perform the power calculations. {Hint: Follow the Example 8.3.10.}
8.3.21 (Exercise 8.3.18 Continued) Let us denote X’ = (X , X ) where X is
1 2
assumed to have the bivariate normal distribution, Here,
we consider µ(∈ ℜ) as the unknown parameter. Derive the MP level α test for
H : µ = µ versus H : µ = µ (< µ ). Evaluate the power of the test. {Hint:
1
0
1
0
0
Write down the likelihood function along the line of (8.3.33) and proceed
accordingly.}
8.4.1 Let X , ..., X be iid having the common Laplace pdf f(x;b) =
n
1
½b exp(- | x - a |/b)I(x ∈ ℜ) where b(> 0) is an unknown parameter but
–1
a(∈ ℜ) is assumed known. Show that the family of distributions has the MLR
increasing property in T, the sufficient statistic for b.
8.4.2 Let X , ..., X be iid having the Weibull distribution with the common
1
n
b-1
pdf f(x;a) = a bx exp(-x /a)I(x > 0) where a(> 0) is an unknown parameter
-1
b
but b(> 0) is assumed known. Show that the family of distributions has the
MLR increasing property in T, the sufficient statistic for a.
8.4.3 Let X , ..., X be iid random variables with the Poisson(λ) distribu-
1
n
+
tion where λ ∈ ℜ is the unknown parameter. Show that the family of distri-
butions has the MLR increasing property in T, the sufficient statistic for λ.
8.4.4 Suppose that X , ..., X are iid having the Uniform(–θ, θ) distribution
n
1
with unknown θ(> 0). Show that the family of distributions has the MLR
increasing property in T = | X |, the sufficient statistic for θ.
n:n