TEST TOPICS AND SAMPLE PROBLEMS FOR SECOND IN-CLASS TEST, Fall 2014
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I promised that the test topics and problems would be taken from
those covered in class and in the homework problems from the first test
until now, including material up through the Neyman-Pearson Lemma.
For the first test, we went up through the end of Chapter 5. For this
test, the coverage is Chapter 6 up through Sec.8.2.2 and 8.3.1--8.3.2.,
plus the handouts on Conjugate Priors in natural exponential families,
and a bit of extra material on decision theory, with Rohatgi and
Bickel-Doksum as references.
A brief summary of the topics for Test 2 are:
(i) Sufficient statistics: definition, factorization theorem, minimality
and Theorem giving criterion for minimality. Definition of completeness
and methods for checking completeness and incompleteness. Ancillarity
and Basu's Theorem.
(ii) Rao-Blackwell Theorem, Lehmann-Scheffe' Theorem, UMVUE estimators.
Criterion for Rest Unbiased Estimators (orthogonality to all unbiased
estimators of 0).
(iii) Exponential families: basic definitions including rank and natural
parameter space; natural exponential families, criteria for minimality
and for completeness. Families of conjugate priors in the setting of
natural exponential families.
(iv) Maximum Likelihood estimators, generalized method of moments
estimators and equivalence in setting of natural exponential families.
Bayes (posterior-expectation) estimators.
(v) Fisher information and its properties. Cramer-Rao inequality including
necessary conditions for equality.
(vi) Decision theory: definitions of action space, loss function, prior,
admissibility, minimax tests. Bayes-optimal estimators with respect to
squared-error, absolute-error, zero-one, and related loss functions.
Admissibility of unique minimax and Bayes tests.
(vii) Definitions associated with hypothesis testing (size, level, power,
UMP test), plus statement, proof and examples of Neyman-Pearson Lemma.
Auxiliary randomization to achieve exact signficance levels.
(viii) Stochastic simulation of multivariate random variables: accept-
reject methods and algorithms based on conditional probability integral
transform.
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Sample test problems:
(I) Problems (1) and (4) from the in-class test from Fall 2009, 10/28/09 that
can be found in the first of the three links in the "Handouts & Other Links"
part (I), at http://www.math.umd.edu/~evs/s700.F09/TestF09.pdf.
(II) Problems (1)-(4) and (6) on the Sample Final from 12/12/2001.
available as the second link in the "Handouts & Other Links" part (I), at
http://www.math.umd.edu/~evs/s700.F09/SmpFinl.pdf.
(III) Problem (2) on the Makeup for Test 1 from 11/6/13, at
http://www.math.umd.edu/~evs/s700.F13/Test1F13Makeup.pdf.
(IV) Problems (I) and (II) on Stat 700 Take-Home Test from 12/6/2008, at
http://www.math.umd.edu/~evs/s700.F09/TkHTestF08.pdf.
(V) Problems (1), (2), (5), and (6) on Stat 700 Sample Test from Fall 2009, at
http://www.math.umd.edu/~evs/s700.F09/SampTestF09.pdf.
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(VI) Here a few more sample problems, which are in scope for the Test but
vary in difficulty. (Numbering continues from the sample problems given for
the first F14 test.)
Problem F14.7: Prove that when there is a sufficient statistic T(X) for a parameter
theta based on a sample X_1,...,X_n from density f(x,theta), then
(a) if any MLE exists, there is an MLE depending onthe data only through T(X);
(b) if L(theta,a) is any loss function for estimation, and pi(theta) any prior,
then the posterior density for theta given the sample depends on the data only
through T(X), and therefore a Bayes-optimal estimator can always be found that
depends on the data only through T(X).
Problem F14.8: Suppose that Y = (Y_1,...,Y_n) is a sample from a density f(y,theta)
or probability mass function p(y,theta) that depends smoothly on the parameter theta, and
let gamma = g(theta) be a known and smooth and smoothly invertible transformation of
the parameter theta.
(a) Find the relation between the information I_Y(gamma) in the sample about gamma
and the information I_Y(theta) about theta.
(b) Find the MLE for gamma in terms of the sample from the MLE for theta. Is the
smoothness of g and its inverse necessary for this result ? What property of g IS
necessary for the result ?
Problem F14.9: [This one is only slightly harder, but many qualifier problems of this
sort requiring reasoning with the Factorization Theorem have been asked over the years.]
Suppose that a dataset consisting of a random vector X with an always-positive density
f(x,beta, gamma) has the properties that
(i) statistic T_1(X) is sufficient for gamma when beta is fixed at SOME known value beta_0, and
(ii) statistic T_2(X) is sufficient for beta for whenever gamma is fixed at ANY value gamma_0.
Show that in these circumstances, (T_1(X),T_2(X)) is sufficient for (beta,gamma).
Problem F14.10: Suppose that X, Y are positive random variables with the joint density
f(x,y) = C exp(-x-2y-3xy) , x,y > 0
where C = 3.8655. Give any method you can to simulate from pseudo-random independent uniform
random variables U_1, U_2, ... a pair of random variables with the same density as (X,Y).