Regression with an Evaporating Logarithmic Trend

Save this PDF as:
 WORD  PNG  TXT  JPG

Size: px
Start display at page:

Download "Regression with an Evaporating Logarithmic Trend"

Transcription

1 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, 00 Abstract Liear regressio o a itercept ad a evaporatig logarithmic tred is show to be asymptotically colliear to the secod order. Cosistecy results for least squares are give, rates of covergece are obtaied ad asymptotic ormality is established for short memory errors. JEL Classificatio: C Key words ad phrases: Asymptotic theory, asymptotic expasios, colliear regressors, liear process, liear regressio, logarithmic factors, logarithmic itegral. Phillips thaks the NSF for research support uder Grat Nos. SBR ad SES Su thaks the Cowles Foudatio for support uder a Cowles Prize Fellowship. The paper was typed by the authors i SW.5. Computig was doe i GAUSS.

2 . Problem Part A The time series X t is geerated by the model X t α + β l t + u t, t,..., where α ad β are ukow parameters whose least squares regressio estimates are deoted by α ad β, respectively. The error u t i is assumed to be iid 0, σ with fiite fourth momet.. Show that α ad β are strogly cosistet for α ad β as.. Fid the asymptotic distributio of α ad β. Part B Suppose that u t i is the liear process u t c i ε t i, i0 with j c j <, j0 ad where ε t is iid 0, σ with fiite fourth momet. Explai how you would modify your derivatios i Part A to allow for such a error process i the regressio model.

3 . Solutio Part A Let z t. The l t l t β β α α z t u t, u t β l t β. We start with β ad fid a asymptotic represetatio of the compoets / l t ad / l t that appear i z t. Usig Euler summatio which justifies 3 below ad partial itegratio which justifies 4 below we obtai a asymptotic series represetatio as follows: l t m dx + O l x 3 k! l k k! l k + m! l m+ dx + O x 4 k m k k! l k k + m! l m+ dx + O. 5 x To show that 5 is a valid asymptotic series, we ca igore the O term ad by a further applicatio of partial itegratio to the secod term we see that the remaider is proportioal to R m+ l m+ x dx Let L α for some α 0, ad write But L l m+ x dx L l m+ x dx l L L It follows from 6-8 that l m+ x dx + L l m+ l m+ + m + l m+ x dx O α + L l m+ x dx l L l m+ dx. 6 x l m+ dx. 7 x l m+ x dx l L R m+, 8 R m+ l m+ + m + l m+ x dx l m+ + m + l L R m+ + O α 3 L l m+ + m + l m+ x dx + m + l L R m+

4 so that ad thus R m+ O l t m k showig that 5 is a valid asymptotic series. I a similar way we may establish that l t m k l m+, 9 k! l k + O l m+, 0 l dx + O x k! l k + O l m+ is a valid asymptotic series. Combiig 0 ad, we get l t l t l x dx l x dx + O 3 l + l l 4 + O l 5 4 { l + l + } l 3 + O l 4 l 4 + O l 5 a.s., 5 observig that we have to go to the third order terms i expasio 4 to avoid degeeracy. We ca obtai higher order asymptotic expasios by takig the above process to further terms, leadig to the followig explicit expressio to order / l 0 l l + 56 l l l l l 6 + O l. 6 It is immediately apparet from the size of the coefficiets i 6 that very large values of are required before these approximatios ca be expected to work well. Of course, such approximatios are hardly ecessary sice is ameable to direct calculatio or to direct approximatio usig the logarithmic itegrals give i 3 ad above. The latter ca be evaluated from the followig well kow series represetatio of the logarithmic itegral e.g., Gradshtey ad Ryzhik, 967, 8.3., p. 96 liy y 0 l x dx γ + l l y + 4 k l y k, for y >, 7 kk!

5 where γ is Euler s costat. I the preset case we have ad l x dx l dx li li, l x l + l x dx l + [li li ]. l Observe that β β is the same as the error i the OLS estimator of β i the regressio X t z t β + u t, where {u t } is a martigale differece sequece with respect to the atural filtratio F t. The persistet excitatio coditio 5 holds i this regressio ad so β a.s. β ad β is strogly cosistet. Now observe that α α u t β l t β o a.s. + o a.s. O a.s. l o a.s., ad so α a.s. α. Next, tur to the asymptotic distributio of α ad β. We have β β z t u t z t z t u t s l + o, s. Let y t z t u t /s, the y, y 3,..., y are idepedet radom variables with zero meas ad variaces that sum to σ. To apply the Liapouoff cetral limit theorem for y t, we eed to show that E y t 3 0, as. But ad so E y t 3 z t 3 E u t 3 y t s 3 Cost. l 4 3/ o, z t u t s N0, σ. 5

6 Hece Note that l β β z t u t s + o p N0, σ. 8 α α u t β l t β O p / O p / l l O p / l. Thus, the term β β l t domiates the asymptotics of α ad we deduce that l α α l β β + o p d N0, σ. 9 I view of 8 ad 9, we have the followig joit asymptotics: / l α α / l β β N0, Σ with Σ σ. 0 Remarks. The limit distributio 0 is sigular ad the compoets / l α α ad / l β β are perfectly egatively correlated as. Observe that the rate of covergece of α exceeds that of β, by virtue of the fact that the sigal from the itercept is stroger tha the sigal from the evaporatig logarithmic regressor l t.. Result 0 gives first order asymptotics. As is apparet from 6, asymptotic series expasios i the preset model ivolve factors of i cotrast to the l more usual ad, correspodigly, they deliver oly very slow improvemets i the first order asymptotics. The asymptotic variace σ l 4 ad higher order approximatios based o 6 are therefore poor approximatios to the variace of β eve for quite large. To illustrate, for values of the sample size [0, 0 4 ] ad for σ, Fig. provides graphs of the exact variace of β, the asymptotic variace, the two-term ad three-term series approximatios to the variace based o the first few terms of 6, ad direct calculatio of the logarithmic itegral represetatios of the variace obtaied from 3 ad 7. Apparetly, oly the latter are adequate for sample sizes i this rage. 3. The theory developed here is part of a geeral theory of regressio o slowly varyig regressors, a subject that has recetly bee studied i Phillips

7 Figure : σ / t z t ad Asymptotic ad Itegral Approximatios 7

8 Part B Usig the Phillips-Solo 99 device, we have u t CLε t Cε t + ε t ε t, where ε t CLε t, CL u0 C u L u, C u su+ c s. The, we ca write It follows that z t u t z t Cε t + z t ε t z t ε t C z t ε t + z t+ z t ε t + z ε z ε. β β C z t ε t z t + z t+ z t ε t + z ε z ε. Note that the first term satisfies the limit theory give i the earlier part. If we prove that the last three terms coverge strogly to zero, the we are doe with the strog cosistecy of β. To obtai the asymptotic distributio, we eed to cosider / β β. We prove that the last three terms, so ormalized, are o p, ad the the asymptotic distributio is determied by the first term. The followig three results are give first. a z ε z t o a.s ad z ε z t / o p. b Proof. These are immediate sice z t ad z t /. z ε z t o a.s ad z ε z t / o p. Proof. Note that z is bouded ad for ay δ > 0 z ε P > δ z E ε δ O, so z P ε > δ < ad the first result follows. The secod follows z t z because P ε > δ O z t / 0. 8

9 c z t+ z t ε t z t o a.s ad z t+ z t ε t z t / o p Proof. Note that { ε t } has fiite fourth momet because ε t 4 : E ε 4 t /4 t j0 t C t j ε j 4 C t j ε j 4 <. j0 Here we have employed the fact that t j0 C t j <, which follows from the assumptio that j0 j c j <. Next, E ε t E ε 4 t + ε s ε t s, s<t Therefore, for ay δ > 0, Sice P O + z t+ z t ε t z t > δ s, s<t E z t+ z t ε t 4 4 δ 4 E [ [ E ε 4 s ] / [ E ε 4 t ] / O. z t+ z t ε t ] 4 δ 4 z t+ z t O 4 δ 4 4 δ 4. E z t+ z t ε t z t+ z t l t + l t l + t [l t][l t + ] < Cost. t <, we have z t+ z t ε t P > δ O l 8 9

10 ad so z t+ z t ε t / o a.s. Similarly, z t+ z t ε t P / > δ E [ z t+ z t ε t ] z t δ E z t+ z t ε t + s>t z t+ z t z s+ z s E ε t ε s z t δ E z t+ z t ε t + s>t + z t z s+ z s E ε t / E ε s / δ Cost. z t+ z t z t δ l Cost. δ 0. The last equality follows as z t+ z t O /t Ol. Hece / z t+ z t ε t op. Combiig results a, b, ad c, we have β β o a.s., ad the / β β / C z t ε t + o p d N0, C σ. With, ad u t / a.s. 0 e.g. Phillips ad Solo, 99, the previous argumets ca ow be repeated to obtai the strog cosistecy ad the asymptotic distributio of α. Specifically / l α α / l β β 3. Refereces N0, Σ with Σ C σ Gradshtey I. S. ad I. M. Ryzhik 965. Tables of Itegrals, Series ad Products. New York: Academic Press. Phillips, P. C. B Regressio with Slowly Varyig Regressors, Cowles Foudatio Discussio Paper #30, Yale Uiversity. Phillips, P. C. B. ad V. Solo 99. Asymptotics for Liear Processes, Aals of Statistics 0,

INFINITE SEQUENCES AND SERIES

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

More information

Chapter 3. Strong convergence. 3.1 Definition of almost sure convergence

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

More information

First Year Quantitative Comp Exam Spring, Part I - 203A. f X (x) = 0 otherwise

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: >

More information

UNIT ROOT MODEL SELECTION PETER C. B. PHILLIPS COWLES FOUNDATION PAPER NO. 1231

UNIT ROOT MODEL SELECTION PETER C. B. PHILLIPS COWLES FOUNDATION PAPER NO. 1231 UNIT ROOT MODEL SELECTION BY PETER C. B. PHILLIPS COWLES FOUNDATION PAPER NO. 1231 COWLES FOUNDATION FOR RESEARCH IN ECONOMICS YALE UNIVERSITY Box 28281 New Have, Coecticut 652-8281 28 http://cowles.eco.yale.edu/

More information

LECTURE 11 LINEAR PROCESSES III: ASYMPTOTIC RESULTS

LECTURE 11 LINEAR PROCESSES III: ASYMPTOTIC RESULTS PRIL 7, 9 where LECTURE LINER PROCESSES III: SYMPTOTIC RESULTS (Phillips ad Solo (99) ad Phillips Lecture Notes o Statioary ad Nostatioary Time Series) I this lecture, we discuss the LLN ad CLT for a liear

More information

Asymptotic Results for the Linear Regression Model

Asymptotic Results for the Linear Regression Model Asymptotic Results for the Liear Regressio Model C. Fli November 29, 2000 1. Asymptotic Results uder Classical Assumptios The followig results apply to the liear regressio model y = Xβ + ε, where X is

More information

Mathematics 170B Selected HW Solutions.

Mathematics 170B Selected HW Solutions. Mathematics 17B Selected HW Solutios. F 4. Suppose X is B(,p). (a)fidthemometgeeratigfuctiom (s)of(x p)/ p(1 p). Write q = 1 p. The MGF of X is (pe s + q), sice X ca be writte as the sum of idepedet Beroulli

More information

Solutions to HW Assignment 1

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.

More information

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

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

More information

Singular Continuous Measures by Michael Pejic 5/14/10

Singular Continuous Measures by Michael Pejic 5/14/10 Sigular Cotiuous Measures by Michael Peic 5/4/0 Prelimiaries Give a set X, a σ-algebra o X is a collectio of subsets of X that cotais X ad ad is closed uder complemetatio ad coutable uios hece, coutable

More information

Løsningsførslag i 4M

Løsningsførslag i 4M Norges tekisk aturviteskapelige uiversitet Istitutt for matematiske fag Side 1 av 6 Løsigsførslag i 4M Oppgave 1 a) A sketch of the graph of the give f o the iterval [ 3, 3) is as follows: The Fourier

More information

Statistical Theory MT 2009 Problems 1: Solution sketches

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

More information

A note on self-normalized Dickey-Fuller test for unit root in autoregressive time series with GARCH errors

A note on self-normalized Dickey-Fuller test for unit root in autoregressive time series with GARCH errors Appl. Math. J. Chiese Uiv. 008, 3(): 97-0 A ote o self-ormalized Dickey-Fuller test for uit root i autoregressive time series with GARCH errors YANG Xiao-rog ZHANG Li-xi Abstract. I this article, the uit

More information

2.2. Central limit theorem.

2.2. Central limit theorem. 36.. Cetral limit theorem. The most ideal case of the CLT is that the radom variables are iid with fiite variace. Although it is a special case of the more geeral Lideberg-Feller CLT, it is most stadard

More information

A Risk Comparison of Ordinary Least Squares vs Ridge Regression

A Risk Comparison of Ordinary Least Squares vs Ridge Regression Joural of Machie Learig Research 14 (2013) 1505-1511 Submitted 5/12; Revised 3/13; Published 6/13 A Risk Compariso of Ordiary Least Squares vs Ridge Regressio Paramveer S. Dhillo Departmet of Computer

More information

Comparison Study of Series Approximation. and Convergence between Chebyshev. and Legendre Series

Comparison Study of Series Approximation. and Convergence between Chebyshev. and Legendre Series Applied Mathematical Scieces, Vol. 7, 03, o. 6, 3-337 HIKARI Ltd, www.m-hikari.com http://d.doi.org/0.988/ams.03.3430 Compariso Study of Series Approimatio ad Covergece betwee Chebyshev ad Legedre Series

More information

Solutions to Tutorial 3 (Week 4)

Solutions to Tutorial 3 (Week 4) The Uiversity of Sydey School of Mathematics ad Statistics Solutios to Tutorial Week 4 MATH2962: Real ad Complex Aalysis Advaced Semester 1, 2017 Web Page: http://www.maths.usyd.edu.au/u/ug/im/math2962/

More information

Lecture 10 October Minimaxity and least favorable prior sequences

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

More information

Solutions to Math 347 Practice Problems for the final

Solutions to Math 347 Practice Problems for the final Solutios to Math 347 Practice Problems for the fial 1) True or False: a) There exist itegers x,y such that 50x + 76y = 6. True: the gcd of 50 ad 76 is, ad 6 is a multiple of. b) The ifiimum of a set is

More information

Probability and Statistics

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,

More information

Dirichlet s Theorem on Arithmetic Progressions

Dirichlet s Theorem on Arithmetic Progressions Dirichlet s Theorem o Arithmetic Progressios Athoy Várilly Harvard Uiversity, Cambridge, MA 0238 Itroductio Dirichlet s theorem o arithmetic progressios is a gem of umber theory. A great part of its beauty

More information

Generalization of Samuelson s inequality and location of eigenvalues

Generalization of Samuelson s inequality and location of eigenvalues Proc. Idia Acad. Sci. Math. Sci.) Vol. 5, No., February 05, pp. 03. c Idia Academy of Scieces Geeralizatio of Samuelso s iequality ad locatio of eigevalues R SHARMA ad R SAINI Departmet of Mathematics,

More information

Basic Sets. Functions. MTH299 - Examples. Example 1. Let S = {1, {2, 3}, 4}. Indicate whether each statement is true or false. (a) S = 4. (e) 2 S.

Basic Sets. Functions. MTH299 - Examples. Example 1. Let S = {1, {2, 3}, 4}. Indicate whether each statement is true or false. (a) S = 4. (e) 2 S. Basic Sets Example 1. Let S = {1, {2, 3}, 4}. Idicate whether each statemet is true or false. (a) S = 4 (b) {1} S (c) {2, 3} S (d) {1, 4} S (e) 2 S. (f) S = {1, 4, {2, 3}} (g) S Example 2. Compute the

More information

The standard deviation of the mean

The standard deviation of the mean Physics 6C Fall 20 The stadard deviatio of the mea These otes provide some clarificatio o the distictio betwee the stadard deviatio ad the stadard deviatio of the mea.. The sample mea ad variace Cosider

More information

Math 2784 (or 2794W) University of Connecticut

Math 2784 (or 2794W) University of Connecticut ORDERS OF GROWTH PAT SMITH Math 2784 (or 2794W) Uiversity of Coecticut Date: Mar. 2, 22. ORDERS OF GROWTH. Itroductio Gaiig a ituitive feel for the relative growth of fuctios is importat if you really

More information

STAT331. Example of Martingale CLT with Cox s Model

STAT331. Example of Martingale CLT with Cox s Model STAT33 Example of Martigale CLT with Cox s Model I this uit we illustrate the Martigale Cetral Limit Theorem by applyig it to the partial likelihood score fuctio from Cox s model. For simplicity of presetatio

More information

Solutions to Tutorial 5 (Week 6)

Solutions to Tutorial 5 (Week 6) The Uiversity of Sydey School of Mathematics ad Statistics Solutios to Tutorial 5 (Wee 6 MATH2962: Real ad Complex Aalysis (Advaced Semester, 207 Web Page: http://www.maths.usyd.edu.au/u/ug/im/math2962/

More information

Discrete-Time Signals and Systems. Discrete-Time Signals and Systems. Signal Symmetry. Elementary Discrete-Time Signals.

Discrete-Time Signals and Systems. Discrete-Time Signals and Systems. Signal Symmetry. Elementary Discrete-Time Signals. Discrete-ime Sigals ad Systems Discrete-ime Sigals ad Systems Dr. Deepa Kudur Uiversity of oroto Referece: Sectios. -.5 of Joh G. Proakis ad Dimitris G. Maolakis, Digital Sigal Processig: Priciples, Algorithms,

More information

1 Covariance Estimation

1 Covariance Estimation Eco 75 Lecture 5 Covariace Estimatio ad Optimal Weightig Matrices I this lecture, we cosider estimatio of the asymptotic covariace matrix B B of the extremum estimator b : Covariace Estimatio Lemma 4.

More information

Shannon s noiseless coding theorem

Shannon s noiseless coding theorem 18.310 lecture otes May 4, 2015 Shao s oiseless codig theorem Lecturer: Michel Goemas I these otes we discuss Shao s oiseless codig theorem, which is oe of the foudig results of the field of iformatio

More information

Chapter 6 Principles of Data Reduction

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

More information

Lecture 3 The Lebesgue Integral

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

More information

Lecture 11 October 27

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..

More information

Gamma Distribution and Gamma Approximation

Gamma Distribution and Gamma Approximation Gamma Distributio ad Gamma Approimatio Xiaomig Zeg a Fuhua (Frak Cheg b a Xiame Uiversity, Xiame 365, Chia mzeg@jigia.mu.edu.c b Uiversity of Ketucky, Leigto, Ketucky 456-46, USA cheg@cs.uky.edu Abstract

More information

A RANK STATISTIC FOR NON-PARAMETRIC K-SAMPLE AND CHANGE POINT PROBLEMS

A RANK STATISTIC FOR NON-PARAMETRIC K-SAMPLE AND CHANGE POINT PROBLEMS J. Japa Statist. Soc. Vol. 41 No. 1 2011 67 73 A RANK STATISTIC FOR NON-PARAMETRIC K-SAMPLE AND CHANGE POINT PROBLEMS Yoichi Nishiyama* We cosider k-sample ad chage poit problems for idepedet data i a

More information

Solutions to Odd Numbered End of Chapter Exercises: Chapter 4

Solutions to Odd Numbered End of Chapter Exercises: Chapter 4 Itroductio to Ecoometrics (3 rd Updated Editio) by James H. Stock ad Mark W. Watso Solutios to Odd Numbered Ed of Chapter Exercises: Chapter 4 (This versio July 2, 24) Stock/Watso - Itroductio to Ecoometrics

More information

Section 11.8: Power Series

Section 11.8: Power Series Sectio 11.8: Power Series 1. Power Series I this sectio, we cosider geeralizig the cocept of a series. Recall that a series is a ifiite sum of umbers a. We ca talk about whether or ot it coverges ad i

More information

On matchings in hypergraphs

On matchings in hypergraphs O matchigs i hypergraphs Peter Frakl Tokyo, Japa peter.frakl@gmail.com Tomasz Luczak Adam Mickiewicz Uiversity Faculty of Mathematics ad CS Pozań, Polad ad Emory Uiversity Departmet of Mathematics ad CS

More information

62. Power series Definition 16. (Power series) Given a sequence {c n }, the series. c n x n = c 0 + c 1 x + c 2 x 2 + c 3 x 3 +

62. Power series Definition 16. (Power series) Given a sequence {c n }, the series. c n x n = c 0 + c 1 x + c 2 x 2 + c 3 x 3 + 62. Power series Defiitio 16. (Power series) Give a sequece {c }, the series c x = c 0 + c 1 x + c 2 x 2 + c 3 x 3 + is called a power series i the variable x. The umbers c are called the coefficiets of

More information

Sequences I. Chapter Introduction

Sequences I. Chapter Introduction Chapter 2 Sequeces I 2. Itroductio A sequece is a list of umbers i a defiite order so that we kow which umber is i the first place, which umber is i the secod place ad, for ay atural umber, we kow which

More information

OPERATOR PROBABILITY THEORY

OPERATOR PROBABILITY THEORY OPERATOR PROBABILITY THEORY Sta Gudder Departmet of Mathematics Uiversity of Dever Dever, Colorado 80208 sta.gudder@sm.du.edu Abstract This article presets a overview of some topics i operator probability

More information

MASSACHUSETTS INSTITUTE OF TECHNOLOGY 6.265/15.070J Fall 2013 Lecture 6 9/23/2013. Brownian motion. Introduction

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

More information

arxiv: v1 [math.pr] 13 Oct 2011

arxiv: v1 [math.pr] 13 Oct 2011 A tail iequality for quadratic forms of subgaussia radom vectors Daiel Hsu, Sham M. Kakade,, ad Tog Zhag 3 arxiv:0.84v math.pr] 3 Oct 0 Microsoft Research New Eglad Departmet of Statistics, Wharto School,

More information

Economics 102C: Advanced Topics in Econometrics 4 - Asymptotics & Large Sample Properties of OLS

Economics 102C: Advanced Topics in Econometrics 4 - Asymptotics & Large Sample Properties of OLS Ecoomics 102C: Advaced Topics i Ecoometrics 4 - Asymptotics & Large Sample Properties of OLS Michael Best Sprig 2015 Asymptotics So far we have looked at the fiite sample properties of OLS Relied heavily

More information

ON POINTWISE BINOMIAL APPROXIMATION

ON POINTWISE BINOMIAL APPROXIMATION Iteratioal Joural of Pure ad Applied Mathematics Volume 71 No. 1 2011, 57-66 ON POINTWISE BINOMIAL APPROXIMATION BY w-functions K. Teerapabolar 1, P. Wogkasem 2 Departmet of Mathematics Faculty of Sciece

More information

Statisticians use the word population to refer the total number of (potential) observations under consideration

Statisticians use the word population to refer the total number of (potential) observations under consideration 6 Samplig Distributios Statisticias use the word populatio to refer the total umber of (potetial) observatios uder cosideratio The populatio is just the set of all possible outcomes i our sample space

More information

An alternative proof of a theorem of Aldous. concerning convergence in distribution for martingales.

An alternative proof of a theorem of Aldous. concerning convergence in distribution for martingales. A alterative proof of a theorem of Aldous cocerig covergece i distributio for martigales. Maurizio Pratelli Dipartimeto di Matematica, Uiversità di Pisa. Via Buoarroti 2. I-56127 Pisa, Italy e-mail: pratelli@dm.uipi.it

More information

arxiv: v1 [math.fa] 3 Apr 2016

arxiv: v1 [math.fa] 3 Apr 2016 Aticommutator Norm Formula for Proectio Operators arxiv:164.699v1 math.fa] 3 Apr 16 Sam Walters Uiversity of Norther British Columbia ABSTRACT. We prove that for ay two proectio operators f, g o Hilbert

More information

Lecture 01: the Central Limit Theorem. 1 Central Limit Theorem for i.i.d. random variables

Lecture 01: the Central Limit Theorem. 1 Central Limit Theorem for i.i.d. random variables CSCI-B609: A Theorist s Toolkit, Fall 06 Aug 3 Lecture 0: the Cetral Limit Theorem Lecturer: Yua Zhou Scribe: Yua Xie & Yua Zhou Cetral Limit Theorem for iid radom variables Let us say that we wat to aalyze

More information

The variance of a sum of independent variables is the sum of their variances, since covariances are zero. Therefore. V (xi )= n n 2 σ2 = σ2.

The variance of a sum of independent variables is the sum of their variances, since covariances are zero. Therefore. V (xi )= n n 2 σ2 = σ2. SAMPLE STATISTICS A radom sample x 1,x,,x from a distributio f(x) is a set of idepedetly ad idetically variables with x i f(x) for all i Their joit pdf is f(x 1,x,,x )=f(x 1 )f(x ) f(x )= f(x i ) The sample

More information

MA131 - Analysis 1. Workbook 2 Sequences I

MA131 - Analysis 1. Workbook 2 Sequences I MA3 - Aalysis Workbook 2 Sequeces I Autum 203 Cotets 2 Sequeces I 2. Itroductio.............................. 2.2 Icreasig ad Decreasig Sequeces................ 2 2.3 Bouded Sequeces..........................

More information

The Ratio Test. THEOREM 9.17 Ratio Test Let a n be a series with nonzero terms. 1. a. n converges absolutely if lim. n 1

The Ratio Test. THEOREM 9.17 Ratio Test Let a n be a series with nonzero terms. 1. a. n converges absolutely if lim. n 1 460_0906.qxd //04 :8 PM Page 69 SECTION 9.6 The Ratio ad Root Tests 69 Sectio 9.6 EXPLORATION Writig a Series Oe of the followig coditios guaratees that a series will diverge, two coditios guaratee that

More information

Solution of Linear Constant-Coefficient Difference Equations

Solution of Linear Constant-Coefficient Difference Equations ECE 38-9 Solutio of Liear Costat-Coefficiet Differece Equatios Z. Aliyazicioglu Electrical ad Computer Egieerig Departmet Cal Poly Pomoa Solutio of Liear Costat-Coefficiet Differece Equatios Example: Determie

More information

The Gamma function Michael Taylor. Abstract. This material is excerpted from 18 and Appendix J of [T].

The Gamma function Michael Taylor. Abstract. This material is excerpted from 18 and Appendix J of [T]. The Gamma fuctio Michael Taylor Abstract. This material is excerpted from 8 ad Appedix J of [T]. The Gamma fuctio has bee previewed i 5.7 5.8, arisig i the computatio of a atural Laplace trasform: 8. ft

More information

PAijpam.eu ON TENSOR PRODUCT DECOMPOSITION

PAijpam.eu ON TENSOR PRODUCT DECOMPOSITION Iteratioal Joural of Pure ad Applied Mathematics Volume 103 No 3 2015, 537-545 ISSN: 1311-8080 (prited versio); ISSN: 1314-3395 (o-lie versio) url: http://wwwijpameu doi: http://dxdoiorg/1012732/ijpamv103i314

More information

11.6 Absolute Convergence and the Ratio and Root Tests

11.6 Absolute Convergence and the Ratio and Root Tests .6 Absolute Covergece ad the Ratio ad Root Tests The most commo way to test for covergece is to igore ay positive or egative sigs i a series, ad simply test the correspodig series of positive terms. Does

More information

A sequence of numbers is a function whose domain is the positive integers. We can see that the sequence

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

More information

IIT JAM Mathematical Statistics (MS) 2006 SECTION A

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

More information

Analysis of the Expected Number of Bit Comparisons Required by Quickselect

Analysis of the Expected Number of Bit Comparisons Required by Quickselect Aalysis of the Expected Number of Bit Comparisos Required by Quickselect James Alle Fill Takéhiko Nakama Abstract Whe algorithms for sortig ad searchig are applied to keys that are represeted as bit strigs,

More information

Parameter, Statistic and Random Samples

Parameter, Statistic and Random Samples Parameter, Statistic ad Radom Samples A parameter is a umber that describes the populatio. It is a fixed umber, but i practice we do ot kow its value. A statistic is a fuctio of the sample data, i.e.,

More information

Monte Carlo Integration

Monte Carlo Integration Mote Carlo Itegratio I these otes we first review basic umerical itegratio methods (usig Riema approximatio ad the trapezoidal rule) ad their limitatios for evaluatig multidimesioal itegrals. Next we itroduce

More information

The Distribution of the Concentration Ratio for Samples from a Uniform Population

The Distribution of the Concentration Ratio for Samples from a Uniform Population Applied Mathematics, 05, 6, 57-70 Published Olie Jauary 05 i SciRes. http://www.scirp.or/joural/am http://dx.doi.or/0.436/am.05.6007 The Distributio of the Cocetratio Ratio for Samples from a Uiform Populatio

More information

Output Analysis and Run-Length Control

Output Analysis and Run-Length Control IEOR E4703: Mote Carlo Simulatio Columbia Uiversity c 2017 by Marti Haugh Output Aalysis ad Ru-Legth Cotrol I these otes we describe how the Cetral Limit Theorem ca be used to costruct approximate (1 α%

More information

Introduction to Econometrics (3 rd Updated Edition) Solutions to Odd- Numbered End- of- Chapter Exercises: Chapter 4

Introduction to Econometrics (3 rd Updated Edition) Solutions to Odd- Numbered End- of- Chapter Exercises: Chapter 4 Itroductio to Ecoometrics (3 rd Updated Editio) by James H. Stock ad Mark W. Watso Solutios to Odd- Numbered Ed- of- Chapter Exercises: Chapter 4 (This versio August 7, 204) 205 Pearso Educatio, Ic. Stock/Watso

More information

Introduction to Machine Learning DIS10

Introduction to Machine Learning DIS10 CS 189 Fall 017 Itroductio to Machie Learig DIS10 1 Fu with Lagrage Multipliers (a) Miimize the fuctio such that f (x,y) = x + y x + y = 3. Solutio: The Lagragia is: L(x,y,λ) = x + y + λ(x + y 3) Takig

More information

Stat 200 -Testing Summary Page 1

Stat 200 -Testing Summary Page 1 Stat 00 -Testig Summary Page 1 Mathematicias are like Frechme; whatever you say to them, they traslate it ito their ow laguage ad forthwith it is somethig etirely differet Goethe 1 Large Sample Cofidece

More information

A goodness-of-fit test based on the empirical characteristic function and a comparison of tests for normality

A goodness-of-fit test based on the empirical characteristic function and a comparison of tests for normality A goodess-of-fit test based o the empirical characteristic fuctio ad a compariso of tests for ormality J. Marti va Zyl Departmet of Mathematical Statistics ad Actuarial Sciece, Uiversity of the Free State,

More information

Numerical Methods for Ordinary Differential Equations

Numerical Methods for Ordinary Differential Equations Numerical Methods for Ordiary Differetial Equatios Braislav K. Nikolić Departmet of Physics ad Astroomy, Uiversity of Delaware, U.S.A. PHYS 460/660: Computatioal Methods of Physics http://www.physics.udel.edu/~bikolic/teachig/phys660/phys660.html

More information

Matrix Representation of Data in Experiment

Matrix Representation of Data in Experiment Matrix Represetatio of Data i Experimet Cosider a very simple model for resposes y ij : y ij i ij, i 1,; j 1,,..., (ote that for simplicity we are assumig the two () groups are of equal sample size ) Y

More information

CARLESON MEASURE, ATOMIC DECOMPOSITION AND FREE INTERPOLATION FROM BLOCH SPACE

CARLESON MEASURE, ATOMIC DECOMPOSITION AND FREE INTERPOLATION FROM BLOCH SPACE Aales Academiæ Scietiarum Feicæ Series A. I. Mathematica Volume 9, 994, 35 46 CARLESON MEASURE, ATOMIC ECOMPOSITION AN FREE INTERPOLATION FROM BLOCH SPACE Jie Xiao Pekig Uiversity, epartmet of Mathematics

More information

Additive processes. Chapter 6

Additive processes. Chapter 6 Chapter 6 Additive processes Discoveries of heavy-tailed pheomea are quite ofte viewed with surprise, as if heavy-tailed distributios are merely a probabilistic curiosity. I large part, this is a cosequece

More information

Ada Boost, Risk Bounds, Concentration Inequalities. 1 AdaBoost and Estimates of Conditional Probabilities

Ada Boost, Risk Bounds, Concentration Inequalities. 1 AdaBoost and Estimates of Conditional Probabilities CS8B/Stat4B Sprig 008) Statistical Learig Theory Lecture: Ada Boost, Risk Bouds, Cocetratio Iequalities Lecturer: Peter Bartlett Scribe: Subhrasu Maji AdaBoost ad Estimates of Coditioal Probabilities We

More information

On the Asymptotics of ADF Tests for Unit Roots 1

On the Asymptotics of ADF Tests for Unit Roots 1 O the Asymptotics of ADF Tests for Uit Roots Yoosoo Chag Departmet of Ecoomics Rice Uiversity ad Joo Y. Park School of Ecoomics Seoul Natioal Uiversity Abstract I this paper, we derive the asymptotic distributios

More information

Sequences and Limits

Sequences and Limits Chapter Sequeces ad Limits Let { a } be a sequece of real or complex umbers A ecessary ad sufficiet coditio for the sequece to coverge is that for ay ɛ > 0 there exists a iteger N > 0 such that a p a q

More information

The Jordan Normal Form: A General Approach to Solving Homogeneous Linear Systems. Mike Raugh. March 20, 2005

The Jordan Normal Form: A General Approach to Solving Homogeneous Linear Systems. Mike Raugh. March 20, 2005 The Jorda Normal Form: A Geeral Approach to Solvig Homogeeous Liear Sstems Mike Raugh March 2, 25 What are we doig here? I this ote, we describe the Jorda ormal form of a matrix ad show how it ma be used

More information

A PROOF OF THE TWIN PRIME CONJECTURE AND OTHER POSSIBLE APPLICATIONS

A PROOF OF THE TWIN PRIME CONJECTURE AND OTHER POSSIBLE APPLICATIONS A PROOF OF THE TWI PRIME COJECTURE AD OTHER POSSIBLE APPLICATIOS by PAUL S. BRUCKMA 38 Frot Street, #3 aaimo, BC V9R B8 (Caada) e-mail : pbruckma@hotmail.com ABSTRACT : A elemetary proof of the Twi Prime

More information

Probabilistic and Average Linear Widths in L -Norm with Respect to r-fold Wiener Measure

Probabilistic and Average Linear Widths in L -Norm with Respect to r-fold Wiener Measure joural of approximatio theory 84, 3140 (1996) Article No. 0003 Probabilistic ad Average Liear Widths i L -Norm with Respect to r-fold Wieer Measure V. E. Maiorov Departmet of Mathematics, Techio, Haifa,

More information

(a) (b) All real numbers. (c) All real numbers. (d) None. to show the. (a) 3. (b) [ 7, 1) (c) ( 7, 1) (d) At x = 7. (a) (b)

(a) (b) All real numbers. (c) All real numbers. (d) None. to show the. (a) 3. (b) [ 7, 1) (c) ( 7, 1) (d) At x = 7. (a) (b) Chapter 0 Review 597. E; a ( + )( + ) + + S S + S + + + + + + S lim + l. D; a diverges by the Itegral l k Test sice d lim [(l ) ], so k l ( ) does ot coverge absolutely. But it coverges by the Alteratig

More information

TRIGONOMETRIC POLYNOMIALS WITH MANY REAL ZEROS AND A LITTLEWOOD-TYPE PROBLEM. Peter Borwein and Tamás Erdélyi. 1. Introduction

TRIGONOMETRIC POLYNOMIALS WITH MANY REAL ZEROS AND A LITTLEWOOD-TYPE PROBLEM. Peter Borwein and Tamás Erdélyi. 1. Introduction TRIGONOMETRIC POLYNOMIALS WITH MANY REAL ZEROS AND A LITTLEWOOD-TYPE PROBLEM Peter Borwei ad Tamás Erdélyi Abstract. We examie the size of a real trigoometric polyomial of degree at most havig at least

More information

Sequences of Definite Integrals, Factorials and Double Factorials

Sequences of Definite Integrals, Factorials and Double Factorials 47 6 Joural of Iteger Sequeces, Vol. 8 (5), Article 5.4.6 Sequeces of Defiite Itegrals, Factorials ad Double Factorials Thierry Daa-Picard Departmet of Applied Mathematics Jerusalem College of Techology

More information

18.657: Mathematics of Machine Learning

18.657: Mathematics of Machine Learning 8.657: Mathematics of Machie Learig Lecturer: Philippe Rigollet Lecture 4 Scribe: Cheg Mao Sep., 05 I this lecture, we cotiue to discuss the effect of oise o the rate of the excess risk E(h) = R(h) R(h

More information

The Sample Variance Formula: A Detailed Study of an Old Controversy

The Sample Variance Formula: A Detailed Study of an Old Controversy The Sample Variace Formula: A Detailed Study of a Old Cotroversy Ky M. Vu PhD. AuLac Techologies Ic. c 00 Email: kymvu@aulactechologies.com Abstract The two biased ad ubiased formulae for the sample variace

More information

Dimension-free PAC-Bayesian bounds for the estimation of the mean of a random vector

Dimension-free PAC-Bayesian bounds for the estimation of the mean of a random vector Dimesio-free PAC-Bayesia bouds for the estimatio of the mea of a radom vector Olivier Catoi CREST CNRS UMR 9194 Uiversité Paris Saclay olivier.catoi@esae.fr Ilaria Giulii Laboratoire de Probabilités et

More information

ON THE LEHMER CONSTANT OF FINITE CYCLIC GROUPS

ON THE LEHMER CONSTANT OF FINITE CYCLIC GROUPS ON THE LEHMER CONSTANT OF FINITE CYCLIC GROUPS NORBERT KAIBLINGER Abstract. Results of Lid o Lehmer s problem iclude the value of the Lehmer costat of the fiite cyclic group Z/Z, for 5 ad all odd. By complemetary

More information

Topics in Probability Theory and Stochastic Processes Steven R. Dunbar. Stirling s Formula Derived from the Gamma Function

Topics in Probability Theory and Stochastic Processes Steven R. Dunbar. Stirling s Formula Derived from the Gamma Function Steve R. Dubar Departmet of Mathematics 23 Avery Hall Uiversity of Nebraska-Licol Licol, NE 68588-3 http://www.math.ul.edu Voice: 42-472-373 Fax: 42-472-8466 Topics i Probability Theory ad Stochastic Processes

More information

A new iterative algorithm for reconstructing a signal from its dyadic wavelet transform modulus maxima

A new iterative algorithm for reconstructing a signal from its dyadic wavelet transform modulus maxima ol 46 No 6 SCIENCE IN CHINA (Series F) December 3 A ew iterative algorithm for recostructig a sigal from its dyadic wavelet trasform modulus maxima ZHANG Zhuosheg ( u ), LIU Guizhog ( q) & LIU Feg ( )

More information

Seunghee Ye Ma 8: Week 5 Oct 28

Seunghee Ye Ma 8: Week 5 Oct 28 Week 5 Summary I Sectio, we go over the Mea Value Theorem ad its applicatios. I Sectio 2, we will recap what we have covered so far this term. Topics Page Mea Value Theorem. Applicatios of the Mea Value

More information

Probability, Expectation Value and Uncertainty

Probability, Expectation Value and Uncertainty Chapter 1 Probability, Expectatio Value ad Ucertaity We have see that the physically observable properties of a quatum system are represeted by Hermitea operators (also referred to as observables ) such

More information

Rademacher Complexity

Rademacher Complexity EECS 598: Statistical Learig Theory, Witer 204 Topic 0 Rademacher Complexity Lecturer: Clayto Scott Scribe: Ya Deg, Kevi Moo Disclaimer: These otes have ot bee subjected to the usual scrutiy reserved for

More information

THE LIM;I,TING BEHAVIOUR OF THE EMPIRICAL KERNEL DISTRIBDTI'ON FUNCTION. Pranab Kumar Sen

THE LIM;I,TING BEHAVIOUR OF THE EMPIRICAL KERNEL DISTRIBDTI'ON FUNCTION. Pranab Kumar Sen -e ON THE LIM;I,TING BEHAVIOUR OF THE EMPIRICAL KERNEL DISTRIBDTI'ON FUNCTION By Praab Kumar Se Departmet of Biostatistics Uiversity of North Carolia at Chapel Hill Istitute of Statistic~ Mimeo Series

More information

Discrete Orthogonal Moment Features Using Chebyshev Polynomials

Discrete Orthogonal Moment Features Using Chebyshev Polynomials Discrete Orthogoal Momet Features Usig Chebyshev Polyomials R. Mukuda, 1 S.H.Og ad P.A. Lee 3 1 Faculty of Iformatio Sciece ad Techology, Multimedia Uiversity 75450 Malacca, Malaysia. Istitute of Mathematical

More information

ARIMA Models. Dan Saunders. y t = φy t 1 + ɛ t

ARIMA Models. Dan Saunders. y t = φy t 1 + ɛ t ARIMA Models Da Sauders I will discuss models with a depedet variable y t, a potetially edogeous error term ɛ t, ad a exogeous error term η t, each with a subscript t deotig time. With just these three

More information

3/3/2014. CDS M Phil Econometrics. Types of Relationships. Types of Relationships. Types of Relationships. Vijayamohanan Pillai N.

3/3/2014. CDS M Phil Econometrics. Types of Relationships. Types of Relationships. Types of Relationships. Vijayamohanan Pillai N. 3/3/04 CDS M Phil Old Least Squares (OLS) Vijayamohaa Pillai N CDS M Phil Vijayamoha CDS M Phil Vijayamoha Types of Relatioships Oly oe idepedet variable, Relatioship betwee ad is Liear relatioships Curviliear

More information

SEMIPARAMETRIC SINGLE-INDEX MODELS. Joel L. Horowitz Department of Economics Northwestern University

SEMIPARAMETRIC SINGLE-INDEX MODELS. Joel L. Horowitz Department of Economics Northwestern University SEMIPARAMETRIC SINGLE-INDEX MODELS by Joel L. Horowitz Departmet of Ecoomics Northwester Uiversity INTRODUCTION Much of applied ecoometrics ad statistics ivolves estimatig a coditioal mea fuctio: E ( Y

More information

SEQUENCES AND SERIES

SEQUENCES AND SERIES 9 SEQUENCES AND SERIES INTRODUCTION Sequeces have may importat applicatios i several spheres of huma activities Whe a collectio of objects is arraged i a defiite order such that it has a idetified first

More information

REGRESSION (Physics 1210 Notes, Partial Modified Appendix A)

REGRESSION (Physics 1210 Notes, Partial Modified Appendix A) REGRESSION (Physics 0 Notes, Partial Modified Appedix A) HOW TO PERFORM A LINEAR REGRESSION Cosider the followig data poits ad their graph (Table I ad Figure ): X Y 0 3 5 3 7 4 9 5 Table : Example Data

More information

MOMENT-METHOD ESTIMATION BASED ON CENSORED SAMPLE

MOMENT-METHOD ESTIMATION BASED ON CENSORED SAMPLE Vol. 8 o. Joural of Systems Sciece ad Complexity Apr., 5 MOMET-METHOD ESTIMATIO BASED O CESORED SAMPLE I Zhogxi Departmet of Mathematics, East Chia Uiversity of Sciece ad Techology, Shaghai 37, Chia. Email:

More information

Frequency Response of FIR Filters

Frequency Response of FIR Filters EEL335: Discrete-Time Sigals ad Systems. Itroductio I this set of otes, we itroduce the idea of the frequecy respose of LTI systems, ad focus specifically o the frequecy respose of FIR filters.. Steady-state

More information

5.1 A mutual information bound based on metric entropy

5.1 A mutual information bound based on metric entropy Chapter 5 Global Fao Method I this chapter, we exted the techiques of Chapter 2.4 o Fao s method the local Fao method) to a more global costructio. I particular, we show that, rather tha costructig a local

More information

EDGEWORTH SIZE CORRECTED W, LR AND LM TESTS IN THE FORMATION OF THE PRELIMINARY TEST ESTIMATOR

EDGEWORTH SIZE CORRECTED W, LR AND LM TESTS IN THE FORMATION OF THE PRELIMINARY TEST ESTIMATOR Joural of Statistical Research 26, Vol. 37, No. 2, pp. 43-55 Bagladesh ISSN 256-422 X EDGEORTH SIZE CORRECTED, AND TESTS IN THE FORMATION OF THE PRELIMINARY TEST ESTIMATOR Zahirul Hoque Departmet of Statistics

More information