8.1 Introduction. 8. Nonparametric Inference Using Orthogonal Functions
|
|
- Lilian Pope
- 5 years ago
- Views:
Transcription
1 8. Noparametric Iferece Usig Orthogoal Fuctios 1. Itroductio. Noparametric Regressio 3. Irregular Desigs 4. Desity Estimatio 5. Compariso of Methods 8.1 Itroductio Use a orthogoal basis to covert oparametric regressio/desity estimatio ito may Normal meas problem Costruct estimates ad cofidece ball/sets usig the miimax theory. For regressio problems, orthogoal methods produce a liear smoother Noparametric Regressio Model: Y i = rx i + σɛ i ɛ i N0, 1 are iid. r L 0, 1. Covert Noparametric Iferece ito May Normal Meas estimatio Oe ca approximate r by the projectio of r: r x = θ j φ j x. Estimatig r is equivalet to estimatig θ = θ 1,..., θ Assume a regular desig: x i = i for i = 1,...,. A aive estimator of θ j =< f, φ j >: φ 1, φ,... : orthoormal basis Oe ca expad r as rx = θ j φ j x Z j N θ j, σ Z j = 1. Y i φ j x i, j = 1,...,. θ j = 1 0 φ jxrxdx. 3 4
2 Defie the itegrated squared bias of size B θ = r r = 1 0 rx r x dx = Lemma 1. Let Θm, c be a Sobolev ellipsoid. The I particular, if m > 1/, the sup B θ = O m θ Θ sup B θ = o 1. θ Θ j=+1 θ j. Modulatio Estimators James-Stei estimator: miimize the SURE over the class of liear estimator bz = bz 1,..., bz b [0, 1]. Wat to miimize the SURE over a larger class of estimators. Modulators: a vector such that 0 b j 1, j = 1,...,. Modulatio estimator: a compoetwise liear estimator. θ = bz = b 1 Z 1, b Z,..., b Z b = b 1,..., b is a modulator. Proof. For θ Θm, c, j=+1 θ j 1 m j=+1 j m θ j c m. 5 6 Examples of Modulators Set of costat modulators M CONS : b M CONS b = b,..., b. Set of ested subset selectio modulator M NSS : b M NSS b = 1,..., 1, 0,..., 0. Set of mootoe modulator M MON : b M MON 1 b 1... b 0. The fuctio estimator r is r t = θ j φ j x = l i t = 1 b j Z j φ j t = b j φ j tφ j x i Y i l i t How to choose b? Use the SURE formula to estimator the risk fuctio ad fid b that miimizes SURE over some class of liear estimators 7 8
3 Theorem 1. The risk of a modulator b is Rb = 1 b j θj + σ The modified SURE estimator of Rb is Rb = 1 b j Zj σ σ is a cosistet estimator of σ. + b j, + σ b j, Lemma. Let σ = J i= J +1 J ad J as. The σ is a cosistet estimate of σ Proof. For large j, θ j 0 ad hece, Zj = θj + σ σ ɛ j ɛ j Z i Therefore, E σ j = J J +1 E Z i J Note: as a default value, J = /4. J +1 σ E ɛ i = σ 9 10 Theorem. Let M be oe of M CONS, M NSS, M MON. Let Rb deote the true risk of the estimator bz = b 1 Z 1,..., b Z. Let ad let The, as, b = argmi Rb b M b = argmi b M Rb. R b Rb 0. For M = M MON, the estimator θ = b 1 Z 1,..., b Z achieves the Pisker boud. How to choose a optimal modulator? For b M NSS, Fidig b = argmi b MNSS Rb is equivalet to fidig Ĵ which miimizes RJ = J σ j + Z j σ + Set r = j=j+1 bj Z j φ j x 11 1
4 For b M MON, Rb ca be writte as Rb = g i = 1 Z i Fidig b = argmi b MMON b = argmi b M MON b i g i + σ Z i gi, σ Rb is equivalet to fidig b i g i Zi It is a weighted least squares problem subject to a mootoicity costrait. Use Pooled-Adjacet-Violators PAV algorithm Theorem 3. Let θ be the MON or NSS estimator ad let σ be a cosistet estimator of σ. Let B = θ = θ 1,..., θ θ j θ j s τ = σ4 s = R b τ + z α b j σ The, for ay c > 0 ad m > 1/, Z j σ 1 b j. lim sup Prθ B 1 α = 0 θ Θ Proof. Defie the pivot process B b: We will use the followig strategy. B b Lb Rb 1. By the fuctioal cetral limt theorem, oe ca show that B b coverges waekly to a mea 0 Gaussia process with covariace kerel Ks, t. 3. Show that τ is a cosistet estimate of K b, b 4. From step 1-3, lim if Pr θ Θm,c lim L b R b τ z α = if Prθ B 1 α. θ Θm,c. By stochastic equicotiuity of the pivot process, oe ca show that B b is stochastically very close to B b. Hece B b has a Guassia limit
5 REACT Cofidece Sets Set Cofidece bad for f: r x = bj Z j φ j x Ix = r x c α σ lx, r x + c α σ lx c α is from the tube formula ad lx 1 b jφ jx Cofidece sets for T f,fuctioals of f: Ix = if T f, sup T f θ B θ B 8.3 Irregular Desigs For a irregular desig, oe ca use a orthoormal basis {φ 1,..., φ } w.r.t {x 1,..., x }. For j = 1,...,, defie Z j = 1 Y i φ j x i. Margially Z j has a asymptotic Normal distributio. Z j N θ j, σ Choose basis for L P P = 1 δ i ad δ i is a poit mass at x i. For j = 1,...,, φ j = φ j xdp x = 1 φ x i = 1 For 1 j < k, < φ j, φ k >= φ j xφ k xdp x = 1 φ j x i φ k x i = 0 Gram-Schmidt orthogoalizatio Costruct a orthoormal basis by Gram-Schmidt orthogoalizatio Let g 1,..., g be a orthoormal basis for R. Let For r defie φ 1 x = ψ 1x ψ r ψ 1x = g 1 x φ r x = ψ rx ψ r ψ r 1 rx = g r x a r,j φ j x a r,j =< g r, φ j >. φ 1,..., φ r form a orthoormal basis w.r.t P. 19 0
6 8.4 Desity Estimatio X 1,..., X are iid from desity f Assume f L 0, 1. Oe ca expad f with a orthoormal basis set {φ 1, φ,...}: fx = θ j φ j x θ j = fxφ j xdx. Oe ca approximate f by the projectio of f: m f m = θ j φ j x. m ad m / 0 as. Defie m f m x = θ j φ j x. Defie Z j as a aive estimator of θ j = Eφ j X Z j = 1 φ j X i, for j = 1,...,. 1 Mea of Z j : E Z j = φ j xfxdx = E φ j X = θ j Variace of Z j : Var Z j = 1 Var φ jx = 1 σ j = Var φ jx. φ jxfxdx θ j σ j Modulatio estimator Modulatio estimator θ: θ = bz = b 1 Z 1,..., b m Z m The risk fuctio of θ is m m Rb = b jσ,j + 1 b jθj. Margially, Z j Nθ j, σ,j, σ,j = sigma j /. But Z j ad Z k are ot idepedet! 3 4
7 The SURE formula yields m m Rb = b j σ,j + 1 b jzj σ,j +, σ,j = σ j = 1 The optimal desity estimator is Z j φ j X i. m f m x = bj Z j φ j x, The desity estimator ca be egative remove the egative part reormalize it to itegrate to 1 Ulike regressio problems, desity estimatio problems are coverted ito may Normal meas with covariace Curretly m = o 1/3 but might be improved up to O 1/ Jag et al. 004 b = b 1,..., b m = argmi b M Rb ad M is a class of modulators Compariso of Methods Local polyomial smoothers have the advatage to correct boudary bias. Orthogoal methods ca easily covert oparametric iferece ito may Normal meas problem. Orthogoal methods ca be cosidered as kerel smoothig with a particular kerel ad vice versa. Härdle et al Use all methods for the problem ad see if they agree. Otherwise, the why? REFERENCES 1. Bera, R REACT scatterplot smoothers: Superefficiecy through basis ecoomy. Joural of the America Statistical Associatio Bera, R. ad Dümbge, L Modulatio of estimators ad cofidece sets. Aals of Statistics Efromovich, S Noparametric Curve Estimatio: Methods, Theory ad Applicatios. Spriger. New York. 4. Härdle, W. ad Kerkyacharia, G. Picard, D. ad Tsybakov, A Wavelets, Approximatio, ad Statistical Applicatio, Spriger. New York. 5. Jag, W. Geovese, C. ad Wasserma, L Noparametric cofidece sets for desities. Upublished mauscript. 7 8
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.
More informationStudy 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
More information32 estimating the cumulative distribution function
32 estimatig the cumulative distributio fuctio 4.6 types of cofidece itervals/bads Let F be a class of distributio fuctios F ad let θ be some quatity of iterest, such as the mea of F or the whole fuctio
More informationWeek 10. f2 j=2 2 j k ; j; k 2 Zg is an orthonormal basis for L 2 (R). This function is called mother wavelet, which can be often constructed
Wee 0 A Itroductio to Wavelet regressio. De itio: Wavelet is a fuctio such that f j= j ; j; Zg is a orthoormal basis for L (R). This fuctio is called mother wavelet, which ca be ofte costructed from father
More informationDirection: 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
More information17. Joint distributions of extreme order statistics Lehmann 5.1; Ferguson 15
17. Joit distributios of extreme order statistics Lehma 5.1; Ferguso 15 I Example 10., we derived the asymptotic distributio of the maximum from a radom sample from a uiform distributio. We did this usig
More informationLet 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,
More informationRandom Variables, Sampling and Estimation
Chapter 1 Radom Variables, Samplig ad Estimatio 1.1 Itroductio This chapter will cover the most importat basic statistical theory you eed i order to uderstad the ecoometric material that will be comig
More informationECE 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
More informationTopic 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
More informationSTATISTICS 593C: Spring, Model Selection and Regularization
STATISTICS 593C: Sprig, 27 Model Selectio ad Regularizatio Jo A. Weller Lecture 2 (March 29): Geeral Notatio ad Some Examples Here is some otatio ad termiology that I will try to use (more or less) systematically
More informationResampling 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
More informationLECTURE 2 LEAST SQUARES CROSS-VALIDATION FOR KERNEL DENSITY ESTIMATION
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
More informationWavelet linear estimation of a density and its derivatives from observations of mixtures under quadrant dependence
ProbStat Forum, Volume 5, April, Pages 38 46 ISSN 974-335 ProbStat Forum is a e-joural For details please visit wwwprobstatorgi Wavelet liear estimatio of a desity ad its derivatives from observatios of
More informationLecture 24: Variable selection in linear models
Lecture 24: Variable selectio i liear models Cosider liear model X = Z β + ε, β R p ad Varε = σ 2 I. Like the LSE, the ridge regressio estimator does ot give 0 estimate to a compoet of β eve if that compoet
More informationAsymptotics. Hypothesis Testing UMP. Asymptotic Tests and p-values
of the secod half Biostatistics 6 - Statistical Iferece Lecture 6 Fial Exam & Practice Problems for the Fial Hyu Mi Kag Apil 3rd, 3 Hyu Mi Kag Biostatistics 6 - Lecture 6 Apil 3rd, 3 / 3 Rao-Blackwell
More informationEstimation of the essential supremum of a regression function
Estimatio of the essetial supremum of a regressio fuctio Michael ohler, Adam rzyżak 2, ad Harro Walk 3 Fachbereich Mathematik, Techische Uiversität Darmstadt, Schlossgartestr. 7, 64289 Darmstadt, Germay,
More informationLinear 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
More informationLocal Polynomial Regression
Local Polyomial Regressio Joh Hughes October 2, 2013 Recall that the oparametric regressio model is Y i f x i ) + ε i, where f is the regressio fuctio ad the ε i are errors such that Eε i 0. The Nadaraya-Watso
More informationIt should be unbiased, or approximately unbiased. Variance of the variance estimator should be small. That is, the variance estimator is stable.
Chapter 10 Variace Estimatio 10.1 Itroductio Variace estimatio is a importat practical problem i survey samplig. Variace estimates are used i two purposes. Oe is the aalytic purpose such as costructig
More informationLecture 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.
More information1 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 =,
More informationTopic 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
More informationProbability 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
More informationSummary. Recap ... Last Lecture. Summary. Theorem
Last Lecture Biostatistics 602 - Statistical Iferece Lecture 23 Hyu Mi Kag April 11th, 2013 What is p-value? What is the advatage of p-value compared to hypothesis testig procedure with size α? How ca
More informationEcon 325 Notes on Point Estimator and Confidence Interval 1 By Hiro Kasahara
Poit Estimator Eco 325 Notes o Poit Estimator ad Cofidece Iterval 1 By Hiro Kasahara Parameter, Estimator, ad Estimate The ormal probability desity fuctio is fully characterized by two costats: populatio
More informationLecture 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
More informationECONOMETRIC 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
More informationEstimation of the Mean and the ACVF
Chapter 5 Estimatio of the Mea ad the ACVF A statioary process {X t } is characterized by its mea ad its autocovariace fuctio γ ), ad so by the autocorrelatio fuctio ρ ) I this chapter we preset the estimators
More informationA Constrained Risk Inequality With Applications to Nonparametric Functional Estimation
Uiversity of Pesylvaia ScholarlyCommos Statistics Papers Wharto Faculty Research 1996 A Costraied Risk Iequality With Applicatios to Noparametric Fuctioal Estimatio Lawrece D. Brow Uiversity of Pesylvaia
More informationCS434a/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
More informationTopic 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
More informationSieve Estimators: Consistency and Rates of Convergence
EECS 598: Statistical Learig Theory, Witer 2014 Topic 6 Sieve Estimators: Cosistecy ad Rates of Covergece Lecturer: Clayto Scott Scribe: Julia Katz-Samuels, Brado Oselio, Pi-Yu Che Disclaimer: These otes
More informationA survey on penalized empirical risk minimization Sara A. van de Geer
A survey o pealized empirical risk miimizatio Sara A. va de Geer We address the questio how to choose the pealty i empirical risk miimizatio. Roughly speakig, this pealty should be a good boud for the
More informationRates of Convergence by Moduli of Continuity
Rates of Covergece by Moduli of Cotiuity Joh Duchi: Notes for Statistics 300b March, 017 1 Itroductio I this ote, we give a presetatio showig the importace, ad relatioship betwee, the modulis of cotiuity
More information62. 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 informationLecture 12: September 27
36-705: Itermediate Statistics Fall 207 Lecturer: Siva Balakrisha Lecture 2: September 27 Today we will discuss sufficiecy i more detail ad the begi to discuss some geeral strategies for costructig estimators.
More informationSummary and Discussion on Simultaneous Analysis of Lasso and Dantzig Selector
Summary ad Discussio o Simultaeous Aalysis of Lasso ad Datzig Selector STAT732, Sprig 28 Duzhe Wag May 4, 28 Abstract This is a discussio o the work i Bickel, Ritov ad Tsybakov (29). We begi with a short
More information4 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
More informationStatistical and Mathematical Methods DS-GA 1002 December 8, Sample Final Problems Solutions
Statistical ad Mathematical Methods DS-GA 00 December 8, 05. Short questios Sample Fial Problems Solutios a. Ax b has a solutio if b is i the rage of A. The dimesio of the rage of A is because A has liearly-idepedet
More informationLecture 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
More informationMATH 320: Probability and Statistics 9. Estimation and Testing of Parameters. Readings: Pruim, Chapter 4
MATH 30: Probability ad Statistics 9. Estimatio ad Testig of Parameters Estimatio ad Testig of Parameters We have bee dealig situatios i which we have full kowledge of the distributio of a radom variable.
More information5.1 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 information10/ 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
More informationFirst 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 informationKernel 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
More informationSection 14. Simple linear regression.
Sectio 14 Simple liear regressio. Let us look at the cigarette dataset from [1] (available to dowload from joural s website) ad []. The cigarette dataset cotais measuremets of tar, icotie, weight ad carbo
More informationSTA Object Data Analysis - A List of Projects. January 18, 2018
STA 6557 Jauary 8, 208 Object Data Aalysis - A List of Projects. Schoeberg Mea glaucomatous shape chages of the Optic Nerve Head regio i aimal models 2. Aalysis of VW- Kedall ati-mea shapes with a applicatio
More informationPrecise Rates in Complete Moment Convergence for Negatively Associated Sequences
Commuicatios of the Korea Statistical Society 29, Vol. 16, No. 5, 841 849 Precise Rates i Complete Momet Covergece for Negatively Associated Sequeces Dae-Hee Ryu 1,a a Departmet of Computer Sciece, ChugWoo
More informationChapter 7 Isoperimetric problem
Chapter 7 Isoperimetric problem Recall that the isoperimetric problem (see the itroductio its coectio with ido s proble) is oe of the most classical problem of a shape optimizatio. It ca be formulated
More informationTopics. Homework Problems. MATH 301 Introduction to Analysis Chapter Four Sequences. 1. Definition of convergence of sequences.
MATH 301 Itroductio to Aalysis Chapter Four Sequeces Topics 1. Defiitio of covergece of sequeces. 2. Fidig ad provig the limit of sequeces. 3. Bouded covergece theorem: Theorem 4.1.8. 4. Theorems 4.1.13
More informationAn Introduction to Asymptotic Theory
A Itroductio to Asymptotic Theory Pig Yu School of Ecoomics ad Fiace The Uiversity of Hog Kog Pig Yu (HKU) Asymptotic Theory 1 / 20 Five Weapos i Asymptotic Theory Five Weapos i Asymptotic Theory Pig Yu
More information4.5 Multiple Imputation
45 ultiple Imputatio Itroductio Assume a parametric model: y fy x; θ We are iterested i makig iferece about θ I Bayesia approach, we wat to make iferece about θ from fθ x, y = πθfy x, θ πθfy x, θdθ where
More informationEFFECTIVE WLLN, SLLN, AND CLT IN STATISTICAL MODELS
EFFECTIVE WLLN, SLLN, AND CLT IN STATISTICAL MODELS Ryszard Zieliński Ist Math Polish Acad Sc POBox 21, 00-956 Warszawa 10, Polad e-mail: rziel@impagovpl ABSTRACT Weak laws of large umbers (W LLN), strog
More informationBull. Korean Math. Soc. 36 (1999), No. 3, pp. 451{457 THE STRONG CONSISTENCY OF NONLINEAR REGRESSION QUANTILES ESTIMATORS Seung Hoe Choi and Hae Kyung
Bull. Korea Math. Soc. 36 (999), No. 3, pp. 45{457 THE STRONG CONSISTENCY OF NONLINEAR REGRESSION QUANTILES ESTIMATORS Abstract. This paper provides suciet coditios which esure the strog cosistecy of regressio
More informationParameter, 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 informationEstimation for Complete Data
Estimatio for Complete Data complete data: there is o loss of iformatio durig study. complete idividual complete data= grouped data A complete idividual data is the oe i which the complete iformatio of
More informationThis exam contains 19 pages (including this cover page) and 10 questions. A Formulae sheet is provided with the exam.
Probability ad Statistics FS 07 Secod Sessio Exam 09.0.08 Time Limit: 80 Miutes Name: Studet ID: This exam cotais 9 pages (icludig this cover page) ad 0 questios. A Formulae sheet is provided with the
More informationAgnostic Learning and Concentration Inequalities
ECE901 Sprig 2004 Statistical Regularizatio ad Learig Theory Lecture: 7 Agostic Learig ad Cocetratio Iequalities Lecturer: Rob Nowak Scribe: Aravid Kailas 1 Itroductio 1.1 Motivatio I the last lecture
More informationElement sampling: Part 2
Chapter 4 Elemet samplig: Part 2 4.1 Itroductio We ow cosider uequal probability samplig desigs which is very popular i practice. I the uequal probability samplig, we ca improve the efficiecy of the resultig
More informationMATHEMATICAL SCIENCES PAPER-II
MATHEMATICAL SCIENCES PAPER-II. Let {x } ad {y } be two sequeces of real umbers. Prove or disprove each of the statemets :. If {x y } coverges, ad if {y } is coverget, the {x } is coverget.. {x + y } coverges
More information1 Introduction to reducing variance in Monte Carlo simulations
Copyright c 010 by Karl Sigma 1 Itroductio to reducig variace i Mote Carlo simulatios 11 Review of cofidece itervals for estimatig a mea I statistics, we estimate a ukow mea µ = E(X) of a distributio by
More informationLecture 22: Review for Exam 2. 1 Basic Model Assumptions (without Gaussian Noise)
Lecture 22: Review for Exam 2 Basic Model Assumptios (without Gaussia Noise) We model oe cotiuous respose variable Y, as a liear fuctio of p umerical predictors, plus oise: Y = β 0 + β X +... β p X p +
More information1 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 informationAbstract Vector Spaces. Abstract Vector Spaces
Astract Vector Spaces The process of astractio is critical i egieerig! Physical Device Data Storage Vector Space MRI machie Optical receiver 0 0 1 0 1 0 0 1 Icreasig astractio 6.1 Astract Vector Spaces
More informationG. R. Pasha Department of Statistics Bahauddin Zakariya University Multan, Pakistan
Deviatio of the Variaces of Classical Estimators ad Negative Iteger Momet Estimator from Miimum Variace Boud with Referece to Maxwell Distributio G. R. Pasha Departmet of Statistics Bahauddi Zakariya Uiversity
More informationLecture 15: Learning Theory: Concentration Inequalities
STAT 425: Itroductio to Noparametric Statistics Witer 208 Lecture 5: Learig Theory: Cocetratio Iequalities Istructor: Ye-Chi Che 5. Itroductio Recall that i the lecture o classificatio, we have see that
More informationTAMS24: Notations and Formulas
TAMS4: Notatios ad Formulas Basic otatios ad defiitios X: radom variable stokastiska variabel Mea Vätevärde: µ = X = by Xiagfeg Yag kpx k, if X is discrete, xf Xxdx, if X is cotiuous Variace Varias: =
More informationMaximum likelihood estimation from record-breaking data for the generalized Pareto distribution
METRON - Iteratioal Joural of Statistics 004, vol. LXII,. 3, pp. 377-389 NAGI S. ABD-EL-HAKIM KHALAF S. SULTAN Maximum likelihood estimatio from record-breakig data for the geeralized Pareto distributio
More informationLecture 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
More informationMASSACHUSETTS INSTITUTE OF TECHNOLOGY 6.265/15.070J Fall 2013 Lecture 2 9/9/2013. Large Deviations for i.i.d. Random Variables
MASSACHUSETTS INSTITUTE OF TECHNOLOGY 6.265/15.070J Fall 2013 Lecture 2 9/9/2013 Large Deviatios for i.i.d. Radom Variables Cotet. Cheroff boud usig expoetial momet geeratig fuctios. Properties of a momet
More informationLast Lecture. Biostatistics Statistical Inference Lecture 16 Evaluation of Bayes Estimator. Recap - Example. Recap - Bayes Estimator
Last Lecture Biostatistics 60 - Statistical Iferece Lecture 16 Evaluatio of Bayes Estimator Hyu Mi Kag March 14th, 013 What is a Bayes Estimator? Is a Bayes Estimator the best ubiased estimator? Compared
More informationLecture 6 Chi Square Distribution (χ 2 ) and Least Squares Fitting
Lecture 6 Chi Square Distributio (χ ) ad Least Squares Fittig Chi Square Distributio (χ ) Suppose: We have a set of measuremets {x 1, x, x }. We kow the true value of each x i (x t1, x t, x t ). We would
More informationTHE EFFICIENCY OF BIAS CORRECTED ESTIMATORS FOR NONPARAMETRIC KERNEL ESTIMATION BASED ON LOCAL ESTIMATING EQUATIONS
THE EFFICIENCY OF BIAS CORRECTED ESTIMATORS FOR NONPARAMETRIC KERNEL ESTIMATION BASED ON LOCAL ESTIMATING EQUATIONS Göra Kauerma, Marlee Müller ad Raymod J. Carroll April 18, 1998 Abstract Stuetzle ad
More informationNonparametric 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
More informationIt 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
More informationSolutions: Homework 3
Solutios: Homework 3 Suppose that the radom variables Y,...,Y satisfy Y i = x i + " i : i =,..., IID where x,...,x R are fixed values ad ",...," Normal(0, )with R + kow. Fid ˆ = MLE( ). IND Solutio: Observe
More informationEECS564 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
More informationMa 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
More informationOutline. CSCI-567: Machine Learning (Spring 2019) Outline. Prof. Victor Adamchik. Mar. 26, 2019
Outlie CSCI-567: Machie Learig Sprig 209 Gaussia mixture models Prof. Victor Adamchik 2 Desity estimatio U of Souther Califoria Mar. 26, 209 3 Naive Bayes Revisited March 26, 209 / 57 March 26, 209 2 /
More informationQuantile regression with multilayer perceptrons.
Quatile regressio with multilayer perceptros. S.-F. Dimby ad J. Rykiewicz Uiversite Paris 1 - SAMM 90 Rue de Tolbiac, 75013 Paris - Frace Abstract. We cosider oliear quatile regressio ivolvig multilayer
More information2. The volume of the solid of revolution generated by revolving the area bounded by the
IIT JAM Mathematical Statistics (MS) Solved Paper. A eigevector of the matrix M= ( ) is (a) ( ) (b) ( ) (c) ( ) (d) ( ) Solutio: (a) Eigevalue of M = ( ) is. x So, let x = ( y) be the eigevector. z (M
More informationLecture 6 Chi Square Distribution (χ 2 ) and Least Squares Fitting
Lecture 6 Chi Square Distributio (χ ) ad Least Squares Fittig Chi Square Distributio (χ ) Suppose: We have a set of measuremets {x 1, x, x }. We kow the true value of each x i (x t1, x t, x t ). We would
More informationSingular 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 informationON 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
More informationDepartment of Mathematics
Departmet of Mathematics Ma 3/103 KC Border Itroductio to Probability ad Statistics Witer 2017 Lecture 19: Estimatio II Relevat textbook passages: Larse Marx [1]: Sectios 5.2 5.7 19.1 The method of momets
More informationBayesian Methods: Introduction to Multi-parameter Models
Bayesia Methods: Itroductio to Multi-parameter Models Parameter: θ = ( θ, θ) Give Likelihood p(y θ) ad prior p(θ ), the posterior p proportioal to p(y θ) x p(θ ) Margial posterior ( θ, θ y) is Iterested
More informationSequential Monte Carlo Methods - A Review. Arnaud Doucet. Engineering Department, Cambridge University, UK
Sequetial Mote Carlo Methods - A Review Araud Doucet Egieerig Departmet, Cambridge Uiversity, UK http://www-sigproc.eg.cam.ac.uk/ ad2/araud doucet.html ad2@eg.cam.ac.uk Istitut Heri Poicaré - Paris - 2
More informationLecture 19. sup y 1,..., yn B d n
STAT 06A: Polyomials of adom Variables Lecture date: Nov Lecture 19 Grothedieck s Iequality Scribe: Be Hough The scribes are based o a guest lecture by ya O Doell. I this lecture we prove Grothedieck s
More informationLecture Stat Maximum Likelihood Estimation
Lecture Stat 461-561 Maximum Likelihood Estimatio A.D. Jauary 2008 A.D. () Jauary 2008 1 / 63 Maximum Likelihood Estimatio Ivariace Cosistecy E ciecy Nuisace Parameters A.D. () Jauary 2008 2 / 63 Parametric
More informationLecture 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
More informationA 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 informationJanuary 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
More informationNotes 27 : Brownian motion: path properties
Notes 27 : Browia motio: path properties Math 733-734: Theory of Probability Lecturer: Sebastie Roch Refereces:[Dur10, Sectio 8.1], [MP10, Sectio 1.1, 1.2, 1.3]. Recall: DEF 27.1 (Covariace) Let X = (X
More informationSOLUTIONS TO EXAM 3. Solution: Note that this defines two convergent geometric series with respective radii r 1 = 2/5 < 1 and r 2 = 1/5 < 1.
SOLUTIONS TO EXAM 3 Problem Fid the sum of the followig series 2 + ( ) 5 5 2 5 3 25 2 2 This series diverges Solutio: Note that this defies two coverget geometric series with respective radii r 2/5 < ad
More informationRandom Signals and Noise Winter Semester 2017 Problem Set 12 Wiener Filter Continuation
Radom Sigals ad Noise Witer Semester 7 Problem Set Wieer Filter Cotiuatio Problem (Sprig, Exam A) Give is the sigal W t, which is a Gaussia white oise with expectatio zero ad power spectral desity fuctio
More information7.1 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
More informationki, X(n) lj (n) = (ϱ (n) ij ) 1 i,j d.
APPLICATIONES MATHEMATICAE 22,2 (1994), pp. 193 200 M. WIŚNIEWSKI (Kielce) EXTREME ORDER STATISTICS IN AN EQUALLY CORRELATED GAUSSIAN ARRAY Abstract. This paper cotais the results cocerig the wea covergece
More informationLecture 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 informationAppendix to: Hypothesis Testing for Multiple Mean and Correlation Curves with Functional Data
Appedix to: Hypothesis Testig for Multiple Mea ad Correlatio Curves with Fuctioal Data Ao Yua 1, Hog-Bi Fag 1, Haiou Li 1, Coli O. Wu, Mig T. Ta 1, 1 Departmet of Biostatistics, Bioiformatics ad Biomathematics,
More information