Homework 2 Advanced Statistics, Semeter A 2024/2025
1. Suppose X1 , X2 . . . , Xn are random samples from the probability model
2
θ2k e−θ
pX (k; θ) = , k = 0, 1, 2, · · ·
k!
Find the maximum likelihood estimator θb for θ.
2. Suppose Y1 , Y2 . . . , Yn are random samples from the uniform pdf, fY (y; θ) = 1/θ, 0 ≤ y ≤ θ.
Let Ymin = min(Y1 , Y2 , . . . , Yn ) be the smallest order statistic. Find an unbiased estimator
for θ based on Ymin .
3. A random sample of size 2, Y1 and Y2 , is drawn from the pdf
fY (y; θ) = 2yθ2 , 0 < y < 1/θ
What must c equal if the statistic c (Y1 + 2Y2 ) is to be an unbiased estimator for 1/θ ?
4. Suppose Y1 , Y2 , . . . , Yn ∼ Exponential distribution: fY (y; θ) = λe−λy for y ≥ 0. Let
Y = n1 ni=1 Yi be the sample mean. Is
P
b ∗ := n−1 1
λ ,
n Y
an efficient estimator for λ?
5. Let Y1 , Y2 , · · · , Yn be a random sample of size n from the pdf
1
fY (y; θ) = e−y/θ , y>0
θ
• Show that θ̂1 = Y1 , θ̂2 = Ȳ and θ̂3 = nYmin are all unbiased estimators for θ.
• Find the variances of θ̂i , i = 1, 2, 3.
• Calculate the relative efficiencies of θ̂1 to θ̂3 and θ̂2 to θ̂3 .
Homework 2 Advanced Statistics, Semeter A 2024/2025
6. Suppose X1 , X2 . . . , Xn are random samples from the pdf
fX (x; θ) = θxθ−1 , 0 < x < 1
Find the form of the Likelihood Ratio Test (LRT) for testing H0 : θ = θ0 versus H1 : θ ̸= θ0 .
7. Suppose Y1 , Y2 . . . , Yn are random samples from the probability model
2y
fY (y; θ) = , y ∈ [0, θ]
θ2
(a) Find the form of the Likelihood Ratio Test (LRT) for testing H0 : θ = θ0 versus
H1 : θ < θ0 .
(b) Suppose the significance level of the test is α. Find the specific critical regions of the
LRT (as a function of α and θ).
8. A sample of size 1 is taken from the pdf
fY (y; θ) = (1 + θ)y θ , 0 ≤ y ≤ 1
The hypothesis H0 : θ = 1 is to be rejected in favor of H1 : θ > 1 if y ≥ 0.9. Find
(a) Find α (significance level) as a function of θ.
(b) Find β (Type II error) as a function of θ.
9. Suppose that a random sample of size 5 is drawn from a uniform pdf
fY (y; θ) = 1/θ, 0≤y≤θ
We wish to test H0 : θ = 2 versus H1 : θ > 2 by regecting the null hypothesis if ymax ≥ k.
Find the value of k that makes the probability of committing a Type I error equal to 0.05.
10. If H0 : µ = 200 is to be tested against H1 : µ < 200 at the α = 0.10 level of significance
based on a random sample of size n from a normal distribution where σ = 15.0. What is the
smallest value for n that will make the power equal to at least 0.75 when µ = 197?
