32
estimating the cumulative distribution function
4.6
types of confidence intervals/bands
Let F be a class of distribution functions F and let be some quantity
of interest, such as the mean of F or t
46
estimating statistical functionals of the cdf
5.8
exercies
1. Suppose that Xn , Yn is a sequence of real-valued random variables such that Xn X and Yn Y. Assume also that for each
n, (Xn , Yn ) are
20
introduction to stochastic processes
3.5.1
Donskers theorem
We will now prove that the re-scaled random walk converges weakly
to Brownian motion. Let X1 , . . . , Xn be IID random variables where
e
40
estimating statistical functionals of the cdf
5.3
metrics on distributions
Our overall goal here is to study the asymptotics of statistical functionals, and to do this, we will need notions of cont
5.9 exercies
5.9
exercies
1. In class we had worked out the asymptotics of the Mann-Whitney
statistic using the functional/nonparametric delta method. There
was one piece of the puzzle that still need
1
INTRODUCTION
Two key theorems in probability and statistics are the law of large
numbers and the central limit theorem.
Theorem 1.1 (Strong law of large numbers) Let X1 , X2 , . . . be independent a
3.4 weak convergence on polish spaces
3.4
weak convergence on polish spaces
Let X be a mapping from a probability space (, F, P) into a Polish
space X . In particular, X is a metric space with some me
50
estimating a density nonparametrically
6.5
exercises
1. Prove Theorem 6.28 on page 133 from Wasserman (2006). You
may assume that K is symmetric like I did! Make sure that
you can clearly and expli
54
variable selection using the lasso
7.5
penalized likelihood and scad
7.6
exercises
1. Prove equality (2.2) in Fan and Li (2001, page 1349).
2. Prove (2.5) in Fan and Li (2001, page 1349). Repeat th
6
E S T I M AT I N G A D E N S I T Y N O N PA R A M E T R I C A L LY
6.1
the histogram
6.2
exercises
1. Let fn denote the histogram density estimator. Show that for
x Bk
var(fn (x) =
pk (1 pk )
.
nh2
MATH 6621: ADVANCED THEORETICAL STATISTICS - WINTER 2013
This course is a continuation of MATH 6620. The goal will be to go over some theoretical techniques
frequently used in statistics, while consid
7.7 penalized likelihood and scad: proof of theorems
7.7
penalized likelihood and scad: proof of theorems
7.8
exercises
1. Suppose that we observe a sample X1 , . . . , Xn IID from the normal distribu
2
SOME REVIEW OF PROBABILITY
2.1
probability and random variables
A probability space is a triple (, F, P), where is the state space, F
is the associated -algebra, and P is the probability itself. Rec
8
some review of probability
2.4
our favourite, the gaussian
The normal distribution is central to much of statistical analysis, so
lets review some of its main properties. A random vector Y = (Y1 , .
4
E S T I M AT I N G T H E C U M U L AT I V E D I S T R I B U T I O N
FUNCTION
4.1
the empirical distribution as the nonparametric
mle
The maximum likelihood method has been a cornerstone of statistic
3
INTRODUCTION TO STOCHASTIC PROCESSES
3.1
definition and existence
A stochastic process is dened to be a collection of real-valued random variables cfw_Xt , t T . If the set T is nite, then were real
7
VA R I A B L E S E L E C T I O N U S I N G T H E L A S S O
7.1
introduction to the lasso
Review. Overview based on Tibshirani (1996).
7.2
exercises
1. Show that the Lasso is also a Bayes MAP estimat