Bootstrap Confidence Interval for Nonparametric Regression - repository civitas UGM
Sponsoring societies
Bernoulli Society
fr;r N4;rLiierrrrtieiii \i:ir;'',r.
and Prob:rliilitv
8th World Congress in Probability and Statistics
lstan b ul, 9 -1 4 July, 2012
fointly sponsored by the Bernoulli Society and the lnstitute of Mathematical Statistics
PROGRAM COMMITTEE
ADRIAN BADDELEY, University of Western Australia
VLADIMIR BOGACHEV, Moscow State University
KRZYSZTOF BURDZY University oJ Washington
T. TONY CAl, University of Pennsylvania
ELVAN CEYHAN, Kog University
P
RO BA L
CHAU
D H U Rl, I n d i a n
Stati sti ca I I n stitute
MINE qAe LA& Kot lJniversity
ERHAN qlNLAR, Princeton University
ANTHONY DAVISON,
Ecole Polytechnique F\ddrale de Lausanne
RICK DURRETT, Duke University
ARNOLDO FRlcESSi, l.)niversity of Oslo (Chair)
ALICE GUIONNET, Ecole Normale Supdrieure de Lyon
PETER CLYNN, Stanford University
PETER CUTTORB University of Washington
ON
ES
I
MO
H
ERNAN DE Z, I n stituto
Po
litecn i co
N a ci o n a
I
DMITRY IOFFE Technion-lsrael Institute of Technology
SUSHMITA MITRA, lndian Statistical lnstitute
SHf GE PENG,
ShandongUniversity
DOMINIQUE PICARD Universite Paris Vll
KAVITA RAMANAN, Brown University
SYLVIA RICHARDSON, lm perial College
VLADAS SIDORAVICIUS, Centrum Wiskunde & Informatica lnstituto Nacional de Matematica Pura
Aplicada
MICHAEL S@RENSEN University of Copenhagen
MATTHEW STEPHENT University of Chicago
GU ENTH ER WALTH ER, Stanford U niversity
VICTOR YOHAL Universidad de Buenos Aires
NAKAHIRO YOSHIDA" University oJ Tokyo
THALEIA ZARIPHOPOULOU The university of Texas at Austin
LOCAL ORGANIZI NG COMMITTEE
Fl KRI AKDENiZ Qukurov a U niv ersity
ELVAN CEYHAN Kog University (Co-chair)
MINE qAe LAR Kog LJniversity (Co-chair)
UXU CUnlER" Bitkent L)niversity
$ENNUR ONUR, Turkish Statistical Institute
SULEYMAN OZefiCi, Kog LJniversity
SEMIH SEZER, Sabanu University
FETI H Yl LDI Rl/W Qankay a U niv ersity
SCI ENTI FIC
SECRETARIAT
MEHMET 62, Xog University
ORGAN IZATIONAL SECRETARIAT
CAN OZHAN, Figilr Kongre & Organizasyon
e
Contents
Program Overview
Program
5
11
Abstracts
117
lndex
295
Abbreviations
IS
Invited Session
CS
Contributed Session
Name in boldface
BS GA BS
BS EC&CM
BS ERC BS
Speaker
General AssemblY
IMS BM
IMS CM
IMS ED
IMS EX
IMS PA
BS Executive Committee & Council Meeting
European Regional Committee Meeting
IMS Business Meeting
IMS Council Meeting
IMS Editors'Luncheon
IMS Executive Committee Meeting
IMS Presidential Address
.
f-
Program Overview
Monday July 9
08:00-09:00 Registration
13
09:00:1 0:00 Opening Ceremony
13
1
0:00-1 1 :00 Tukey Lecture
11:30-12:300 Wald Lecture 'l
14:00-15:45
IS 3 Geometric Perspectives on High-Dimensional Data
IS 33 Stochastic Methods for Equilibrium In Financial Markets
IS
Statistical Methods for Neuroscience
IS 30 High-Throughput Genomic Assays
IS l8 Probability and Computer Science
IS l7 MCMC: Adaptive, Likelihood Free, Particle Filters
3l
14:00-15:40
CS Applications to Biology and Medicine
CS 2 Asymptotic Distribution Theory
CS 3 Design of Experiments
I
CS
CS
CS
4 Air and Oceans
5 Likelihood and Her Cousins
6 Inference and Markov Chains
13
13
t3
l4
t4
l4
l5
15
t6
l7
18
l9
t9
20
<
CS
PROCRAM OVERVIEW
7 Measurement
>
Error and Missing Data
fiS 8 Queueing Theory and Traffic
(iS 9 Causality
l6:15-18:00
IS l4 Machine Learning
2l
22
22
23
8 Functional Data Analysis
IS 9 Graphical Models, Networks and Causalify
IS 40 Stochastic Population Genetics and Evolution
24
24
IS 34 Dynamics and Cubature for Stochastic Non-Linear Systems
25
IS
25
16:15-17:55
CS 10 Bayesian Statistics I
CS l1 Copula Models
CS l2 Classificationand Clustering
CS 13 Image Analysis
CS 14 Model Selection I
CS 15 Stochastic Differential Equations
CS 16 Biostatistics I
CS l7 Inference for Stochastic Processes I
CS 18 Theory I
CS 19 Time Series I
CS 20 Theory 2
1
9:30 -22:O0
Welcome Reception
25
26
27
28
28
29
30
30
3l
32
33
33
Tuesday 10July
09:00-10:00 Medallion Lecrure
1
10:00-11:00 Wald Lecrure 2
"l 1
:3O-12:30 Kol mogorov Lecture
34
34
34
12:30-14:O0 Meeting IMS ED
34
12:30-14:00 Meering
34
BS ES
& CM
13:30-17:30 Meeting IMS Ex
14:00-15.45
IS 39 Stochastic Partial Differential Equations with Applications
IS l5 Modem Applications of Malliavin Calculus
IS l0 High-Dimensional Inference
IS
Random Matrices and Applications
IS 24 Critical2-D Systems and SLE
2l
34
34
35
35
36
36
'14:00-"15:40
CS
CS
CS
21 Event History Analysis
22 Commodity Markets and Finance
23 Graphical Models and Random Graphs
36
37
38
<
CS
CS
CS
CS
CS
CS
PROCRAMOV€RVIEW
>
24 Thresholding and Variable Selection
25 Nonparametric Inference
26 Mode{ Selection 2
27 Geometric Probability and Stochastic Gcottrclrr
28 Gaussian Processes and Sample Path Properties
29 Theory 3
14:00-1 8:00 Poster Session
16:15-18:00
IS 13 Long-Range Dependence and Self-Similarity
IS 37 Stochastic Models of Cancer
IS 5 Data Depth
IS I Bayesian Nonparametrics
IS 26 Sparse Signals
!i
i-'
li
44
,16
,1
li
49
,19
50
16:15-17:55
CS 30 Statistical Genomics
CS 3l Central Limit and Other Weak Theorems I
CS
CS
CS
CS
CS
CS
CS
32 Markov Processes I
33 Factor Analysis and Principal Components
34 Finance
35 Statistical lndustrial Process Control
36 Stochastic Integrals
37 Processes with Independent Increments
38 Distribution Theory I
18:10-19:00
BS
Awards Ceremony and General Assembly
57
Wednesday 11 July
09:00-10:00 Wald Lecture 3
58
0:00-1 1:00 Laplace Lecture
58
1
11:3O-12:30 Medallion Lecture 2
58
14:00-1 5:30 IMS Presidential Address and IMS Awards Ceremony
58
16215-17:55
CS 39 Algorithms
CS 40 Biostatistics 2
Central Limit and other Weak Theorems 2
CS
CS 42 Models for Risk I
CS 43 Detection and Identification
4l
CS
CS
CS
CS
CS
CS
44 Markov Processes 2
45 High Dimensional Data
46 Networks
47 Re.gression I
48 Stochastic Calculus of Variations and Malliavin Calculus
49 Stein's Method and Applications
58
59
60
6l
61
62
63
63
64
6s
65
(S
(S
(-\
(S
i0
ln
ir
[)r stri
ir
il
linrtc Drr. rsrbrlrty' :rnil [-cvv Processes
bution f ltcor"I'uirc Scric. l
66
67
67
Titcorr
68
-1
Thursday "l2luly
70
09:00-1 0:00 Doob Lecture
10:00-11:00 World Congress Public Lecture
1:30-1 2:30 Bernoulli Lecture
70
17:3O-14:OO Meeting BS ERC
70
12:3O-14:3O Meeting IMS CM
70
14:00-1545
IS 27 Spatial Stochastic Models
IS 36 Stochastic Methods and Finance
70
1
l2
7l
Large-Scale Multiple Comparisons
7t
Composite Likelihood Inference
IS 23 Rare-Event Monte Carlo Methods
72
72
IS
IS
2
14:OO-15:40
CS
CS
CS
CS
CS
CS
CS
CS
CS
CS
54
55
56
57
58
59
60
Theory
72
5
Bayesian Statistics 2
73
Models for Risk 2
Robust Methods
Order Statistics
74
75
76
76
77
Quantile Regression
Reliability
6l Stationary Processes
and Gaussian Processes
62 Stable Laws and Fractional Diffusions
63 Theory6
78
79
79
14:00-18:00 Poster Session 2
80
15:45-16:15 Meeting IMS BM
85
16:15-18:00
IS 20 Random Combinatorial Structures
IS 6 Decision Theory, Control Theory and Games
IS 25 Space Time Data
85
85
86
IS 19 Quantile Regression
86
IS 38 Stochastic Networks with Applications
87
16:15-17:55
CS
CS
CS
64 Applications in Biology, Agriculture and Forestry
65 Functional Limit Theorems and Invariance Principles
66 Pair Copula Constructi,ons
87
88
89
ll.'L >
( \ ,r iiyrr4tltcsrs Testing
( \ r'\ l'\'r':rlisetl rcglession.p t l
i \ ,i\r \tr)!ir.rsric Ordinary Difterential Equations
( \ '-{) Il.rindr)n) lle lds
( S I I Rundt)ni Processes with lnteractions
( S 7l Sanrpling and Survey Methods
f
90
90
91
92
93
93
Friday t I July
09:00-10:00 Medallion Lecture
3
95
1:00 Medallion Lecture 4
95
1.1:30-12:30 Medallion Lecture 5
95
123O-'14:OO Meeting BS EC & CM
95
.10:00-1
14:OO-'15:45
IS
IS
i5
Stochastic Geometry and Spatial Point Processes
l6
Markov Processes
IS 28 Statistical Climatology
IS 29 Statistical Inference for Stochastic Differential Equations
IS 22 Random Media
95
96
96
97
97
14:00-15t4O
73 Semiparametric and Nonparametric Bayesian Methods
74 Stochastic Processes I
75 Industrial Applications and Clinical Trials
76 Portfolio Optimisation and Volatility
77 Markov Processes 3
CS 78 Random Matrices
CS 79 Regression 2
CS 80 Stochastic Processes 2
CS
CS
CS
CS
CS
CS
8l
Foundations
16:15-18:00
IS 4 Copula models
IS 32 Statistical methods in finance
IS 7 Extremes for Complex Phenomena
IS 11 Interacting Particle Systems
98
99
99
100
101
102
102
103
104
105
105
106
106
"16:15-17t55
CS
CS
CS
CS
CS
82
83
84
85
Limit Theorems and Random Walks
106
Weather and Climate
101
Mixfures
108
Wavelets
109
86 Stochastic Partial Differential Equations
CS 87 Spatial Models
CS 88 Inference for Stochastic Processes 2
CS 89 Stopping Times and Optirnal Stopping Problems
CS 90 Theory 7
CS 91 Time Series 3 and Longitudinal Data
109
110
111
tt2
113
114
<
PROCRAMOVERVIEW
>
Saturday 14July
09:00-10:00 Le Cam Lecture
115
10:00-1 1:00 L6vy Lecture
r15
1 1 :3O -
12:30
Cl osi n g
Ceremony
115
r< )li(x,RA\,1 p
,+"1*nclay July
I
cs 20 Theory 2
Vrruur: Kristal
CHAIR: YUVAL BENJAMINI, Department of Statistics, University af
Cahfornia Berkeley, Berkeley, CA
16:15 Shuffle estimators: measuring the explainable variance in fMRl experiments
Assrnncr 4zSPACE 235
YUVAL BENJAMINI, Department
oJ
Statistics, University of Cahfornia
Berkeley, Berkeley, CA
BIN YU, Department of Statistics, University oJ Caltfornia Berkeley, Berkeley,
CA
1
6:35 Modified goodness of flt test under selective order statistics
AssrRecr 336 PACE 2o9
FARIDATULAZNA AHMAD SHAHABUDDIN, School of Mathematical
Sciences,
Universiti Kebangsaan, Malaysia
KAMARULZAMAN IBRAHIM, School
oJ
Mathematical
Sciences,
Universiti
Kebangsaan, Malaysia
ABDU L AZIZ JEMAIN , School of Mathematical Sciences, Universiti
Kebangsaan, Malaysia
16:55 Delta method for deriving the consistency of bootstrap estimator for
parameter of autoregressive model
AssrRecr 146 PACE i53
Palembang lndonesia
SURYO GURITNO, Mathematics Department, Gadjahmada University,
Yogyakarta, lndonesia
Rl HA RYAT Ml, M ath e m ati cs
Yogyakarta, lndonesia
S
17:1 5
i'
D ep a
rtment,
Ga di ah m a d a U niv er sity,
Bootstrap confidence interval for nonparametric regression
AssrRecr 3B4Pecrzzz
SRI HARYATMI KARTIKO Department of Mathematics, Cadjah Mada
University
17:35 Test of arch effect based on second-order least square estimation
l.
Assrnect 169 PACE't60
HERNI UTAMI, Department
oJ
Mathematics, Gadjah Mada University,
Yogtakarta, lndonesia
SUBANAR SUBANAR, Department of Mathematics, Gadjah Mada
U niv ersity, Y o gy aka rta, n d o n esi a
DEDI ROSADI, Department of Mathematics, Gadjah Mada University,
I
Yogyakarta, lndonesia
19:3O -22:OO We lcome
reception
VrNue: Kog University
73
WORLDCONG2012 9-14 July 2012 ISTAMBUL, TURKtrY
BOOTSTRAP CONFIDENCE INTERVAL
FOR NONPARAMETRIC REGRESSION
Sri Haryatmi Kartiko
Mathematics Department
Gad.i ahmada University, Yogyakarta, Indonesia
Abstract
In
regression, pairs
objcctive is to estimate
of data (Xt,Yt),.'.,(X,",Y")
EIYIX: r]'
^(") -
a,re observed, and the
Nonparametric techniques allow m to
bc estimatcd without any assumptions on its underlying form. Unfortunately, this
flexibility
ca,n rrra,ke
it difficult to a,scertairt wlretlrer observed leatules are physica,lly
meaningful or the resull, of random va,riabilitv in the data. Confidence inl,ervals gives
a mcchanism to asscss the impact of this variability on the cstimate.
'I'he bootstrap a,pproach to constructing confidence intervals for irernel regression
cstimates is proposcd. First, hernel Nada.raya Watson rcgrcssion estimator is uscd
to calculatc thc residuals, which approximate thc tlue nnol:sclved errors.
Second,
thc r-csiduals arc re sampled, to create "ncw" erroLs. Third, thc re sa.mplcd errors
arc aclded to an estimate of thc regrcssion function, creating a new bootstrap data
set (X1
,yi),.. ,(X,.,Y,i).Finallv,
the bootstrap data set is smoothed, taking into
accolult the bia.s ald used for constructing qua,ntile bootstla,p cortfidertce irrterva.l.
Simulation study is conducted to see that the covcrage probability approximates
its level of significance.
Key wolds : kcrncl firnction. biam. bootstlap, quantilc, covcragc probabilit.v
l
I.
INrRoorrcrroN
I'he goal of regression fitting is to fincl the relationship of inclependent(s) r.ariable
7 and dependent
variable
fronr pair valiables
Y. If
('l,Y),
we havc n indepcndent obscrvations
{(",,y,)}i:1
regression equation can be wlitten as
Y:SQ)*e;i,:1,...,rL
(1 1)
whcre e is a random variable denoting the variation of
regrcssion cwve
Y around .q(f), the
mean
EIYIT : t].
The problem is to approximate the mean rcsponse function g,
s(t)
where
7.
: n(vlr : r, -
f(l,g) is the joint density of (T,Y)
J ttf(t'Y)da
j
f(t)
and
(1.2)
/(l) is thc marginal density of
Thc nonparametric approach fol cstimating thc regression curve, gives morc
Ilexibility since the modcl is no1, spccified such
a,s
in
1,hc
paramel,ric model. From
this point of view, Hardle (1991) said that in this approach we,'let the data speak
for themselves". This regression model
assumcs
that g belongs to sorne infinite
dimensional collection of functions. The choice of g is motivatcd by smoothness
propertics the regression function can be assumed to possess. In this research.g is
smooth of ordcr p.
II.
Esrrrur.q.rroN
The estimator of g is the wcighted avcrage of Y; rvith thc rvcight dcpcnding on
thc distancc of
Z
from
l. formulizc
: r tiw,,i(t;l1.-.,7,,)Y,,
(2 3)
is thc wcight function clcpencling on bandwiclth
ol smoothing paramctcr
o.Q)
r'vhcle
llf";
as
h and samplc Tt . . . ,7,, from thc inclcpcndcnt varitrblc. For
sinplicity W,,;(t;Ty
...
,7,,)
is u'r'itten asl,V,,;(.t). Thc abovc cstimator is a r.vcightcd lcast sqnarcs cstimator as
provcd in thc rrcxt thcorcm.
Theorem
2,L For sent'ipara,m,e,t,r,ic:
lt,,tt)
uhere LDl,'-r!4r.,,r1/)
: I,
'is
rertress'iort,,model 7.1.. th,t: t:st,i,m,ator
-,, 't
-
ll',,,(/)Y'.
(2 +)
a uLeighted least sq,uares estirnator for g(t)
Nrrrlala\a arrd \\'atson(196.1) choose I,I,',,i(t)
- ;+*;i-J.
Iil,
(1)
: iri(il,
1( is a kcrncl function, and g,.(l) is thcn called kerncl estimator', wlittcn
(2.5)
u,hclc lr is bandr,r,idth or smoothing pararneter u'hich defines the degree of smoothncss
of the estimator. The most uscd kerncl functions arc Uniform, Tliangle,
Epancchrrikov and Gaussian.
Bv taking u,,,i(t)
: t+trfi
thc cstimator of 9(l) statect abovc is cquivalcnt
to
g,,(t) - D::tw-i(t)Y.
Sevcral properties of wcight function I'tr/,;(l) (which is also
truc for.r',;(l) with
scvcral adjustment) are:
(1) The Weights tr44",(1) depend on the samplc {Ti}'i':, through the hcrncl dcnsity estimatot ir,(t).
(2) Obscrvations g; obtained morc
ing
(3)
fi
r'vcight in thosc aLeas
t'hclc thc collcspond-
are spalsc.
If /i - 0, thcn I41,;(l) - n for t - Tt.
(4) Fol h
-
oo, the weight function W,,i(t)
-
1 for all
l, hcrcc A(l)
convergcs
rn lltc collstarr frrDctiorr y.
(5) llhc bandrviclth h dctcrmincs thc lcvcl of snoothncss. clccrcasing h lcacls to
a lcss smooth estimatc.
Tlreorem 2.2.
^rsElL,
Xt[ean Squarcd,
Error' .for g.,,(t)
zs
(t)): h581r( l3+ 'i (n" (r) + 2
)') LL3Q{ )
^tft;Q
lo(rth t) + o(tLr)
(2.6)
4
Frorn thc abor.c theolen, lt,ISElQ,,(l)] is of olclcl O(rz l/5). fbr
'l'he fir'st srrrnrna,rrd in'-['heolern 2.2 t]csclibcs
- 11,-1/5.
thc zrsvniptotic valia.rcc ol (t,,(t1.
This fbrnmla for the variancc is a fiurction of the ma.r'ginal dcnsitv
conditior-ral r.ariancc
a2(l) and
11
l(z)
and thc
agrecs u'ith thc assr.rnrption that the lcgrcssiol cnrvc
is more stable in thosc arcas rvhere we havc plcnty of obscrvations. The second
sunrnrand corresponds
to thc squared bias of g.,,(t) and is cithcl doniinated by
the second derivative g" (t) ifwe arc ncar thc local extreme of g(l) or bv the first
dcrivativc A'(l) when we ilre nezrr the inflection point of 9(i).
From Eubank(1988), the use ofseveral kernel will not give abig difference
fbr tlie estirnator. The choicc of bandwidth gives rnorc irifluctrcc. Irr otllcr wolds,
if
the choice of bandwidth is optimized, the different hernel will give almost the same
result. Sevcral distance measurencnts arc:
(1) I\{can Squared
Errol
'lL
MSE(h): ASE(h):
(2) Integrated Squarccl
I
"-'fi:1' .(r')(s.(T") - g(.ri))2
(2-7)
Error
SE(h)
:
fn
J _*utlt)(!1,,(t)
-
s\t))" .f (t)dt
(2.8)
(3) Condition Squared Error
A,,r
AS
E(h)
:
E(AS E(.h)
lTt,. . .,7,,)
(2.e)
(r) Nlcan Integra,ted Squarcd Enor
NrrsE(h): E(rsE(h))
The nonncgatir,-e rveight function
of
X.
u.r
(2.10)
is a tool to dlop observa.tions at the boundaly
Thc abovc distance neasLll'cmcnt lcads asyrnptotica,lly to thc sirmc lcvcl of
smoothing.
Theorem 2.3. (Hardle dan Al[artrn. 1996)
Suppose that
: Al. E(yk'ft : t) < c:t < cc, k - 7.2,.. . ,
: A2. ,f (t) kortt'inu,
Hol,dcr dan positiJ'pado,
su:1tort, d,o.rt
u.
I
A3.
Ii
Ilolder
kontin'Lt,
then. .for kernel
c:st;int,at.or-s.
-I ::3 0 untrr,k l'L
supld(h)AH- AIISEth\l
(2.11)
- cx:.
S E(h)
t, u,, I
wlrere H,,: [,.'r-1,n i]. 0 < 6 < Il2 a,rt d(Lt) is n,e r.f the th,ee d,,istance,meusu,rement except fuI I
S
E.
The theoren implics that cach sequcnce of bandwidths minimizing any of thcsc
threc distance measuremcnts AStr(h), ISE(h), N,IAStr(h) is also asymptotically opti-
mal in rclation to the optimizing bandwidth of the \,Iean Intcgrated Squarcd Error.
Fol convcniencc. u'e will tal1,ive
Amer'ican Stati,sti.cal
A
ss
suroothilg :rlrl
ocio,ti,on 83 (401
Bernoulli Society
fr;r N4;rLiierrrrtieiii \i:ir;'',r.
and Prob:rliilitv
8th World Congress in Probability and Statistics
lstan b ul, 9 -1 4 July, 2012
fointly sponsored by the Bernoulli Society and the lnstitute of Mathematical Statistics
PROGRAM COMMITTEE
ADRIAN BADDELEY, University of Western Australia
VLADIMIR BOGACHEV, Moscow State University
KRZYSZTOF BURDZY University oJ Washington
T. TONY CAl, University of Pennsylvania
ELVAN CEYHAN, Kog University
P
RO BA L
CHAU
D H U Rl, I n d i a n
Stati sti ca I I n stitute
MINE qAe LA& Kot lJniversity
ERHAN qlNLAR, Princeton University
ANTHONY DAVISON,
Ecole Polytechnique F\ddrale de Lausanne
RICK DURRETT, Duke University
ARNOLDO FRlcESSi, l.)niversity of Oslo (Chair)
ALICE GUIONNET, Ecole Normale Supdrieure de Lyon
PETER CLYNN, Stanford University
PETER CUTTORB University of Washington
ON
ES
I
MO
H
ERNAN DE Z, I n stituto
Po
litecn i co
N a ci o n a
I
DMITRY IOFFE Technion-lsrael Institute of Technology
SUSHMITA MITRA, lndian Statistical lnstitute
SHf GE PENG,
ShandongUniversity
DOMINIQUE PICARD Universite Paris Vll
KAVITA RAMANAN, Brown University
SYLVIA RICHARDSON, lm perial College
VLADAS SIDORAVICIUS, Centrum Wiskunde & Informatica lnstituto Nacional de Matematica Pura
Aplicada
MICHAEL S@RENSEN University of Copenhagen
MATTHEW STEPHENT University of Chicago
GU ENTH ER WALTH ER, Stanford U niversity
VICTOR YOHAL Universidad de Buenos Aires
NAKAHIRO YOSHIDA" University oJ Tokyo
THALEIA ZARIPHOPOULOU The university of Texas at Austin
LOCAL ORGANIZI NG COMMITTEE
Fl KRI AKDENiZ Qukurov a U niv ersity
ELVAN CEYHAN Kog University (Co-chair)
MINE qAe LAR Kog LJniversity (Co-chair)
UXU CUnlER" Bitkent L)niversity
$ENNUR ONUR, Turkish Statistical Institute
SULEYMAN OZefiCi, Kog LJniversity
SEMIH SEZER, Sabanu University
FETI H Yl LDI Rl/W Qankay a U niv ersity
SCI ENTI FIC
SECRETARIAT
MEHMET 62, Xog University
ORGAN IZATIONAL SECRETARIAT
CAN OZHAN, Figilr Kongre & Organizasyon
e
Contents
Program Overview
Program
5
11
Abstracts
117
lndex
295
Abbreviations
IS
Invited Session
CS
Contributed Session
Name in boldface
BS GA BS
BS EC&CM
BS ERC BS
Speaker
General AssemblY
IMS BM
IMS CM
IMS ED
IMS EX
IMS PA
BS Executive Committee & Council Meeting
European Regional Committee Meeting
IMS Business Meeting
IMS Council Meeting
IMS Editors'Luncheon
IMS Executive Committee Meeting
IMS Presidential Address
.
f-
Program Overview
Monday July 9
08:00-09:00 Registration
13
09:00:1 0:00 Opening Ceremony
13
1
0:00-1 1 :00 Tukey Lecture
11:30-12:300 Wald Lecture 'l
14:00-15:45
IS 3 Geometric Perspectives on High-Dimensional Data
IS 33 Stochastic Methods for Equilibrium In Financial Markets
IS
Statistical Methods for Neuroscience
IS 30 High-Throughput Genomic Assays
IS l8 Probability and Computer Science
IS l7 MCMC: Adaptive, Likelihood Free, Particle Filters
3l
14:00-15:40
CS Applications to Biology and Medicine
CS 2 Asymptotic Distribution Theory
CS 3 Design of Experiments
I
CS
CS
CS
4 Air and Oceans
5 Likelihood and Her Cousins
6 Inference and Markov Chains
13
13
t3
l4
t4
l4
l5
15
t6
l7
18
l9
t9
20
<
CS
PROCRAM OVERVIEW
7 Measurement
>
Error and Missing Data
fiS 8 Queueing Theory and Traffic
(iS 9 Causality
l6:15-18:00
IS l4 Machine Learning
2l
22
22
23
8 Functional Data Analysis
IS 9 Graphical Models, Networks and Causalify
IS 40 Stochastic Population Genetics and Evolution
24
24
IS 34 Dynamics and Cubature for Stochastic Non-Linear Systems
25
IS
25
16:15-17:55
CS 10 Bayesian Statistics I
CS l1 Copula Models
CS l2 Classificationand Clustering
CS 13 Image Analysis
CS 14 Model Selection I
CS 15 Stochastic Differential Equations
CS 16 Biostatistics I
CS l7 Inference for Stochastic Processes I
CS 18 Theory I
CS 19 Time Series I
CS 20 Theory 2
1
9:30 -22:O0
Welcome Reception
25
26
27
28
28
29
30
30
3l
32
33
33
Tuesday 10July
09:00-10:00 Medallion Lecrure
1
10:00-11:00 Wald Lecrure 2
"l 1
:3O-12:30 Kol mogorov Lecture
34
34
34
12:30-14:O0 Meeting IMS ED
34
12:30-14:00 Meering
34
BS ES
& CM
13:30-17:30 Meeting IMS Ex
14:00-15.45
IS 39 Stochastic Partial Differential Equations with Applications
IS l5 Modem Applications of Malliavin Calculus
IS l0 High-Dimensional Inference
IS
Random Matrices and Applications
IS 24 Critical2-D Systems and SLE
2l
34
34
35
35
36
36
'14:00-"15:40
CS
CS
CS
21 Event History Analysis
22 Commodity Markets and Finance
23 Graphical Models and Random Graphs
36
37
38
<
CS
CS
CS
CS
CS
CS
PROCRAMOV€RVIEW
>
24 Thresholding and Variable Selection
25 Nonparametric Inference
26 Mode{ Selection 2
27 Geometric Probability and Stochastic Gcottrclrr
28 Gaussian Processes and Sample Path Properties
29 Theory 3
14:00-1 8:00 Poster Session
16:15-18:00
IS 13 Long-Range Dependence and Self-Similarity
IS 37 Stochastic Models of Cancer
IS 5 Data Depth
IS I Bayesian Nonparametrics
IS 26 Sparse Signals
!i
i-'
li
44
,16
,1
li
49
,19
50
16:15-17:55
CS 30 Statistical Genomics
CS 3l Central Limit and Other Weak Theorems I
CS
CS
CS
CS
CS
CS
CS
32 Markov Processes I
33 Factor Analysis and Principal Components
34 Finance
35 Statistical lndustrial Process Control
36 Stochastic Integrals
37 Processes with Independent Increments
38 Distribution Theory I
18:10-19:00
BS
Awards Ceremony and General Assembly
57
Wednesday 11 July
09:00-10:00 Wald Lecture 3
58
0:00-1 1:00 Laplace Lecture
58
1
11:3O-12:30 Medallion Lecture 2
58
14:00-1 5:30 IMS Presidential Address and IMS Awards Ceremony
58
16215-17:55
CS 39 Algorithms
CS 40 Biostatistics 2
Central Limit and other Weak Theorems 2
CS
CS 42 Models for Risk I
CS 43 Detection and Identification
4l
CS
CS
CS
CS
CS
CS
44 Markov Processes 2
45 High Dimensional Data
46 Networks
47 Re.gression I
48 Stochastic Calculus of Variations and Malliavin Calculus
49 Stein's Method and Applications
58
59
60
6l
61
62
63
63
64
6s
65
(S
(S
(-\
(S
i0
ln
ir
[)r stri
ir
il
linrtc Drr. rsrbrlrty' :rnil [-cvv Processes
bution f ltcor"I'uirc Scric. l
66
67
67
Titcorr
68
-1
Thursday "l2luly
70
09:00-1 0:00 Doob Lecture
10:00-11:00 World Congress Public Lecture
1:30-1 2:30 Bernoulli Lecture
70
17:3O-14:OO Meeting BS ERC
70
12:3O-14:3O Meeting IMS CM
70
14:00-1545
IS 27 Spatial Stochastic Models
IS 36 Stochastic Methods and Finance
70
1
l2
7l
Large-Scale Multiple Comparisons
7t
Composite Likelihood Inference
IS 23 Rare-Event Monte Carlo Methods
72
72
IS
IS
2
14:OO-15:40
CS
CS
CS
CS
CS
CS
CS
CS
CS
CS
54
55
56
57
58
59
60
Theory
72
5
Bayesian Statistics 2
73
Models for Risk 2
Robust Methods
Order Statistics
74
75
76
76
77
Quantile Regression
Reliability
6l Stationary Processes
and Gaussian Processes
62 Stable Laws and Fractional Diffusions
63 Theory6
78
79
79
14:00-18:00 Poster Session 2
80
15:45-16:15 Meeting IMS BM
85
16:15-18:00
IS 20 Random Combinatorial Structures
IS 6 Decision Theory, Control Theory and Games
IS 25 Space Time Data
85
85
86
IS 19 Quantile Regression
86
IS 38 Stochastic Networks with Applications
87
16:15-17:55
CS
CS
CS
64 Applications in Biology, Agriculture and Forestry
65 Functional Limit Theorems and Invariance Principles
66 Pair Copula Constructi,ons
87
88
89
ll.'L >
( \ ,r iiyrr4tltcsrs Testing
( \ r'\ l'\'r':rlisetl rcglession.p t l
i \ ,i\r \tr)!ir.rsric Ordinary Difterential Equations
( \ '-{) Il.rindr)n) lle lds
( S I I Rundt)ni Processes with lnteractions
( S 7l Sanrpling and Survey Methods
f
90
90
91
92
93
93
Friday t I July
09:00-10:00 Medallion Lecture
3
95
1:00 Medallion Lecture 4
95
1.1:30-12:30 Medallion Lecture 5
95
123O-'14:OO Meeting BS EC & CM
95
.10:00-1
14:OO-'15:45
IS
IS
i5
Stochastic Geometry and Spatial Point Processes
l6
Markov Processes
IS 28 Statistical Climatology
IS 29 Statistical Inference for Stochastic Differential Equations
IS 22 Random Media
95
96
96
97
97
14:00-15t4O
73 Semiparametric and Nonparametric Bayesian Methods
74 Stochastic Processes I
75 Industrial Applications and Clinical Trials
76 Portfolio Optimisation and Volatility
77 Markov Processes 3
CS 78 Random Matrices
CS 79 Regression 2
CS 80 Stochastic Processes 2
CS
CS
CS
CS
CS
CS
8l
Foundations
16:15-18:00
IS 4 Copula models
IS 32 Statistical methods in finance
IS 7 Extremes for Complex Phenomena
IS 11 Interacting Particle Systems
98
99
99
100
101
102
102
103
104
105
105
106
106
"16:15-17t55
CS
CS
CS
CS
CS
82
83
84
85
Limit Theorems and Random Walks
106
Weather and Climate
101
Mixfures
108
Wavelets
109
86 Stochastic Partial Differential Equations
CS 87 Spatial Models
CS 88 Inference for Stochastic Processes 2
CS 89 Stopping Times and Optirnal Stopping Problems
CS 90 Theory 7
CS 91 Time Series 3 and Longitudinal Data
109
110
111
tt2
113
114
<
PROCRAMOVERVIEW
>
Saturday 14July
09:00-10:00 Le Cam Lecture
115
10:00-1 1:00 L6vy Lecture
r15
1 1 :3O -
12:30
Cl osi n g
Ceremony
115
r< )li(x,RA\,1 p
,+"1*nclay July
I
cs 20 Theory 2
Vrruur: Kristal
CHAIR: YUVAL BENJAMINI, Department of Statistics, University af
Cahfornia Berkeley, Berkeley, CA
16:15 Shuffle estimators: measuring the explainable variance in fMRl experiments
Assrnncr 4zSPACE 235
YUVAL BENJAMINI, Department
oJ
Statistics, University of Cahfornia
Berkeley, Berkeley, CA
BIN YU, Department of Statistics, University oJ Caltfornia Berkeley, Berkeley,
CA
1
6:35 Modified goodness of flt test under selective order statistics
AssrRecr 336 PACE 2o9
FARIDATULAZNA AHMAD SHAHABUDDIN, School of Mathematical
Sciences,
Universiti Kebangsaan, Malaysia
KAMARULZAMAN IBRAHIM, School
oJ
Mathematical
Sciences,
Universiti
Kebangsaan, Malaysia
ABDU L AZIZ JEMAIN , School of Mathematical Sciences, Universiti
Kebangsaan, Malaysia
16:55 Delta method for deriving the consistency of bootstrap estimator for
parameter of autoregressive model
AssrRecr 146 PACE i53
Palembang lndonesia
SURYO GURITNO, Mathematics Department, Gadjahmada University,
Yogyakarta, lndonesia
Rl HA RYAT Ml, M ath e m ati cs
Yogyakarta, lndonesia
S
17:1 5
i'
D ep a
rtment,
Ga di ah m a d a U niv er sity,
Bootstrap confidence interval for nonparametric regression
AssrRecr 3B4Pecrzzz
SRI HARYATMI KARTIKO Department of Mathematics, Cadjah Mada
University
17:35 Test of arch effect based on second-order least square estimation
l.
Assrnect 169 PACE't60
HERNI UTAMI, Department
oJ
Mathematics, Gadjah Mada University,
Yogtakarta, lndonesia
SUBANAR SUBANAR, Department of Mathematics, Gadjah Mada
U niv ersity, Y o gy aka rta, n d o n esi a
DEDI ROSADI, Department of Mathematics, Gadjah Mada University,
I
Yogyakarta, lndonesia
19:3O -22:OO We lcome
reception
VrNue: Kog University
73
WORLDCONG2012 9-14 July 2012 ISTAMBUL, TURKtrY
BOOTSTRAP CONFIDENCE INTERVAL
FOR NONPARAMETRIC REGRESSION
Sri Haryatmi Kartiko
Mathematics Department
Gad.i ahmada University, Yogyakarta, Indonesia
Abstract
In
regression, pairs
objcctive is to estimate
of data (Xt,Yt),.'.,(X,",Y")
EIYIX: r]'
^(") -
a,re observed, and the
Nonparametric techniques allow m to
bc estimatcd without any assumptions on its underlying form. Unfortunately, this
flexibility
ca,n rrra,ke
it difficult to a,scertairt wlretlrer observed leatules are physica,lly
meaningful or the resull, of random va,riabilitv in the data. Confidence inl,ervals gives
a mcchanism to asscss the impact of this variability on the cstimate.
'I'he bootstrap a,pproach to constructing confidence intervals for irernel regression
cstimates is proposcd. First, hernel Nada.raya Watson rcgrcssion estimator is uscd
to calculatc thc residuals, which approximate thc tlue nnol:sclved errors.
Second,
thc r-csiduals arc re sampled, to create "ncw" erroLs. Third, thc re sa.mplcd errors
arc aclded to an estimate of thc regrcssion function, creating a new bootstrap data
set (X1
,yi),.. ,(X,.,Y,i).Finallv,
the bootstrap data set is smoothed, taking into
accolult the bia.s ald used for constructing qua,ntile bootstla,p cortfidertce irrterva.l.
Simulation study is conducted to see that the covcrage probability approximates
its level of significance.
Key wolds : kcrncl firnction. biam. bootstlap, quantilc, covcragc probabilit.v
l
I.
INrRoorrcrroN
I'he goal of regression fitting is to fincl the relationship of inclependent(s) r.ariable
7 and dependent
variable
fronr pair valiables
Y. If
('l,Y),
we havc n indepcndent obscrvations
{(",,y,)}i:1
regression equation can be wlitten as
Y:SQ)*e;i,:1,...,rL
(1 1)
whcre e is a random variable denoting the variation of
regrcssion cwve
Y around .q(f), the
mean
EIYIT : t].
The problem is to approximate the mean rcsponse function g,
s(t)
where
7.
: n(vlr : r, -
f(l,g) is the joint density of (T,Y)
J ttf(t'Y)da
j
f(t)
and
(1.2)
/(l) is thc marginal density of
Thc nonparametric approach fol cstimating thc regression curve, gives morc
Ilexibility since the modcl is no1, spccified such
a,s
in
1,hc
paramel,ric model. From
this point of view, Hardle (1991) said that in this approach we,'let the data speak
for themselves". This regression model
assumcs
that g belongs to sorne infinite
dimensional collection of functions. The choice of g is motivatcd by smoothness
propertics the regression function can be assumed to possess. In this research.g is
smooth of ordcr p.
II.
Esrrrur.q.rroN
The estimator of g is the wcighted avcrage of Y; rvith thc rvcight dcpcnding on
thc distancc of
Z
from
l. formulizc
: r tiw,,i(t;l1.-.,7,,)Y,,
(2 3)
is thc wcight function clcpencling on bandwiclth
ol smoothing paramctcr
o.Q)
r'vhcle
llf";
as
h and samplc Tt . . . ,7,, from thc inclcpcndcnt varitrblc. For
sinplicity W,,;(t;Ty
...
,7,,)
is u'r'itten asl,V,,;(.t). Thc abovc cstimator is a r.vcightcd lcast sqnarcs cstimator as
provcd in thc rrcxt thcorcm.
Theorem
2,L For sent'ipara,m,e,t,r,ic:
lt,,tt)
uhere LDl,'-r!4r.,,r1/)
: I,
'is
rertress'iort,,model 7.1.. th,t: t:st,i,m,ator
-,, 't
-
ll',,,(/)Y'.
(2 +)
a uLeighted least sq,uares estirnator for g(t)
Nrrrlala\a arrd \\'atson(196.1) choose I,I,',,i(t)
- ;+*;i-J.
Iil,
(1)
: iri(il,
1( is a kcrncl function, and g,.(l) is thcn called kerncl estimator', wlittcn
(2.5)
u,hclc lr is bandr,r,idth or smoothing pararneter u'hich defines the degree of smoothncss
of the estimator. The most uscd kerncl functions arc Uniform, Tliangle,
Epancchrrikov and Gaussian.
Bv taking u,,,i(t)
: t+trfi
thc cstimator of 9(l) statect abovc is cquivalcnt
to
g,,(t) - D::tw-i(t)Y.
Sevcral properties of wcight function I'tr/,;(l) (which is also
truc for.r',;(l) with
scvcral adjustment) are:
(1) The Weights tr44",(1) depend on the samplc {Ti}'i':, through the hcrncl dcnsity estimatot ir,(t).
(2) Obscrvations g; obtained morc
ing
(3)
fi
r'vcight in thosc aLeas
t'hclc thc collcspond-
are spalsc.
If /i - 0, thcn I41,;(l) - n for t - Tt.
(4) Fol h
-
oo, the weight function W,,i(t)
-
1 for all
l, hcrcc A(l)
convergcs
rn lltc collstarr frrDctiorr y.
(5) llhc bandrviclth h dctcrmincs thc lcvcl of snoothncss. clccrcasing h lcacls to
a lcss smooth estimatc.
Tlreorem 2.2.
^rsElL,
Xt[ean Squarcd,
Error' .for g.,,(t)
zs
(t)): h581r( l3+ 'i (n" (r) + 2
)') LL3Q{ )
^tft;Q
lo(rth t) + o(tLr)
(2.6)
4
Frorn thc abor.c theolen, lt,ISElQ,,(l)] is of olclcl O(rz l/5). fbr
'l'he fir'st srrrnrna,rrd in'-['heolern 2.2 t]csclibcs
- 11,-1/5.
thc zrsvniptotic valia.rcc ol (t,,(t1.
This fbrnmla for the variancc is a fiurction of the ma.r'ginal dcnsitv
conditior-ral r.ariancc
a2(l) and
11
l(z)
and thc
agrecs u'ith thc assr.rnrption that the lcgrcssiol cnrvc
is more stable in thosc arcas rvhere we havc plcnty of obscrvations. The second
sunrnrand corresponds
to thc squared bias of g.,,(t) and is cithcl doniinated by
the second derivative g" (t) ifwe arc ncar thc local extreme of g(l) or bv the first
dcrivativc A'(l) when we ilre nezrr the inflection point of 9(i).
From Eubank(1988), the use ofseveral kernel will not give abig difference
fbr tlie estirnator. The choicc of bandwidth gives rnorc irifluctrcc. Irr otllcr wolds,
if
the choice of bandwidth is optimized, the different hernel will give almost the same
result. Sevcral distance measurencnts arc:
(1) I\{can Squared
Errol
'lL
MSE(h): ASE(h):
(2) Integrated Squarccl
I
"-'fi:1' .(r')(s.(T") - g(.ri))2
(2-7)
Error
SE(h)
:
fn
J _*utlt)(!1,,(t)
-
s\t))" .f (t)dt
(2.8)
(3) Condition Squared Error
A,,r
AS
E(h)
:
E(AS E(.h)
lTt,. . .,7,,)
(2.e)
(r) Nlcan Integra,ted Squarcd Enor
NrrsE(h): E(rsE(h))
The nonncgatir,-e rveight function
of
X.
u.r
(2.10)
is a tool to dlop observa.tions at the boundaly
Thc abovc distance neasLll'cmcnt lcads asyrnptotica,lly to thc sirmc lcvcl of
smoothing.
Theorem 2.3. (Hardle dan Al[artrn. 1996)
Suppose that
: Al. E(yk'ft : t) < c:t < cc, k - 7.2,.. . ,
: A2. ,f (t) kortt'inu,
Hol,dcr dan positiJ'pado,
su:1tort, d,o.rt
u.
I
A3.
Ii
Ilolder
kontin'Lt,
then. .for kernel
c:st;int,at.or-s.
-I ::3 0 untrr,k l'L
supld(h)AH- AIISEth\l
(2.11)
- cx:.
S E(h)
t, u,, I
wlrere H,,: [,.'r-1,n i]. 0 < 6 < Il2 a,rt d(Lt) is n,e r.f the th,ee d,,istance,meusu,rement except fuI I
S
E.
The theoren implics that cach sequcnce of bandwidths minimizing any of thcsc
threc distance measuremcnts AStr(h), ISE(h), N,IAStr(h) is also asymptotically opti-
mal in rclation to the optimizing bandwidth of the \,Iean Intcgrated Squarcd Error.
Fol convcniencc. u'e will tal1,ive
Amer'ican Stati,sti.cal
A
ss
suroothilg :rlrl
ocio,ti,on 83 (401