LECTURE 2 LEAST SQUARES CROSS-VALIDATION FOR KERNEL DENSITY ESTIMATION
|
|
|
- Andrew Tucker
- 5 years ago
- Views:
Transcription
1 Jauary 3 07 LECTURE LEAST SQUARES CROSS-VALIDATION FOR ERNEL DENSITY ESTIMATION Noparametric kerel estimatio is extremely sesitive to te coice of badwidt as larger values of result i averagig over more observatios smooter estimates ad more bias wile smaller values of result i rouger estimates ad bigger variaces. Tus it is desirable to ave a automatic procedure tat picks te optimal value of i some sese from data. Here we cosider oe suc procedure least squares cross-validatio i te case of a uivariate desity estimatio. As te optimality criterio we use te itegrated MSE IMSE discussed i te previous lecture: IMSE E ˆf x fx dx b c xdx 4 b 4 + c + o f x dx 4 + u udu as c x was defied i Lecture as c x f x u udu; ad c c xdx udu as c x was defied i Lecture as c x fx udu. As sow i Lecture te IMSE-optimal badwidt is give by / c /5 b u du /5 f x dx /5 u u du /5 /5. Te discussio below follows Li ad Racie 007 see Sectio.3. Cross-validatio Ideally oe would like to compute te IMSE for differet values of owever fx appearig i te expressio of IMSE is ukow. Tus we sould attempt to approximate te IMSE usig data. Cosider te itegrated squared differece betwee ˆf ad f: ˆf x fx dx ˆf xdx ˆf xfxdx + f xdx.
2 Te last term does ot deped o ad terefore ca be igored. For te first term we ave ˆf xdx Xi x dx i Xi x Xj x dx i j u + u du i j i j we used te cage of variable u X i x/ x X i u X j x/ X j X i / + u dx du ad defied v uv + udu. Te fuctio v is called te covolutio kerel ad ca be computed for ay value v sice is kow ad cose by te ecoometricia. Te secod term i ivolves te ukow f ad terefore must be estimated. Sice for a fuctio ψx ψxfxdx EψX i oe ca estimate ψxfxdx by i ψx i. Hece a aive estimator of ˆf xfxdx would be i ˆf X i i j Tis approac owever does ot approximate ˆf xfxdx well as te expressio above cotais terms wit 0 wic produces 0. Tus te expressio must be modified to elimiate te terms wit 0: ˆf i X i i j i i Xj x ˆf i x j i.
3 is te so called leave-oe-out estimator of fx. Oe ca sow tat usig te leave-oe-out estimator of f produces a ubiased estimator of te leadig term of E ˆf xfxdx we data are iid ad te badwidt is fixed: E ˆf i E ˆf i X i i EE ˆf i X i X... X i X i+... X E ˆf i xf Xi X...X i X i+...x x X... X i X i+... X dx E ˆf i xfxdx by idepedece of X... X E ˆf i xfxdx E ˆf xfxdx te last equality olds sice E ˆf x E X j x/ does ot deped o te umber of observatios we is fixed. Te least squares cross-validatio criterio becomes CV i j i j i wic must be miimized umerically w.r.t to fid te optimal badwidt. Oe ca sow tat te leadig term of E CV captures tat of IMSE i. Lemma. Suppose tat Assumptios o te kerel fuctio i Lecture olds. Suppose tat data are iid wit te PDF f wic is four times cotiuously differetiable wit uiformly bouded derivatives. Te E CV f xdx + b 4 + c + o 4 +. Remark. i Note tat miimizig CV is equivalet to miimizig CV + f xdx. Hece te leadig term of E CV captures exactly te leadig term of IMSE E ˆf x fx dx. ii If oe does ot use te leave-oe-out estimator te tird term i te expressio for E CV becomes c 0/ wic is egative for typical kerels. As a result miimizig CV would produce 0 see Exercise.6 i Li ad Racie 007. Proof. Tis is a part of Exercise.6 i Li ad Racie 007. First re-write CV as CV 0 + CV + R 3 3
4 CV i j i i j i R i j i j i. i j i Note tat te first term i 3 captures oe of te leadig terms of te IMSE as 0 udu c. Sice u 0 oe ca easily sow tat R O p : P R > ɛ E R /ɛ wile E R E u v fufvdudv yfv + yfvdydv O. We ca re-write CV term as CV κ i j i κu u u. Next by te usual cage of variable argumet E CV κyfv + yfvdvdy 3 κy fv + j! f j vy j j + 4! f 4 v yy 4 4 fvdvdy j v y v y ad f j x dj fx dx j. 4
5 Usig te domiated covergece teorem ad te assumptio tat f 4 is uiformly bouded i.e. sup f 4 x < c x for some costat c > 0 we obtai: E CV 4 j0 j y j κydy j! f j vfvdv + o 4. Te proof ca ow be completed by sowig tat 0 j f j f x dx j vfvdv 0 j 3 f x dx j 4 wic ca be sow usig itegratio by parts ad tat j 0 0 j y j κydy 0 j 0 j 3 6 y ydy j 4 te result for j 4 ca be sow usig te biomial formula: for ay positive iteger p p p a + b p a i b b i. i i0 Lemma ca be used to sow tat CV E CV + s.o.p 4 s.o.p stads for smaller-order terms i probability. Te formula for CV ivolves expressios wit double summatios over i j: H X i X j. i j Suc statistics are called U-statistics. Teorem 4 i te Appedix sows ow U-statistics ca be approximated by usual averages of observatios ivolvig oly a sigle sum i wic i tur ca be used to sow 4. 5
6 A Appedix: U-statistics Let X... X be iid radom variables. Let H be a symmetric fuctio i.e. H : R R H a b Hb ad H is allowed to cage wit. A secod-order U-statistic is defied as U H X i X j i<j!!! deotes te umber of distict combiatios of two elemets out of elemets. Note tat a secod-order U-statistic is costructed by cosiderig all suc combiatios computig H X i X j for eac combiatio ad averagig over te combiatios sice te sum is take over i < j. Note also tat te order of X i ad X j is uimportat as H is symmetric. We are iterested i limitig teorems WLLNs ad CLTs for U-statistics. Te key step is trasformig a U-statistic ito averages of te form i g X i for some fuctio g : R R after wic usual WLLNs ad CLTs ca be applied. Suc a trasformatio is kow as Hájek s projectio. Te fuctio g x is costructed as g x E H X i X j X i x EH x X j EH X i x te secod equality olds by idepedece of X i ad X j ad te tird equality olds by te symmetry of H. Moreover µ Eg X i EE H X i X j X i EH X i X j. Projectig te U-statistic U i o observatio X i produces: E U X i E H X k X l X i. k<l Cosider E H X k X l X i for some fixed i. We i k ad i l E H X l X k X i EH X k X l µ. Te umber of suc terms is. 3 Te remaiig terms cotai X i ad terefore satisfy E H X l X k X i g X i. Hece E U X i µ + g X i µ + g X i µ + g X i µ. 5 6
7 Te projectio Û of U is defied as Û E U X i EU i µ + g X i µ. 6 i Note tat EÛ EU is acieved by subtractio of EU. Te scalig by i te secod term esures tat V arû/v aru as we will see from te followig lemma. Lemma. Suppose tat tere is c > 0 suc tat ξ V arg X i > c ξ ad ξ V arh X X are fiite ad as Te V arû/v aru. ξ ξ 0. Remark. Te coditio V arg X i > 0 rules out degeeracy of Û. A U-statistic satisfyig suc a coditio is called o-degeerate. Proof. First V arû 4 V arg X i 4 ξ. 7 For te U-statistic V aru i<j k<l Cov H X i X j H X k X l. Te covariace terms are zero we i j k l are all differet by idepedece. Tus a covariace term ca be o-zero we i te two variables i H X i X j coicide wit te two variables i H X k X l i.e. X i X k ad X j X l or ii H X i X j ad H X k X l ave oly oe variable i commo. Te cotributio from i to te variace of U is V arh X X V arh X X. To fid te cotributio from ii for eac i j select k i wic leaves coice for l j. Hece Cov H X i X j H X k X l E H X i X j µ H X i X l µ k<l EH X i X j H X i X l µ for some l j. Tus te cotributio from ii to te variace of U is 4 EH X i X j H X i X l µ. 7
8 Next EH X i X j H X i X l µ EE H X i X j H X i X l X i µ E E H X i X j X i E H X i X l X i µ Eg X i µ V arg X i. We ave ad terefore V aru V arh X X + 4 V arg X i ξ + 4 ξ V arû V aru 4ξ / ξ / + 4ξ / ξ / ξ + o +. Te asymptotic equivalece of te variaces of te U-statistic ad its projectio togeter wit equality of teir meas implies te asymptotic equivalece of U ad Û. Lemma 3. Suppose tat V arû/v aru. Te R U EU V aru Û EÛ V ar Û o p. Proof. By costructio ER 0. Tus if we sow tat V arr 0 te result follows by Cebysev s Markov s iequality. CovU V arr Û V aru V ar Û. 8 Cosider E U Û Û µ i i E U Û g X i EE U Û X i g X i 0 9 8
9 te last equality follows by 5 ad sice by costructio Te result i 9 implies tat EÛ X i µ + g X i µ. 0 EU Û µ EÛÛ µ CovU Û V arû. 0 It follows ow from 8 ad 0 tat V arû V arr V aru 0. We ca ow state te desired projectio result wic we ca be used to establis LLNs ad CLTs for te U-statistic U. Teorem 4 Projectio of o-degeerate secod-order U-statistics. Suppose tat te coditios of Lemma old. Te / U EU / g X i Eg X i + o p. i Moreover if ξ < d for some d > 0 ad all large U EU + O p /. Proof. Te first result follows from Lemmas ad 3 sice te result of Lemma 3 ca be re-writte as / U EU ξ + 4ξ / Û EÛ 4ξ + o p. Te secod result follows from te first by Cebysev s iequality ad sice V ar / g X i Eg X i ξ < d for all large. i 9
Kernel density estimator
Jauary, 07 NONPARAMETRIC ERNEL DENSITY ESTIMATION I this lecture, we discuss kerel estimatio of probability desity fuctios PDF Noparametric desity estimatio is oe of the cetral problems i statistics I
4 Conditional Distribution Estimation
4 Coditioal Distributio Estimatio 4. Estimators Te coditioal distributio (CDF) of y i give X i = x is F (y j x) = P (y i y j X i = x) = E ( (y i y) j X i = x) : Tis is te coditioal mea of te radom variable
Notes On Nonparametric Density Estimation. James L. Powell Department of Economics University of California, Berkeley
Notes O Noparametric Desity Estimatio James L Powell Departmet of Ecoomics Uiversity of Califoria, Berkeley Uivariate Desity Estimatio via Numerical Derivatives Cosider te problem of estimatig te desity
Convergence of random variables. (telegram style notes) P.J.C. Spreij
Covergece of radom variables (telegram style otes).j.c. Spreij this versio: September 6, 2005 Itroductio As we kow, radom variables are by defiitio measurable fuctios o some uderlyig measurable space
MASSACHUSETTS INSTITUTE OF TECHNOLOGY 6.436J/15.085J Fall 2008 Lecture 19 11/17/2008 LAWS OF LARGE NUMBERS II THE STRONG LAW OF LARGE NUMBERS
MASSACHUSTTS INSTITUT OF TCHNOLOGY 6.436J/5.085J Fall 2008 Lecture 9 /7/2008 LAWS OF LARG NUMBRS II Cotets. The strog law of large umbers 2. The Cheroff boud TH STRONG LAW OF LARG NUMBRS While the weak
7.1 Convergence of sequences of random variables
Chapter 7 Limit Theorems Throughout this sectio we will assume a probability space (, F, P), i which is defied a ifiite sequece of radom variables (X ) ad a radom variable X. The fact that for every ifiite
EECS564 Estimation, Filtering, and Detection Hwk 2 Solns. Winter p θ (z) = (2θz + 1 θ), 0 z 1
EECS564 Estimatio, Filterig, ad Detectio Hwk 2 Sols. Witer 25 4. Let Z be a sigle observatio havig desity fuctio where. p (z) = (2z + ), z (a) Assumig that is a oradom parameter, fid ad plot the maximum
Lecture 7: Density Estimation: k-nearest Neighbor and Basis Approach
STAT 425: Itroductio to Noparametric Statistics Witer 28 Lecture 7: Desity Estimatio: k-nearest Neighbor ad Basis Approach Istructor: Ye-Chi Che Referece: Sectio 8.4 of All of Noparametric Statistics.
Lecture 15: Density estimation
Lecture 15: Desity estimatio Why do we estimate a desity? Suppose that X 1,...,X are i.i.d. radom variables from F ad that F is ukow but has a Lebesgue p.d.f. f. Estimatio of F ca be doe by estimatig f.
10/ Statistical Machine Learning Homework #1 Solutions
Caregie Mello Uiversity Departet of Statistics & Data Sciece 0/36-70 Statistical Macie Learig Hoework # Solutios Proble [40 pts.] DUE: February, 08 Let X,..., X P were X i [0, ] ad P as desity p. Let p
Study the bias (due to the nite dimensional approximation) and variance of the estimators
2 Series Methods 2. Geeral Approach A model has parameters (; ) where is ite-dimesioal ad is oparametric. (Sometimes, there is o :) We will focus o regressio. The fuctio is approximated by a series a ite
Lecture 19: Convergence
Lecture 19: Covergece Asymptotic approach I statistical aalysis or iferece, a key to the success of fidig a good procedure is beig able to fid some momets ad/or distributios of various statistics. I may
Nonparametric regression: minimax upper and lower bounds
Capter 4 Noparametric regressio: miimax upper ad lower bouds 4. Itroductio We cosider oe of te two te most classical o-parametric problems i tis example: estimatig a regressio fuctio o a subset of te real
Lecture 7: Properties of Random Samples
Lecture 7: Properties of Radom Samples 1 Cotiued From Last Class Theorem 1.1. Let X 1, X,...X be a radom sample from a populatio with mea µ ad variace σ
Lecture 9: Regression: Regressogram and Kernel Regression
STAT 425: Itroductio to Noparametric Statistics Witer 208 Lecture 9: Regressio: Regressogram ad erel Regressio Istructor: Ye-Ci Ce Referece: Capter 5 of All of oparametric statistics 9 Itroductio Let X,
Let us give one more example of MLE. Example 3. The uniform distribution U[0, θ] on the interval [0, θ] has p.d.f.
![Let us give one more example of MLE. Example 3. The uniform distribution U[0, θ] on the interval [0, θ] has p.d.f.](/thumbs/91/105380228.jpg "Let us give one more example of MLE. Example 3. The uniform distribution U[0, θ] on the interval [0, θ] has p.d.f.")
Lecture 5 Let us give oe more example of MLE. Example 3. The uiform distributio U[0, ] o the iterval [0, ] has p.d.f. { 1 f(x =, 0 x, 0, otherwise The likelihood fuctio ϕ( = f(x i = 1 I(X 1,..., X [0,
INFINITE SEQUENCES AND SERIES
11 INFINITE SEQUENCES AND SERIES INFINITE SEQUENCES AND SERIES 11.4 The Compariso Tests I this sectio, we will lear: How to fid the value of a series by comparig it with a kow series. COMPARISON TESTS
Point Estimation: properties of estimators 1 FINITE-SAMPLE PROPERTIES. finite-sample properties (CB 7.3) large-sample properties (CB 10.
Poit Estimatio: properties of estimators fiite-sample properties CB 7.3) large-sample properties CB 10.1) 1 FINITE-SAMPLE PROPERTIES How a estimator performs for fiite umber of observatios. Estimator:
Exponential Families and Bayesian Inference
Computer Visio Expoetial Families ad Bayesia Iferece Lecture Expoetial Families A expoetial family of distributios is a d-parameter family f(x; havig the followig form: f(x; = h(xe g(t T (x B(, (. where
Since X n /n P p, we know that X n (n. Xn (n X n ) Using the asymptotic result above to obtain an approximation for fixed n, we obtain
Assigmet 9 Exercise 5.5 Let X biomial, p, where p 0, 1 is ukow. Obtai cofidece itervals for p i two differet ways: a Sice X / p d N0, p1 p], the variace of the limitig distributio depeds oly o p. Use the
1 The Haar functions and the Brownian motion
1 The Haar fuctios ad the Browia motio 1.1 The Haar fuctios ad their completeess The Haar fuctios The basic Haar fuctio is 1 if x < 1/2, ψx) = 1 if 1/2 x < 1, otherwise. 1.1) It has mea zero 1 ψx)dx =,
Chapter 3. Strong convergence. 3.1 Definition of almost sure convergence
Chapter 3 Strog covergece As poited out i the Chapter 2, there are multiple ways to defie the otio of covergece of a sequece of radom variables. That chapter defied covergece i probability, covergece i
Linear regression. Daniel Hsu (COMS 4771) (y i x T i β)2 2πσ. 2 2σ 2. 1 n. (x T i β y i ) 2. 1 ˆβ arg min. β R n d
Liear regressio Daiel Hsu (COMS 477) Maximum likelihood estimatio Oe of the simplest liear regressio models is the followig: (X, Y ),..., (X, Y ), (X, Y ) are iid radom pairs takig values i R d R, ad Y
CONCENTRATION INEQUALITIES
CONCENTRATION INEQUALITIES MAXIM RAGINSKY I te previous lecture, te followig result was stated witout proof. If X 1,..., X are idepedet Beroulliθ radom variables represetig te outcomes of a sequece of
Lecture 13: Maximum Likelihood Estimation
ECE90 Sprig 007 Statistical Learig Theory Istructor: R. Nowak Lecture 3: Maximum Likelihood Estimatio Summary of Lecture I the last lecture we derived a risk (MSE) boud for regressio problems; i.e., select
7.1 Convergence of sequences of random variables
Chapter 7 Limit theorems Throughout this sectio we will assume a probability space (Ω, F, P), i which is defied a ifiite sequece of radom variables (X ) ad a radom variable X. The fact that for every ifiite
Topic 9: Sampling Distributions of Estimators
Topic 9: Samplig Distributios of Estimators Course 003, 2016 Page 0 Samplig distributios of estimators Sice our estimators are statistics (particular fuctios of radom variables), their distributio ca be
Unbiased Estimation. February 7-12, 2008
Ubiased Estimatio February 7-2, 2008 We begi with a sample X = (X,..., X ) of radom variables chose accordig to oe of a family of probabilities P θ where θ is elemet from the parameter space Θ. For radom
Chapter 6 Infinite Series
Chapter 6 Ifiite Series I the previous chapter we cosidered itegrals which were improper i the sese that the iterval of itegratio was ubouded. I this chapter we are goig to discuss a topic which is somewhat
LECTURE 8: ASYMPTOTICS I
LECTURE 8: ASYMPTOTICS I We are iterested i the properties of estimators as. Cosider a sequece of radom variables {, X 1}. N. M. Kiefer, Corell Uiversity, Ecoomics 60 1 Defiitio: (Weak covergece) A sequece
Lecture 18: Sampling distributions
Lecture 18: Samplig distributios I may applicatios, the populatio is oe or several ormal distributios (or approximately). We ow study properties of some importat statistics based o a radom sample from
ST5215: Advanced Statistical Theory
ST525: Advaced Statistical Theory Departmet of Statistics & Applied Probability Tuesday, September 7, 2 ST525: Advaced Statistical Theory Lecture : The law of large umbers The Law of Large Numbers The
ON LOCAL LINEAR ESTIMATION IN NONPARAMETRIC ERRORS-IN-VARIABLES MODELS 1
Teory of Stocastic Processes Vol2 28, o3-4, 2006, pp*-* SILVELYN ZWANZIG ON LOCAL LINEAR ESTIMATION IN NONPARAMETRIC ERRORS-IN-VARIABLES MODELS Local liear metods are applied to a oparametric regressio
Statistical Theory MT 2008 Problems 1: Solution sketches
Statistical Theory MT 008 Problems : Solutio sketches. Which of the followig desities are withi a expoetial family? Explai your reasoig. a) Let 0 < θ < ad put fx, θ) = θ)θ x ; x = 0,,,... b) c) where α
Lecture 33: Bootstrap
Lecture 33: ootstrap Motivatio To evaluate ad compare differet estimators, we eed cosistet estimators of variaces or asymptotic variaces of estimators. This is also importat for hypothesis testig ad cofidece
ECE 901 Lecture 14: Maximum Likelihood Estimation and Complexity Regularization
ECE 90 Lecture 4: Maximum Likelihood Estimatio ad Complexity Regularizatio R Nowak 5/7/009 Review : Maximum Likelihood Estimatio We have iid observatios draw from a ukow distributio Y i iid p θ, i,, where
Lecture 7: October 18, 2017
Iformatio ad Codig Theory Autum 207 Lecturer: Madhur Tulsiai Lecture 7: October 8, 207 Biary hypothesis testig I this lecture, we apply the tools developed i the past few lectures to uderstad the problem
Statistical Theory MT 2009 Problems 1: Solution sketches
Statistical Theory MT 009 Problems : Solutio sketches. Which of the followig desities are withi a expoetial family? Explai your reasoig. (a) Let 0 < θ < ad put f(x, θ) = ( θ)θ x ; x = 0,,,... (b) (c) where
Week 6. Intuition: Let p (x, y) be the joint density of (X, Y ) and p (x) be the marginal density of X. Then. h dy =
Week 6 Lecture Kerel regressio ad miimax rates Model: Observe Y i = f (X i ) + ξ i were ξ i, i =,,,, iid wit Eξ i = 0 We ofte assume X i are iid or X i = i/ Nadaraya Watso estimator Let w i (x) = K ( X
Math 341 Lecture #31 6.5: Power Series
Math 341 Lecture #31 6.5: Power Series We ow tur our attetio to a particular kid of series of fuctios, amely, power series, f(x = a x = a 0 + a 1 x + a 2 x 2 + where a R for all N. I terms of a series
University of Colorado Denver Dept. Math. & Stat. Sciences Applied Analysis Preliminary Exam 13 January 2012, 10:00 am 2:00 pm. Good luck!
Uiversity of Colorado Dever Dept. Math. & Stat. Scieces Applied Aalysis Prelimiary Exam 13 Jauary 01, 10:00 am :00 pm Name: The proctor will let you read the followig coditios before the exam begis, ad
Lecture 8: Convergence of transformations and law of large numbers
Lecture 8: Covergece of trasformatios ad law of large umbers Trasformatio ad covergece Trasformatio is a importat tool i statistics. If X coverges to X i some sese, we ofte eed to check whether g(x ) coverges
It is often useful to approximate complicated functions using simpler ones. We consider the task of approximating a function by a polynomial.
Taylor Polyomials ad Taylor Series It is ofte useful to approximate complicated fuctios usig simpler oes We cosider the task of approximatig a fuctio by a polyomial If f is at least -times differetiable
Fall 2013 MTH431/531 Real analysis Section Notes
Fall 013 MTH431/531 Real aalysis Sectio 8.1-8. Notes Yi Su 013.11.1 1. Defiitio of uiform covergece. We look at a sequece of fuctios f (x) ad study the coverget property. Notice we have two parameters
MASSACHUSETTS INSTITUTE OF TECHNOLOGY 6.265/15.070J Fall 2013 Lecture 6 9/23/2013. Brownian motion. Introduction
MASSACHUSETTS INSTITUTE OF TECHNOLOGY 6.265/5.070J Fall 203 Lecture 6 9/23/203 Browia motio. Itroductio Cotet.. A heuristic costructio of a Browia motio from a radom walk. 2. Defiitio ad basic properties
MATH4822E FOURIER ANALYSIS AND ITS APPLICATIONS
MATH48E FOURIER ANALYSIS AND ITS APPLICATIONS 7.. Cesàro summability. 7. Summability methods Arithmetic meas. The followig idea is due to the Italia geometer Eresto Cesàro (859-96). He shows that eve if
Direction: This test is worth 250 points. You are required to complete this test within 50 minutes.
Term Test October 3, 003 Name Math 56 Studet Number Directio: This test is worth 50 poits. You are required to complete this test withi 50 miutes. I order to receive full credit, aswer each problem completely
6. Uniform distribution mod 1
6. Uiform distributio mod 1 6.1 Uiform distributio ad Weyl s criterio Let x be a seuece of real umbers. We may decompose x as the sum of its iteger part [x ] = sup{m Z m x } (i.e. the largest iteger which
Lecture 3 The Lebesgue Integral
Lecture 3: The Lebesgue Itegral 1 of 14 Course: Theory of Probability I Term: Fall 2013 Istructor: Gorda Zitkovic Lecture 3 The Lebesgue Itegral The costructio of the itegral Uless expressly specified
Chapter 6 Principles of Data Reduction
Chapter 6 for BST 695: Special Topics i Statistical Theory. Kui Zhag, 0 Chapter 6 Priciples of Data Reductio Sectio 6. Itroductio Goal: To summarize or reduce the data X, X,, X to get iformatio about a
Advanced Stochastic Processes.
Advaced Stochastic Processes. David Gamarik LECTURE 2 Radom variables ad measurable fuctios. Strog Law of Large Numbers (SLLN). Scary stuff cotiued... Outlie of Lecture Radom variables ad measurable fuctios.
Math 216A Notes, Week 5
Math 6A Notes, Week 5 Scribe: Ayastassia Sebolt Disclaimer: These otes are ot early as polished (ad quite possibly ot early as correct) as a published paper. Please use them at your ow risk.. Thresholds
Uniformly Consistency of the Cauchy-Transformation Kernel Density Estimation Underlying Strong Mixing
Appl. Mat. If. Sci. 7, No. L, 5-9 (203) 5 Applied Matematics & Iformatio Scieces A Iteratioal Joural c 203 NSP Natural Scieces Publisig Cor. Uiformly Cosistecy of te Caucy-Trasformatio Kerel Desity Estimatio
Sequences and Series of Functions
Chapter 6 Sequeces ad Series of Fuctios 6.1. Covergece of a Sequece of Fuctios Poitwise Covergece. Defiitio 6.1. Let, for each N, fuctio f : A R be defied. If, for each x A, the sequece (f (x)) coverges
Lecture 3: August 31
36-705: Itermediate Statistics Fall 018 Lecturer: Siva Balakrisha Lecture 3: August 31 This lecture will be mostly a summary of other useful expoetial tail bouds We will ot prove ay of these i lecture,
ALLOCATING SAMPLE TO STRATA PROPORTIONAL TO AGGREGATE MEASURE OF SIZE WITH BOTH UPPER AND LOWER BOUNDS ON THE NUMBER OF UNITS IN EACH STRATUM
ALLOCATING SAPLE TO STRATA PROPORTIONAL TO AGGREGATE EASURE OF SIZE WIT BOT UPPER AND LOWER BOUNDS ON TE NUBER OF UNITS IN EAC STRATU Lawrece R. Erst ad Cristoper J. Guciardo Erst_L@bls.gov, Guciardo_C@bls.gov
First Year Quantitative Comp Exam Spring, Part I - 203A. f X (x) = 0 otherwise
First Year Quatitative Comp Exam Sprig, 2012 Istructio: There are three parts. Aswer every questio i every part. Questio I-1 Part I - 203A A radom variable X is distributed with the margial desity: >
Lecture 10 October Minimaxity and least favorable prior sequences
STATS 300A: Theory of Statistics Fall 205 Lecture 0 October 22 Lecturer: Lester Mackey Scribe: Brya He, Rahul Makhijai Warig: These otes may cotai factual ad/or typographic errors. 0. Miimaxity ad least
Problem Set 4 Due Oct, 12
EE226: Radom Processes i Systems Lecturer: Jea C. Walrad Problem Set 4 Due Oct, 12 Fall 06 GSI: Assae Gueye This problem set essetially reviews detectio theory ad hypothesis testig ad some basic otios
ECE 901 Lecture 12: Complexity Regularization and the Squared Loss
ECE 90 Lecture : Complexity Regularizatio ad the Squared Loss R. Nowak 5/7/009 I the previous lectures we made use of the Cheroff/Hoeffdig bouds for our aalysis of classifier errors. Hoeffdig s iequality
Stat410 Probability and Statistics II (F16)
Some Basic Cocepts of Statistical Iferece (Sec 5.) Suppose we have a rv X that has a pdf/pmf deoted by f(x; θ) or p(x; θ), where θ is called the parameter. I previous lectures, we focus o probability problems
Regression with an Evaporating Logarithmic Trend
Regressio with a Evaporatig Logarithmic Tred Peter C. B. Phillips Cowles Foudatio, Yale Uiversity, Uiversity of Aucklad & Uiversity of York ad Yixiao Su Departmet of Ecoomics Yale Uiversity October 5,
Lecture 7 Testing Nonlinear Inequality Restrictions 1
Eco 75 Lecture 7 Testig Noliear Iequality Restrictios I Lecture 6, we discussed te testig problems were te ull ypotesis is de ed by oliear equality restrictios: H : ( ) = versus H : ( ) 6= : () We sowed
Lecture 3. Properties of Summary Statistics: Sampling Distribution
Lecture 3 Properties of Summary Statistics: Samplig Distributio Mai Theme How ca we use math to justify that our umerical summaries from the sample are good summaries of the populatio? Lecture Summary
Ma 4121: Introduction to Lebesgue Integration Solutions to Homework Assignment 5
Ma 42: Itroductio to Lebesgue Itegratio Solutios to Homework Assigmet 5 Prof. Wickerhauser Due Thursday, April th, 23 Please retur your solutios to the istructor by the ed of class o the due date. You
A Proof of Birkhoff s Ergodic Theorem
A Proof of Birkhoff s Ergodic Theorem Joseph Hora September 2, 205 Itroductio I Fall 203, I was learig the basics of ergodic theory, ad I came across this theorem. Oe of my supervisors, Athoy Quas, showed
Polynomials with Rational Roots that Differ by a Non-zero Constant. Generalities
Polyomials with Ratioal Roots that Differ by a No-zero Costat Philip Gibbs The problem of fidig two polyomials P(x) ad Q(x) of a give degree i a sigle variable x that have all ratioal roots ad differ by
Solutions to HW Assignment 1
Solutios to HW: 1 Course: Theory of Probability II Page: 1 of 6 Uiversity of Texas at Austi Solutios to HW Assigmet 1 Problem 1.1. Let Ω, F, {F } 0, P) be a filtered probability space ad T a stoppig time.
Semiparametric Mixtures of Nonparametric Regressions
Semiparametric Mixtures of Noparametric Regressios Sijia Xiag, Weixi Yao Proofs I tis sectio, te coditios required by Teorems, 2, 3 ad 4 are listed. Tey are ot te weakest sufficiet coditios, but could
Application to Random Graphs
A Applicatio to Radom Graphs Brachig processes have a umber of iterestig ad importat applicatios. We shall cosider oe of the most famous of them, the Erdős-Réyi radom graph theory. 1 Defiitio A.1. Let
Linear Support Vector Machines
Liear Support Vector Machies David S. Roseberg The Support Vector Machie For a liear support vector machie (SVM), we use the hypothesis space of affie fuctios F = { f(x) = w T x + b w R d, b R } ad evaluate
MASSACHUSETTS INSTITUTE OF TECHNOLOGY 6.265/15.070J Fall 2013 Lecture 21 11/27/2013
MASSACHUSETTS INSTITUTE OF TECHNOLOGY 6.265/15.070J Fall 2013 Lecture 21 11/27/2013 Fuctioal Law of Large Numbers. Costructio of the Wieer Measure Cotet. 1. Additioal techical results o weak covergece
Lecture 11 October 27
STATS 300A: Theory of Statistics Fall 205 Lecture October 27 Lecturer: Lester Mackey Scribe: Viswajith Veugopal, Vivek Bagaria, Steve Yadlowsky Warig: These otes may cotai factual ad/or typographic errors..
Probability and Statistics
ICME Refresher Course: robability ad Statistics Staford Uiversity robability ad Statistics Luyag Che September 20, 2016 1 Basic robability Theory 11 robability Spaces A probability space is a triple (Ω,
LIMITS AND DERIVATIVES
Capter LIMITS AND DERIVATIVES. Overview.. Limits of a fuctio Let f be a fuctio defied i a domai wic we take to be a iterval, say, I. We sall study te cocept of it of f at a poit a i I. We say f ( ) is
MA Advanced Econometrics: Properties of Least Squares Estimators
MA Advaced Ecoometrics: Properties of Least Squares Estimators Karl Whela School of Ecoomics, UCD February 5, 20 Karl Whela UCD Least Squares Estimators February 5, 20 / 5 Part I Least Squares: Some Fiite-Sample
Sequences A sequence of numbers is a function whose domain is the positive integers. We can see that the sequence
Sequeces A sequece of umbers is a fuctio whose domai is the positive itegers. We ca see that the sequece 1, 1, 2, 2, 3, 3,... is a fuctio from the positive itegers whe we write the first sequece elemet
4. Partial Sums and the Central Limit Theorem
1 of 10 7/16/2009 6:05 AM Virtual Laboratories > 6. Radom Samples > 1 2 3 4 5 6 7 4. Partial Sums ad the Cetral Limit Theorem The cetral limit theorem ad the law of large umbers are the two fudametal theorems
Discrete Mathematics for CS Spring 2008 David Wagner Note 22
CS 70 Discrete Mathematics for CS Sprig 2008 David Wager Note 22 I.I.D. Radom Variables Estimatig the bias of a coi Questio: We wat to estimate the proportio p of Democrats i the US populatio, by takig
Probability 2 - Notes 10. Lemma. If X is a random variable and g(x) 0 for all x in the support of f X, then P(g(X) 1) E[g(X)].
![Probability 2 - Notes 10. Lemma. If X is a random variable and g(x) 0 for all x in the support of f X, then P(g(X) 1) E[g(X)].](/thumbs/90/101606445.jpg "Probability 2 - Notes 10. Lemma. If X is a random variable and g(x) 0 for all x in the support of f X, then P(g(X) 1) E[g(X)].")
Probability 2 - Notes 0 Some Useful Iequalities. Lemma. If X is a radom variable ad g(x 0 for all x i the support of f X, the P(g(X E[g(X]. Proof. (cotiuous case P(g(X Corollaries x:g(x f X (xdx x:g(x
Economics 241B Relation to Method of Moments and Maximum Likelihood OLSE as a Maximum Likelihood Estimator
Ecoomics 24B Relatio to Method of Momets ad Maximum Likelihood OLSE as a Maximum Likelihood Estimator Uder Assumptio 5 we have speci ed the distributio of the error, so we ca estimate the model parameters
Empirical Processes: Glivenko Cantelli Theorems
Empirical Processes: Gliveko Catelli Theorems Mouliath Baerjee Jue 6, 200 Gliveko Catelli classes of fuctios The reader is referred to Chapter.6 of Weller s Torgo otes, Chapter??? of VDVW ad Chapter 8.3
Maximum Likelihood Estimation and Complexity Regularization
ECE90 Sprig 004 Statistical Regularizatio ad Learig Theory Lecture: 4 Maximum Likelihood Estimatio ad Complexity Regularizatio Lecturer: Rob Nowak Scribe: Pam Limpiti Review : Maximum Likelihood Estimatio
Econ 325/327 Notes on Sample Mean, Sample Proportion, Central Limit Theorem, Chi-square Distribution, Student s t distribution 1.
Eco 325/327 Notes o Sample Mea, Sample Proportio, Cetral Limit Theorem, Chi-square Distributio, Studet s t distributio 1 Sample Mea By Hiro Kasahara We cosider a radom sample from a populatio. Defiitio
This section is optional.
4 Momet Geeratig Fuctios* This sectio is optioal. The momet geeratig fuctio g : R R of a radom variable X is defied as g(t) = E[e tx ]. Propositio 1. We have g () (0) = E[X ] for = 1, 2,... Proof. Therefore
A sequence of numbers is a function whose domain is the positive integers. We can see that the sequence
Sequeces A sequece of umbers is a fuctio whose domai is the positive itegers. We ca see that the sequece,, 2, 2, 3, 3,... is a fuctio from the positive itegers whe we write the first sequece elemet as
Resampling Methods. X (1/2), i.e., Pr (X i m) = 1/2. We order the data: X (1) X (2) X (n). Define the sample median: ( n.
Jauary 1, 2019 Resamplig Methods Motivatio We have so may estimators with the property θ θ d N 0, σ 2 We ca also write θ a N θ, σ 2 /, where a meas approximately distributed as Oce we have a cosistet estimator
Topic 9: Sampling Distributions of Estimators
Topic 9: Samplig Distributios of Estimators Course 003, 2018 Page 0 Samplig distributios of estimators Sice our estimators are statistics (particular fuctios of radom variables), their distributio ca be
Exercise 4.3 Use the Continuity Theorem to prove the Cramér-Wold Theorem, Theorem. (1) φ a X(1).
Assigmet 7 Exercise 4.3 Use the Cotiuity Theorem to prove the Cramér-Wold Theorem, Theorem 4.12. Hit: a X d a X implies that φ a X (1) φ a X(1). Sketch of solutio: As we poited out i class, the oly tricky
Notes 5 : More on the a.s. convergence of sums
Notes 5 : More o the a.s. covergece of sums Math 733-734: Theory of Probability Lecturer: Sebastie Roch Refereces: Dur0, Sectios.5; Wil9, Sectio 4.7, Shi96, Sectio IV.4, Dur0, Sectio.. Radom series. Three-series
Topic 9: Sampling Distributions of Estimators
Topic 9: Samplig Distributios of Estimators Course 003, 2018 Page 0 Samplig distributios of estimators Sice our estimators are statistics (particular fuctios of radom variables), their distributio ca be
ECE 330:541, Stochastic Signals and Systems Lecture Notes on Limit Theorems from Probability Fall 2002
ECE 330:541, Stochastic Sigals ad Systems Lecture Notes o Limit Theorems from robability Fall 00 I practice, there are two ways we ca costruct a ew sequece of radom variables from a old sequece of radom
IIT JAM Mathematical Statistics (MS) 2006 SECTION A
IIT JAM Mathematical Statistics (MS) 6 SECTION A. If a > for ad lim a / L >, the which of the followig series is ot coverget? (a) (b) (c) (d) (d) = = a = a = a a + / a lim a a / + = lim a / a / + = lim
ECONOMETRIC THEORY. MODULE XIII Lecture - 34 Asymptotic Theory and Stochastic Regressors
ECONOMETRIC THEORY MODULE XIII Lecture - 34 Asymptotic Theory ad Stochastic Regressors Dr. Shalabh Departmet of Mathematics ad Statistics Idia Istitute of Techology Kapur Asymptotic theory The asymptotic
LIMITS AND DERIVATIVES NCERT
. Overview.. Limits of a fuctio Let f be a fuctio defied i a domai wic we take to be a iterval, say, I. We sall study te cocept of it of f at a poit a i I. We say f ( ) is te epected value of f at a give
January 25, 2017 INTRODUCTION TO MATHEMATICAL STATISTICS
Jauary 25, 207 INTRODUCTION TO MATHEMATICAL STATISTICS Abstract. A basic itroductio to statistics assumig kowledge of probability theory.. Probability I a typical udergraduate problem i probability, we
Lecture 2: Monte Carlo Simulation
STAT/Q SCI 43: Itroductio to Resamplig ethods Sprig 27 Istructor: Ye-Chi Che Lecture 2: ote Carlo Simulatio 2 ote Carlo Itegratio Assume we wat to evaluate the followig itegratio: e x3 dx What ca we do?
Expectation and Variance of a random variable
Chapter 11 Expectatio ad Variace of a radom variable The aim of this lecture is to defie ad itroduce mathematical Expectatio ad variace of a fuctio of discrete & cotiuous radom variables ad the distributio
CS434a/541a: Pattern Recognition Prof. Olga Veksler. Lecture 5
CS434a/54a: Patter Recogitio Prof. Olga Veksler Lecture 5 Today Itroductio to parameter estimatio Two methods for parameter estimatio Maimum Likelihood Estimatio Bayesia Estimatio Itroducto Bayesia Decisio