Download STAT 410 Homework 7: Exponential Families and Sufficient Statistics and more Assignments Probability and Statistics in PDF only on Docsity!
STAT 410 HW
Due Wednesday, December 10, 2003 You can turn it in to my office, 116B IH, or mailbox in 101 IH.
- Suppose Xijk’s are independent, i = 1, 2; j = 1, 2; k = 1, 2; with
Xijk = N (μ + αi + βj , 1),
where (μ, α 1 , α 2 , β 1 , β 2 ) ∈ R^5. These data can be thought of as observations from a two-way analysis of variance:
Column 1 Column 2 Row 1 X 111 , X 112 X 121 , X 122 Row 2 X 211 , X 212 X 221 , X 222 Let
Xi++ = Xi 11 + Xi 12 + Xi 21 + Xi 22 , i = 1, 2 (row sums), X+j+ = X 1 j 1 + X 1 j 2 + X 2 j 1 + X 2 j 2 , j = 1, 2 (column sums),
and X+++ be the sum of all eight observations.
(a) Write the pdf of the data as a K = 5 dimensional exponential family, where the natural parameter is (μ, α 1 , α 2 , β 1 , β 2 ). What is the natural sufficient statistic?
(b) Show that the model in part (a) is not minimal. (c) Rewrite the model as a K = 3 dimensional exponential family. (Note that X2++ = X+++ − X1++, and similarly for the columns.) What are the natural parameter and natural sufficient statistic? Is this model minimal?
(d) Is the model in part (c) complete? (What is the space of the natural parameter?) (e) Find the expected values of the natural sufficient statistics from part (c) as functions of (μ, α 1 , α 2 , β 1 , β 2 ).
(f) For each of the following, find an unbiased estimator if possible, and if you can, say whether it is UMVUE or not. (Two are possible, two not.)
(i) μ (ii) μ + α 1 + β 2 (iii) β 1 (iv) β 1 − β 2
- Here are the data on snoring and heart disease:
Heart Disease? i Snores... Y es N o
- N ever 24 1355
- Occasionally 35 603
- N early every night 21 192
- Every night 30 224 (This is from Table 4.1 in An Introduction to Categorical Data Analysis by Alan Agresti. Wiley, 1996). Suppose the X 1 , X 2 , X 3 , X 4 are independent, with Xi ∼ Binomial(ni, pi), e.g., x 1 = 24 and n 1 = 24 + 1355 = 1379. The model we will consider sets
log
( pi 1 − pi
) = α + β × i
for i = 1, 2 , 3 , 4, where (α, β) ∈ R^2.
(a) Write the pmf of (X 1 , X 2 , X 3 , X 4 ) as an exponential family with natural parameter being (α, β). What are the natural sufficient statistics T 1 and T 2?
(b) Find the observed values of T 1 and T 2 , and find their expected values as functions of α and β.
(c) The MLE’s of α and β are − 4 .4319 and 0.6545, respectively. Verify that they satisfy the appropriate estimating equations (up to round-off error),
- Recall the fruit fly data from HW#5, Problem 1. Let
N 00 = #{(yi 1 , yi 2 ) = (0, 0)}, N 01 = #{(yi 1 , yi 2 ) = (0, 1)}, N 10 = #{(yi 1 , yi 2 ) = (1, 0)}, N 11 = #{(yi 1 , yi 2 ) = (1, 1)}.
(a) Write the model as a K = 2 dimensional exponential family. (Note that the N (^) ij′ s sum to n.) Does the space of the natural parameter look like it contains a 2-dimensional nonempty open rectangle? (You don’t have to prove anything; just take an educated guess.)
(b) Find the Fisher Information as a function of θ. (c) Find the Cram´er-Rao Lower Bound for unbiased estimates of θ. (d) Find Cov(Yi 1 , Yi 2 ). (Yi 1 and Yi 2 are not independent, because they come from the same Father.)
(e) Find V ar(Yi 1 + Yi 2 ).
