IEOR 165 Homework 3
Due March 19, 2015
Electronic copies will not be accepted except under extenuating circumstances. Each student must turn in their own homework solutions.
Question 1. Use One Sample Kolmogorov Smirnov Test at significance level 0.05 to test
whether the observations come from a standard uniform distribution.
0.276 0.612 0.19 0.452 0.966 0.89 0.483 0.682
Discuss the advantages and the limitations of Kolmogorov Smirnov test.
Question 2. Let X1 , X2 , ..., Xn be i.i.d random variables, each with the same cumulative
distribution function FX (x) = P (Xi < x). Let Xmax = max{X1 , X2 , ..., Xn }. What is the
cdf of Xmax ?
Question 3. Suppose that X1 , X2 , ..., Xn form a random sample from a uniform distribution
on the interval (0, θ), and that the following hypotheses are to be tested:
H0 : θ ≥ 2
H1 : θ < 2
Let Xmax = max{X1 , X2 , ..., Xn }, and consider a test whose rejection region contains all the
outcomes for which Xmax ≤ 1.5.
a. Determine the power function of the test.
b. Determine the size of the test.
Question 4. Let Ai denote the categorical variables. Draw the simplicial complex associated to the null hypothesis below.
a. A1 , A2 and A3 are pairwise dependent but not jointly dependent.
b. A1 , A2 , A3 and A4 are jointly dependent and A5 is independent of A1 , A2 , A3 and A4 .
c. A1 and A2 are dependent and independent of A3 , A4 and A5 . A3 , A4 and A5 are jointly
dependent.
d. A1 , A2 , A3 , A4 and A5 are independent.
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Question 5.
a. Let X1 , ..., Xn be iid with density
Pθ (X = x) = θx (1 − θ)1−x
for x = 0, 1 and 0 ≤ θ ≤ 1/2
Find the MLE of θ.
b. Suppose Xi ∼ Uniform(0, θ). Find the MLE of θ.
Question 6. Suppose iid data is from Xi ∼ N (µ, σ 2 ) where σ 2 = 1. Consider the null
hypothesis
H0 : µ = 2
H1 : µ = 1
Assume the power of the test is 0.90 and the significance level is 0.05, use a Monte Carlo
algorithm to determine the threshold k.
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