Due: Thursday Sept 17 by 3:00
1. Let X have mean . We say that X is sub-Gaussian if there exists 2 such that
log E[et(X) ]
for all t.
(i) Show that X is sub-Gaussian if and only if X is sub-Gaussian.
(ii) Show that if X is sub-Ga
Lecture Notes 6
The Likelihood Function
Denition. Let X n = (X1 , , Xn ) have joint density p(xn ; ) = p(x1 , . . . , xn ; ) where
. The likelihood function L : [0, ) is dened by
L() L(; xn ) = p(xn ; )
where xn is xed and varies in . The log-likelihoo
Lecture Notes 5
(Chapter 6.) A statistical model P is a collection of probability distributions (or a collection of densities). Examples of nonparametric models are
(p (x)2 dx < ,
and P =
all distributions on Rd .
A parametric m
Lecture Notes 8
Suppose we want to estimate a parameter using data X n = (X1 , . . . , Xn ). What is the
best possible estimator = (X1 , . . . , Xn ) of ? Minimax theory provides a framework for
answering this question.
Lecture Notes 9
Asymptotic (Large Sample) Theory (Chapter 9)
Review of o, O, etc.
1. an = o(1) mean an 0 as n .
2. A random sequence An is op (1) if An 0 as n .
3. A random sequence An is op (bn ) if An /bn 0 as n .
4. nb op (1) = op (nb ), so n o
10-705/36-705: Intermediate Statistics, Fall 2015
Baker Hall 132F
Mon-Wed-Fri 12:30 - 1:20
Baker Hall A51