Docsity
Log inSign up
Docsity
Prepare for your exams
Prepare for your exams

Study with the several resources on Docsity

Earn points to download
Earn points to download

Earn points by helping other students or get them with a premium plan

Guidelines and tips
Guidelines and tips
Sell on Docsity
Sell on Docsity
Log inSign up

Prepare for your exams

Study with the several resources on Docsity

Find documents

Prepare for your exams with the study notes shared by other students like you on Docsity

Search Store documents

The best documents sold by students who completed their studies

Search through all study resources
Docsity AINEW

Summarize your documents, ask them questions, convert them into quizzes and concept maps

Explore questions

Clear up your doubts by reading the answers to questions asked by your fellow students

Earn points to download

Earn points by helping other students or get them with a premium plan

Share documents
download point icon20 Points

For each uploaded document

Answer questions
download point icon5 Points

For each given answer (max 1 per day)

All the ways to get free points
Get points immediately

Choose a premium plan with all the points you need

Study Opportunities

Choose your next study program

Get in touch with the best universities in the world. Search through thousands of universities and official partners

Community

Ask the community

Ask the community for help and clear up your study doubts

University Rankings

Discover the best universities in your country according to Docsity users

Free resources

Our save-the-student-ebooks!

Download our free guides on studying techniques, anxiety management strategies, and thesis advice from Docsity tutors

From our blog

Exams and Study
Go to the blog

STAT 9220 Test #1 Solutions: Exponential Families, Confidence Intervals, Independence - Pr, Exams of Statistics

Medical College of Georgia (MCG)Statistics
Prof. Grzegorz A. Rempala

Solutions to test # 1 of stat 9220, focusing on topics such as exponential families, confidence intervals, and independence in statistics. Students will find answers to problems related to lindeberg and liapunov conditions, sufficient statistics, log transformations, and independence of random variables.

Typology: Exams

Pre 2010

Uploaded on 08/04/2009

koofers-user-x5g
koofers-user-x5g 🇺🇸

10 documents

1 / 3

Toggle sidebar

This page cannot be seen from the preview

Don't miss anything!

bg1
STAT 9220
Test # 1
Mar 3, 2009
Name:
Instructions: Please return for credit by 1 PM Wed, Mar 3rd. Show the justification for your
calculations for full credit and round the answers to two decimal points when appropriate. Solve
any four problems for total of 100 points.
1. (25 points) Let X1, . . . , Xnbe a sequence of iid random variables with an unknown mean ν,
unit variance, and a finite third moment. Show that the (a) Lindeberg and (b) Liapunov
conditions are satisfied. Derive the asymptotic confidence interval for µ.
Answer: Lindeberg’s condition was checked in class. As to Liapunov condition:
Pn
i=1 E|Xi|3
n3/2=n E|X1|3
n3/2→0
The confidence interval for µbased on normal limit theorem is
¯
X±zα
√n.
2. (25 points) Let Pθ={N(θ, θ2) : θ∈R}.
(a) Is Pθan exponential family ? what is its rank ? can it we written as natural exponential
family of full rank ? Justify you answer.
(b) Find the sufficient statistic for θ. Is it complete ? Justify your answer. Be careful, it not
as easy as it seems...
Answer: (a) Since
dPθ
dx (x) = (2π)−1/2exp −x2
2θ2+x
θ−1/2−log θ
hence it is an exponential family of rank one since the dimension of the subspace (θ, θ2) is one.
Natural exponential family parametrized by η1= (2θ)−2, η2=θ−1will not be of full rank since
η2is a function of η1. (b) By factorization theorem (−x2, x) is sufficient. But it is not complete,
take h(x, y) = x+y2.
pf3

Partial preview of the text

Download STAT 9220 Test #1 Solutions: Exponential Families, Confidence Intervals, Independence - Pr and more Exams Statistics in PDF only on Docsity!

STAT 9220

Test # 1

Mar 3, 2009

Name:

Instructions: Please return for credit by 1 PM Wed, Mar 3rd. Show the justification for your calculations for full credit and round the answers to two decimal points when appropriate. Solve any four problems for total of 100 points.

  1. (25 points) Let X 1 ,... , Xn be a sequence of iid random variables with an unknown mean ν, unit variance, and a finite third moment. Show that the (a) Lindeberg and (b) Liapunov conditions are satisfied. Derive the asymptotic confidence interval for μ. Answer: Lindeberg’s condition was checked in class. As to Liapunov condition: ∑n i=1 E|Xi| 3 n^3 /^2

n E|X 1 |^3 n^3 /^2

The confidence interval for μ based on normal limit theorem is

X¯ ± √zα n

  1. (25 points) Let Pθ = {N (θ, θ^2 ) : θ ∈ R}. (a) Is Pθ an exponential family? what is its rank? can it we written as natural exponential family of full rank? Justify you answer. (b) Find the sufficient statistic for θ. Is it complete? Justify your answer. Be careful, it not as easy as it seems... Answer: (a) Since

dPθ dx

(x) = (2π)−^1 /^2 exp

x^2 2 θ^2

x θ

− 1 / 2 − log θ

hence it is an exponential family of rank one since the dimension of the subspace (θ, θ^2 ) is one. Natural exponential family parametrized by η 1 = (2θ)−^2 , η 2 = θ−^1 will not be of full rank since η 2 is a function of η 1. (b) By factorization theorem (−x^2 , x) is sufficient. But it is not complete, take h(x, y) = x + y^2.

STAT 9220

Test # 1

  1. (25 points) Show that if the distribution of a positive random variable X is in a scale family, then log(X) is in a location family. Answer: Consider families of probability distributions Pσ which is a scale family. Let B be any Borel set on R

Pσ(X ∈ B) = Pσ(X/σ ∈ B/σ) since Pσ is a scale family (1) = P (Z ∈ B/σ) where Z = X/σ is a r.v. free of σ (2) = P (log Z ∈ log(B/σ)) (3) = P (log Z ∈ log B − log σ) (4) = Pσ(log X − log σ ∈ log B − log σ) (5) = Pσ(log X ∈ log B). (6)

The result follows from (4) and (6).

  1. (25 points) Let X 1 ,... , Xn be iid with exponential distribution E(a, 1) i.e., with Lebesgue density f (x) = exp(−(x − a))I(a,+∞)(x). Are

(Xi − a) and X(1) independent? Justify your answer. Answer: There is no way they are independent. Basu’s theorem doesn’t apply since

(Xi −a) is not a statistic. Consider the related homework problem. We have shown that

(Xi − X(1)) and X(1) are independent. Thus ∑ (Xi − a) =

(Xi − X(1)) + n(X(1) − a)

and (^) ∑ (Xi − a) and X(1) are correlated, and thus can’t be independent.

To complete the argument we perhaps need to argue that Basu’s theorem applies to

(Xi − X(1)) and X(1). But the distribution of

(Xi − X(1)) is clearly uneffected by adding or sub- tracting a constant, hence it is ancilliary. Sufficiency of X(1) follows from factorization theorem, completeness can be shown exactly like in the case of arguing the completeness of X(n) in the family U (0, θ).

Page 2