Introduction to Empirical Processes and Semiparametric Inference
BIOS 791

Spring 2014
Solutions to Problem Set 1
2.4.1 Let F (s, t) P 1cfw_X s, Y t, F1 (s) = F (s, ), and F2 (t) =
F (, t). For each > 0, choose = s1 < s2 < < sk1 = and
= t1 < t2 < < tk2 = such that F1 (sj ) F1 (sj1 ) < , for
all 1 < j k1 , and F2 (tj ) F2 (tj1 ) < , for all
Introduction to Empirical Processes and Semiparametric Inference
BIOS 791

Spring 2014
Solutions to Problem Set 4
7.5.5 We will apply the extended continuous mapping theorem with the additional assumption that T0 is totally bounded by (which should
have been a condition in the statement of the problem). It suces
to show that suptTn xn (t) s
Introduction to Empirical Processes and Semiparametric Inference
BIOS 791

Spring 2014
Empirical Processes: Lecture 01
Spring, 2010
Introduction to Empirical Processes
and Semiparametric Inference
Lecture 01: Introduction and Overview
Michael R. Kosorok, Ph.D.
Professor and Chair of Biostatistics
Professor of Statistics and Operations Resea
Introduction to Empirical Processes and Semiparametric Inference
BIOS 791

Spring 2014
Empirical Processes: Lecture 05
Spring, 2014
Introduction to Empirical Processes
and Semiparametric Inference
Lecture 05: Overview Continued
Michael R. Kosorok, Ph.D.
Professor and Chair of Biostatistics
Professor of Statistics and Operations Research
Uni
Introduction to Empirical Processes and Semiparametric Inference
BIOS 791

Spring 2014
Empirical Processes: Lecture 03
Spring, 2012
Introduction to Empirical Processes
and Semiparametric Inference
Lecture 03: Overview Continued
Yair Goldberg, Ph.D.
Postdoctoral Fellow, Biostatistics
University of North CarolinaChapel Hill
1
Empirical Proce
Introduction to Empirical Processes and Semiparametric Inference
BIOS 791

Spring 2014
Empirical Processes: Lecture 06
Spring, 2010
Introduction to Empirical Processes
and Semiparametric Inference
Lecture 06: Metric Spaces
Michael R. Kosorok, Ph.D.
Professor and Chair of Biostatistics
Professor of Statistics and Operations Research
Universi
Introduction to Empirical Processes and Semiparametric Inference
BIOS 791

Spring 2014
Empirical Processes: Lecture 02
Spring, 2012
Introduction to Empirical Processes
and Semiparametric Inference
Lecture 02: Overview Continued
Michael R. Kosorok, Ph.D.
Professor and Chair of Biostatistics
Professor of Statistics and Operations Research
Uni
Introduction to Empirical Processes and Semiparametric Inference
BIOS 791

Spring 2014
Empirical Processes: Lecture 04
Spring, 2014
Introduction to Empirical Processes
and Semiparametric Inference
Lecture 04: Overview Continued
Michael R. Kosorok, Ph.D.
Professor and Chair of Biostatistics
Professor of Statistics and Operations Research
Uni
Introduction to Empirical Processes and Semiparametric Inference
BIOS 791

Spring 2014
Empirical Processes: Lecture 07
Spring, 2014
Introduction to Empirical Processes
and Semiparametric Inference
Lecture 07: Metric Spaces
Michael R. Kosorok, Ph.D.
Professor and Chair of Biostatistics
Professor of Statistics and Operations Research
Universi
Introduction to Empirical Processes and Semiparametric Inference
BIOS 791

Spring 2014
Empirical Processes: Lecture 10
Spring, 2014
Introduction to Empirical Processes
and Semiparametric Inference
Lecture 10: Empirical Process Methods
Michael R. Kosorok, Ph.D.
Professor and Chair of Biostatistics
Professor of Statistics and Operations Resea
Introduction to Empirical Processes and Semiparametric Inference
BIOS 791

Spring 2014
Empirical Processes: Lecture 11
Spring, 2014
Introduction to Empirical Processes
and Semiparametric Inference
Lecture 11: Empirical Process
Methods, Continued
Michael R. Kosorok, Ph.D.
Professor and Chair of Biostatistics
Professor of Statistics and Opera
Introduction to Empirical Processes and Semiparametric Inference
BIOS 791

Spring 2014
Empirical Processes: Lecture 08
Spring, 2012
Introduction to Empirical Processes
and Semiparametric Inference
Lecture 08: Stochastic Convergence
Michael R. Kosorok, Ph.D.
Professor and Chair of Biostatistics
Professor of Statistics and Operations Research
Introduction to Empirical Processes and Semiparametric Inference
BIOS 791

Spring 2014
Empirical Processes: Lecture 09
Spring, 2014
Introduction to Empirical Processes
and Semiparametric Inference
Lecture 09: Stochastic Convergence,
Continued
Michael R. Kosorok, Ph.D.
Professor and Chair of Biostatistics
Professor of Statistics and Operatio
Introduction to Empirical Processes and Semiparametric Inference
BIOS 791

Spring 2014
Empirical Processes: Lecture 12
Spring, 2010
Introduction to Empirical Processes
and Semiparametric Inference
Lecture 13: Entropy Calculations
Michael R. Kosorok, Ph.D.
Professor and Chair of Biostatistics
Professor of Statistics and Operations Research
U
Introduction to Empirical Processes and Semiparametric Inference
BIOS 791

Spring 2014
Empirical Processes: Lecture 12
Spring, 2010
Introduction to Empirical Processes
and Semiparametric Inference
Lecture 12: GlivenkoCantelli and
Donsker Results
Michael R. Kosorok, Ph.D.
Professor and Chair of Biostatistics
Professor of Statistics and Oper
Introduction to Empirical Processes and Semiparametric Inference
BIOS 791

Spring 2014
Empirical Processes: Lecture 15
Spring, 2010
Introduction to Empirical Processes
and Semiparametric Inference
Lecture 15: The Bootstrap and the
Delta Method
Michael R. Kosorok, Ph.D.
Professor and Chair of Biostatistics
Professor of Statistics and Operati
Introduction to Empirical Processes and Semiparametric Inference
BIOS 791

Spring 2014
Empirical Processes: Lecture 12
Spring, 2010
Introduction to Empirical Processes
and Semiparametric Inference
Lecture 14: Entropy Calculations and
the Bootstrap
Michael R. Kosorok, Ph.D.
Professor and Chair of Biostatistics
Professor of Statistics and Ope
Introduction to Empirical Processes and Semiparametric Inference
BIOS 791

Spring 2014
Solutions to Problem Set 10
2
Sn
18.5.3 When
is a consistent estimate of var( nTn ),
asymptotically optimal at level . Now,
(n)Tn
Sn
> z1 is
n sign(x)dFn (x)
R
n(P (x > 0) P (x < 0)
=
=
n(1 2Fn (0)
n
1
=
n(1 2
1cfw_Xi 0);
n i=1
nTn =
4
var( nTn ) = n
Introduction to Empirical Processes and Semiparametric Inference
BIOS 791

Spring 2014
Solutions to Problem Set 7
P ( > u)du. However, this is not norm,
10.5.1 (a) Recall that 2,1 = 0
because it does not satisfy the triangle inequality.
(b)For the rst inequality, if we can prove E( 2 ) = 2 0 P ( > u)udu
2 2,1 2 , then we have 2 = E( 2 )
Introduction to Empirical Processes and Semiparametric Inference
BIOS 791

Spring 2014
Empirical Processes: Lecture 17
Spring, 2010
Introduction to Empirical Processes
and Semiparametric Inference
Lecture 17: ZEstimators
Michael R. Kosorok, Ph.D.
Professor and Chair of Biostatistics
Professor of Statistics and Operations Research
Universit
Introduction to Empirical Processes and Semiparametric Inference
BIOS 791

Spring 2014
Empirical Processes: Lecture 20
Spring, 2010
Introduction to Empirical Processes
and Semiparametric Inference
Lecture 20: Examples
Michael R. Kosorok, Ph.D.
Professor and Chair of Biostatistics
Professor of Statistics and Operations Research
University of
Introduction to Empirical Processes and Semiparametric Inference
BIOS 791

Spring 2014
Empirical Processes: Lecture 22
Spring, 2012
Introduction to Empirical Processes
and Semiparametric Inference
Lecture 22: Preliminaries for
Semiparametric Inference
Michael R. Kosorok, Ph.D.
Professor and Chair of Biostatistics
Professor of Statistics and
Introduction to Empirical Processes and Semiparametric Inference
BIOS 791

Spring 2014
Empirical Processes: Lecture 24
Spring, 2010
Introduction to Empirical Processes
and Semiparametric Inference
Lecture 24: Regularity, Efciency and
Testing
Michael R. Kosorok, Ph.D.
Professor and Chair of Biostatistics
Professor of Statistics and Operation
Introduction to Empirical Processes and Semiparametric Inference
BIOS 791

Spring 2014
Empirical Processes: Lecture 21
Spring, 2010
Introduction to Empirical Processes
and Semiparametric Inference
Lecture 21: Proportional Odds Model,
Continued
Michael R. Kosorok, Ph.D.
Professor and Chair of Biostatistics
Professor of Statistics and Operati
Introduction to Empirical Processes and Semiparametric Inference
BIOS 791

Spring 2014
Empirical Processes: Lecture 23
Spring, 2012
Introduction to Empirical Processes
and Semiparametric Inference
Lecture 22: Semiparametric Models
and Efciency
Michael R. Kosorok, Ph.D.
Professor and Chair of Biostatistics
Professor of Statistics and Operati
Introduction to Empirical Processes and Semiparametric Inference
BIOS 791

Spring 2014
Solutions to Problem Set 5
8.5.1 (a) i) Clearly, X
X
C
= 0, then
= 0, i.e., X = 0 a.s.
ii)
aX
= inf c > 0 : E
= inf c > 0 : E
aX
c
X
c/a
c
> 0 : E
a
= a X .
= a inf
1
1
X
c/a
1
iii) Since is a convex function, then if X1 , X2 H , c1 , c2 > 0
sa
Introduction to Empirical Processes and Semiparametric Inference
BIOS 791

Spring 2014
Empirical Processes: Lecture 26
Spring, 2010
Introduction to Empirical Processes
and Semiparametric Inference
Lecture 26: Semiparametric Maximum
Likelihood Inference
Michael R. Kosorok, Ph.D.
Professor and Chair of Biostatistics
Professor of Statistics an
Introduction to Empirical Processes and Semiparametric Inference
BIOS 791

Spring 2014
Solutions to Problem Set 2
4.6.1 Without loss of generality, assume u v. Then
v
(u) (v) =
u
(e)
(e)de
(e)
=
1cfw_u s v
(e)
(e)de
(e)
1/2
1cfw_u s v(e)de
1/2
= F (u) F (v)
(e)
P
(e)
2
(e)
(e)
1/2
2
(e)de
1/2
.2
4.6.8 Since E[Y Z = z, U = u] = , (u, z)
Introduction to Empirical Processes and Semiparametric Inference
BIOS 791

Spring 2014
Empirical Processes: Lecture 25
Spring, 2010
Introduction to Empirical Processes
and Semiparametric Inference
Lecture 25: Semiparametric Models
Michael R. Kosorok, Ph.D.
Professor and Chair of Biostatistics
Professor of Statistics and Operations Research