Histogram Härdle, Müller, Sperlich, Werwatz, 1995, Nonparametric and Semiparametric Models, An Introduction
|
|
- Anissa Crawford
- 5 years ago
- Views:
Transcription
1 Härdle, Müller, Sperlich, Werwatz, 1995, Nonparametric and Semiparametric Models, An Introduction Tine Buch-Kromann
2 Construction X 1,..., X n iid r.v. with (unknown) density, f. Aim: Estimate the density and display it graphically. Construction: Divide the range into bins B j = [x 0 + (j 1)h, x 0 + jh), j Z with origin x 0 and binwidth h. Count the observations in each B j (=: n j ) Normalize to 1: f j = n j nh (relative frequencies, divided by h). Draw bars with height f j for bin B j.
3 Formula Formula of the histogram: ˆf h (x) = 1 nh n 1 (Xi B j )1 (x Bj ) i=1 Note: Denote by m j the center of the bin B j. The histogram assigns each x in B j = [m j h 2, m j + h 2 ) the same estimate, ˆf h (m j ) for f. j
4 Derivation Motivation of the histogram: The probability of an observation X will fall into the bin B j = [m j h 2, m j + h 2 ) is P(X B j ) = f (u)du B j f (m j ) h Approximate by the relative frequency of observations in the interval: P(X B j ) 1 n #{X i B j } Combining this, we get ˆf h (m j ) = 1 nh #{X i B j }
5 Binwidth The histogram ˆf h (m j ) depends on the binwidth h and the origin x 0. The effect of the choice of binwidth is displayed in the four histograms:
6 Statistical properties (Asymptotic) Statistical properties of the histogram as an estimator of the unknown density. Let X 1,..., X n f. We have Consistency: ˆf h (x) = 1 nh n 1 (Xi B j )1 (x Bj ) i=1 Is ˆf h (x) a consistent estimator of f (x), ie. ˆf h (x) j P f (x)? Suppose the origin x 0 = 0. We want to estimate the density at x B j = [(j 1)h, jh) ˆf h (x) = 1 nh n i=1 1 (Xi B j )
7 Bias and Variance Bias E[ˆf h (x) f (x)] f (m j ) (m j x) Note: The bias is increasing in the slope of f (m j ) and the bias is 0 if x = m j. Variance V[ˆf h (x)] 1 nh f (x) Note: The variance is proportional to f (x) and decreases when nh increases. Bias increases when h increases and variance decreases when h increases. i.e. we have to find a compromise between bias and variance to find an optimal h.
8 Mean Square Error (MSE) Mean Square Error MSE[ˆf h (x)] = E[ˆf h (x) f (x)] 2 = Variance + Bias 2 (general result) 1 nh f (x) + [ f (m j ) ] 2 (mj x) 2 Note: The histogram converges in mean square to f(x) if h 0 and nh. That means more and more observations and smaller and smaller binwidth, but not too fast. Convergence in mean square implies convergence i probability: ˆf h (x) is a consistent estimator of f (x).
9 Bias, variance and MSE for a histogram Squared bias: Thin solid line. Variance: Dashed line. MSE: Thick line.
10 Mean Integrated Squared Error (MISE) MSE measures the accuracy of ˆf h (x) as an estimator of f in a single point. But we want a global quality measure: MISE [ ] 2 MISE(ˆf h ) = E (ˆf h (x) f (x)) dx [ ) ] 2 = E (ˆf h (x) f (x) dx =. where f 2 2 = f (x) 2 dx ] MSE [ˆfh (x) dx 1 nh + h2 12 f 2 2 = AMISE(ˆf h )
11 Optimal Binwidth Criterion for selecting an optimal binwidth h: Select h that minimizes AMISE. Hence AMISE(ˆf h ) h = 1 nh h f 2 2 = 0 ( ) 6 1/3 h 0 = n f 2 n 1/3 2
12 Rule-of-thumb binwidth Problem: f is unknown, so we cannot calculate f 2 2!!! Solution: Assume that f follows a special distribution, ex. standard normal distribution, then: f 2 2 = 1 4 π Therefore we get a rule-of-thumb binwidth: ( ) 1/3 6 h 0 = n 1 3.5n 1/3 4 π
13 Origin The histogram depends on the origin
14 Drawbacks of the histogram Constant over interval (step function) Results depend on origin Binwidth choice Slow rate of convergence. Solution to the dependence on the origin x 0 : Averaged Shifted (ASH)
15 Averaged shifted histogram (idea) ASH is obtained by averaging over histograms correspondig to different origins. It seems to correspond to a smaller binwidth than the histogram from which it is constructed. But it is not an ordinary histogram with smaller binwidth.
16 Averaged shifted histogram with origin x 0 = 0, and bins B j = [(j 1)h, jh), j Z Generate M 1 new bin grids by shifting each B j by the amount kh/m to the right [( B jk = j 1 k ) ( h, j + k ) ) h, k {1,..., M 1} M M Calculate a histogram for each bin grid ˆf h,k (x) = 1 n 1 nh (Xi B jk )1 (x Bjk ) i=1 j
17 Averaged shifted histogram Compute an average over these estimates ˆf h (x) = 1 M 1 1 n 1 M nh (Xi B jk )1 (x Bjk ) k=0 i=1 j = 1 n 1 M 1 1 n Mh (Xi B jk )1 (x Bjk ) i=1 k=0 Note: As M, ASH does not depend on the origin ie. step function continuous function. j Motivation for kernel density estimation.
18 Summary (1) The formula of the histogram with binwidth h and origin x 0 : ˆf h (x) = 1 n 1 nh (Xi B j )1 (x Bj ) i=1 where B j = [x 0 + (j 1)h, x 0 + jh) and j Z. Bias E[ˆf h (x) f (x)] f (m j ) (m j x) j Variance V[ˆf h (x)] 1 nh f (x) The asymptotic MISE AMISE = 1 nh + h2 12 f 2 2
19 Summary (2) The optimal binwidth h 0 that minimizes AMISE ( ) 6 1/3 h 0 = n f 2 n 1/3 2 The optimal binwidth h 0 that minimizes AMISE for N(0,1) (Rule-of-thumb) h 0 3.5n 1/3 The averaged shifted histogram (ASH) ˆf h (x) = 1 n 1 M 1 1 n Mh (Xi B jk )1 (x Bjk ) i=1 k=0 j
Quantitative Economics for the Evaluation of the European Policy. Dipartimento di Economia e Management
Quantitative Economics for the Evaluation of the European Policy Dipartimento di Economia e Management Irene Brunetti 1 Davide Fiaschi 2 Angela Parenti 3 9 ottobre 2015 1 ireneb@ec.unipi.it. 2 davide.fiaschi@unipi.it.
More informationNonparametric Density Estimation (Multidimension)
Nonparametric Density Estimation (Multidimension) Härdle, Müller, Sperlich, Werwarz, 1995, Nonparametric and Semiparametric Models, An Introduction Tine Buch-Kromann February 19, 2007 Setup One-dimensional
More informationNonparametric Density Estimation
Nonparametric Density Estimation Econ 690 Purdue University Justin L. Tobias (Purdue) Nonparametric Density Estimation 1 / 29 Density Estimation Suppose that you had some data, say on wages, and you wanted
More informationKernel density estimation for heavy-tailed distributions...
Kernel density estimation for heavy-tailed distributions using the Champernowne transformation Buch-Larsen, Nielsen, Guillen, Bolance, Kernel density estimation for heavy-tailed distributions using the
More informationAdaptive Nonparametric Density Estimators
Adaptive Nonparametric Density Estimators by Alan J. Izenman Introduction Theoretical results and practical application of histograms as density estimators usually assume a fixed-partition approach, where
More informationKernel density estimation
Kernel density estimation Patrick Breheny October 18 Patrick Breheny STA 621: Nonparametric Statistics 1/34 Introduction Kernel Density Estimation We ve looked at one method for estimating density: histograms
More informationNonparametric Methods
Nonparametric Methods Michael R. Roberts Department of Finance The Wharton School University of Pennsylvania July 28, 2009 Michael R. Roberts Nonparametric Methods 1/42 Overview Great for data analysis
More informationNonparametric Regression Härdle, Müller, Sperlich, Werwarz, 1995, Nonparametric and Semiparametric Models, An Introduction
Härdle, Müller, Sperlich, Werwarz, 1995, Nonparametric and Semiparametric Models, An Introduction Tine Buch-Kromann Univariate Kernel Regression The relationship between two variables, X and Y where m(
More informationDensity and Distribution Estimation
Density and Distribution Estimation Nathaniel E. Helwig Assistant Professor of Psychology and Statistics University of Minnesota (Twin Cities) Updated 04-Jan-2017 Nathaniel E. Helwig (U of Minnesota) Density
More informationDensity estimation Nonparametric conditional mean estimation Semiparametric conditional mean estimation. Nonparametrics. Gabriel Montes-Rojas
0 0 5 Motivation: Regression discontinuity (Angrist&Pischke) Outcome.5 1 1.5 A. Linear E[Y 0i X i] 0.2.4.6.8 1 X Outcome.5 1 1.5 B. Nonlinear E[Y 0i X i] i 0.2.4.6.8 1 X utcome.5 1 1.5 C. Nonlinearity
More informationNonparametric Density Estimation
Nonparametric Density Estimation Advanced Econometrics Douglas G. Steigerwald UC Santa Barbara D. Steigerwald (UCSB) Density Estimation 1 / 20 Overview Question of interest has wage inequality among women
More informationKernel Density Estimation
Kernel Density Estimation Univariate Density Estimation Suppose tat we ave a random sample of data X 1,..., X n from an unknown continuous distribution wit probability density function (pdf) f(x) and cumulative
More informationAnalysis methods of heavy-tailed data
Institute of Control Sciences Russian Academy of Sciences, Moscow, Russia February, 13-18, 2006, Bamberg, Germany June, 19-23, 2006, Brest, France May, 14-19, 2007, Trondheim, Norway PhD course Chapter
More informationNonparametric Estimation of Luminosity Functions
x x Nonparametric Estimation of Luminosity Functions Chad Schafer Department of Statistics, Carnegie Mellon University cschafer@stat.cmu.edu 1 Luminosity Functions The luminosity function gives the number
More informationNONPARAMETRIC DENSITY ESTIMATION WITH RESPECT TO THE LINEX LOSS FUNCTION
NONPARAMETRIC DENSITY ESTIMATION WITH RESPECT TO THE LINEX LOSS FUNCTION R. HASHEMI, S. REZAEI AND L. AMIRI Department of Statistics, Faculty of Science, Razi University, 67149, Kermanshah, Iran. ABSTRACT
More informationO Combining cross-validation and plug-in methods - for kernel density bandwidth selection O
O Combining cross-validation and plug-in methods - for kernel density selection O Carlos Tenreiro CMUC and DMUC, University of Coimbra PhD Program UC UP February 18, 2011 1 Overview The nonparametric problem
More informationFrom Histograms to Multivariate Polynomial Histograms and Shape Estimation. Assoc Prof Inge Koch
From Histograms to Multivariate Polynomial Histograms and Shape Estimation Assoc Prof Inge Koch Statistics, School of Mathematical Sciences University of Adelaide Inge Koch (UNSW, Adelaide) Poly Histograms
More informationLecture 3: Statistical Decision Theory (Part II)
Lecture 3: Statistical Decision Theory (Part II) Hao Helen Zhang Hao Helen Zhang Lecture 3: Statistical Decision Theory (Part II) 1 / 27 Outline of This Note Part I: Statistics Decision Theory (Classical
More informationECON 721: Lecture Notes on Nonparametric Density and Regression Estimation. Petra E. Todd
ECON 721: Lecture Notes on Nonparametric Density and Regression Estimation Petra E. Todd Fall, 2014 2 Contents 1 Review of Stochastic Order Symbols 1 2 Nonparametric Density Estimation 3 2.1 Histogram
More informationprobability of k samples out of J fall in R.
Nonparametric Techniques for Density Estimation (DHS Ch. 4) n Introduction n Estimation Procedure n Parzen Window Estimation n Parzen Window Example n K n -Nearest Neighbor Estimation Introduction Suppose
More informationNon-parametric Inference and Resampling
Non-parametric Inference and Resampling Exercises by David Wozabal (Last update 3. Juni 2013) 1 Basic Facts about Rank and Order Statistics 1.1 10 students were asked about the amount of time they spend
More informationBoundary Correction Methods in Kernel Density Estimation Tom Alberts C o u(r)a n (t) Institute joint work with R.J. Karunamuni University of Alberta
Boundary Correction Methods in Kernel Density Estimation Tom Alberts C o u(r)a n (t) Institute joint work with R.J. Karunamuni University of Alberta November 29, 2007 Outline Overview of Kernel Density
More information4 Nonparametric Regression
4 Nonparametric Regression 4.1 Univariate Kernel Regression An important question in many fields of science is the relation between two variables, say X and Y. Regression analysis is concerned with the
More informationTime Series and Forecasting Lecture 4 NonLinear Time Series
Time Series and Forecasting Lecture 4 NonLinear Time Series Bruce E. Hansen Summer School in Economics and Econometrics University of Crete July 23-27, 2012 Bruce Hansen (University of Wisconsin) Foundations
More informationChapter 1. Density Estimation
Capter 1 Density Estimation Let X 1, X,..., X n be observations from a density f X x. Te aim is to use only tis data to obtain an estimate ˆf X x of f X x. Properties of f f X x x, Parametric metods f
More informationMinimum Hellinger Distance Estimation in a. Semiparametric Mixture Model
Minimum Hellinger Distance Estimation in a Semiparametric Mixture Model Sijia Xiang 1, Weixin Yao 1, and Jingjing Wu 2 1 Department of Statistics, Kansas State University, Manhattan, Kansas, USA 66506-0802.
More informationNonparametric Density Estimation. October 1, 2018
Nonparametric Density Estimation October 1, 2018 Introduction If we can t fit a distribution to our data, then we use nonparametric density estimation. Start with a histogram. But there are problems with
More informationLocal linear multiple regression with variable. bandwidth in the presence of heteroscedasticity
Local linear multiple regression with variable bandwidth in the presence of heteroscedasticity Azhong Ye 1 Rob J Hyndman 2 Zinai Li 3 23 January 2006 Abstract: We present local linear estimator with variable
More informationI [Xi t] n ˆFn (t) Binom(n, F (t))
Histograms & Densities We have seen in class various pictures of theoretical distribution functions and also some pictures of empirical distribution functions based on data. The definition of this concept
More informationLogistic Kernel Estimator and Bandwidth Selection. for Density Function
International Journal of Contemporary Matematical Sciences Vol. 13, 2018, no. 6, 279-286 HIKARI Ltd, www.m-ikari.com ttps://doi.org/10.12988/ijcms.2018.81133 Logistic Kernel Estimator and Bandwidt Selection
More information3 Nonparametric Density Estimation
3 Nonparametric Density Estimation Example: Income distribution Source: U.K. Family Expenditure Survey (FES) 1968-1995 Approximately 7000 British Households per year For each household many different variables
More informationLog-Density Estimation with Application to Approximate Likelihood Inference
Log-Density Estimation with Application to Approximate Likelihood Inference Martin Hazelton 1 Institute of Fundamental Sciences Massey University 19 November 2015 1 Email: m.hazelton@massey.ac.nz WWPMS,
More informationInstance-based Learning CE-717: Machine Learning Sharif University of Technology. M. Soleymani Fall 2016
Instance-based Learning CE-717: Machine Learning Sharif University of Technology M. Soleymani Fall 2016 Outline Non-parametric approach Unsupervised: Non-parametric density estimation Parzen Windows Kn-Nearest
More informationIntroduction to Regression
Introduction to Regression p. 1/97 Introduction to Regression Chad Schafer cschafer@stat.cmu.edu Carnegie Mellon University Introduction to Regression p. 1/97 Acknowledgement Larry Wasserman, All of Nonparametric
More informationSmooth simultaneous confidence bands for cumulative distribution functions
Journal of Nonparametric Statistics, 2013 Vol. 25, No. 2, 395 407, http://dx.doi.org/10.1080/10485252.2012.759219 Smooth simultaneous confidence bands for cumulative distribution functions Jiangyan Wang
More informationA Novel Nonparametric Density Estimator
A Novel Nonparametric Density Estimator Z. I. Botev The University of Queensland Australia Abstract We present a novel nonparametric density estimator and a new data-driven bandwidth selection method with
More informationMIT Spring 2015
Assessing Goodness Of Fit MIT 8.443 Dr. Kempthorne Spring 205 Outline 2 Poisson Distribution Counts of events that occur at constant rate Counts in disjoint intervals/regions are independent If intervals/regions
More information12 - Nonparametric Density Estimation
ST 697 Fall 2017 1/49 12 - Nonparametric Density Estimation ST 697 Fall 2017 University of Alabama Density Review ST 697 Fall 2017 2/49 Continuous Random Variables ST 697 Fall 2017 3/49 1.0 0.8 F(x) 0.6
More informationNonparametric Econometrics
Applied Microeconometrics with Stata Nonparametric Econometrics Spring Term 2011 1 / 37 Contents Introduction The histogram estimator The kernel density estimator Nonparametric regression estimators Semi-
More informationRewriting Absolute Value Functions as Piece-wise Defined Functions
Rewriting Absolute Value Functions as Piece-wise Defined Functions Consider the absolute value function f ( x) = 2x+ 4-3. Sketch the graph of f(x) using the strategies learned in Algebra II finding the
More informationRight-truncated data. STAT474/STAT574 February 7, / 44
Right-truncated data For this data, only individuals for whom the event has occurred by a given date are included in the study. Right truncation can occur in infectious disease studies. Let T i denote
More informationAdditive Isotonic Regression
Additive Isotonic Regression Enno Mammen and Kyusang Yu 11. July 2006 INTRODUCTION: We have i.i.d. random vectors (Y 1, X 1 ),..., (Y n, X n ) with X i = (X1 i,..., X d i ) and we consider the additive
More informationSemiparametric Regression Based on Multiple Sources
Semiparametric Regression Based on Multiple Sources Benjamin Kedem Department of Mathematics & Inst. for Systems Research University of Maryland, College Park Give me a place to stand and rest my lever
More informationMotivational Example
Motivational Example Data: Observational longitudinal study of obesity from birth to adulthood. Overall Goal: Build age-, gender-, height-specific growth charts (under 3 year) to diagnose growth abnomalities.
More informationChapter 9. Non-Parametric Density Function Estimation
9-1 Density Estimation Version 1.2 Chapter 9 Non-Parametric Density Function Estimation 9.1. Introduction We have discussed several estimation techniques: method of moments, maximum likelihood, and least
More informationNonparametric Function Estimation with Infinite-Order Kernels
Nonparametric Function Estimation with Infinite-Order Kernels Arthur Berg Department of Statistics, University of Florida March 15, 2008 Kernel Density Estimation (IID Case) Let X 1,..., X n iid density
More informationEcon 582 Nonparametric Regression
Econ 582 Nonparametric Regression Eric Zivot May 28, 2013 Nonparametric Regression Sofarwehaveonlyconsideredlinearregressionmodels = x 0 β + [ x ]=0 [ x = x] =x 0 β = [ x = x] [ x = x] x = β The assume
More informationChapter 9. Non-Parametric Density Function Estimation
9-1 Density Estimation Version 1.1 Chapter 9 Non-Parametric Density Function Estimation 9.1. Introduction We have discussed several estimation techniques: method of moments, maximum likelihood, and least
More informationNonparametric Statistics
Nonparametric Statistics Jessi Cisewski Yale University Astrostatistics Summer School - XI Wednesday, June 3, 2015 1 Overview Many of the standard statistical inference procedures are based on assumptions
More informationSemiparametric Regression Based on Multiple Sources
Semiparametric Regression Based on Multiple Sources Benjamin Kedem Department of Mathematics & Inst. for Systems Research University of Maryland, College Park Give me a place to stand and rest my lever
More information41903: Introduction to Nonparametrics
41903: Notes 5 Introduction Nonparametrics fundamentally about fitting flexible models: want model that is flexible enough to accommodate important patterns but not so flexible it overspecializes to specific
More informationIntroduction to Regression
Introduction to Regression Chad M. Schafer May 20, 2015 Outline General Concepts of Regression, Bias-Variance Tradeoff Linear Regression Nonparametric Procedures Cross Validation Local Polynomial Regression
More informationDEPARTMENT MATHEMATIK ARBEITSBEREICH MATHEMATISCHE STATISTIK UND STOCHASTISCHE PROZESSE
Estimating the error distribution in nonparametric multiple regression with applications to model testing Natalie Neumeyer & Ingrid Van Keilegom Preprint No. 2008-01 July 2008 DEPARTMENT MATHEMATIK ARBEITSBEREICH
More informationA PROBABILITY DENSITY FUNCTION ESTIMATION USING F-TRANSFORM
K Y BERNETIKA VOLUM E 46 ( 2010), NUMBER 3, P AGES 447 458 A PROBABILITY DENSITY FUNCTION ESTIMATION USING F-TRANSFORM Michal Holčapek and Tomaš Tichý The aim of this paper is to propose a new approach
More informationON SOME TWO-STEP DENSITY ESTIMATION METHOD
UNIVESITATIS IAGELLONICAE ACTA MATHEMATICA, FASCICULUS XLIII 2005 ON SOME TWO-STEP DENSITY ESTIMATION METHOD by Jolanta Jarnicka Abstract. We introduce a new two-step kernel density estimation method,
More informationA NOTE ON THE CHOICE OF THE SMOOTHING PARAMETER IN THE KERNEL DENSITY ESTIMATE
BRAC University Journal, vol. V1, no. 1, 2009, pp. 59-68 A NOTE ON THE CHOICE OF THE SMOOTHING PARAMETER IN THE KERNEL DENSITY ESTIMATE Daniel F. Froelich Minnesota State University, Mankato, USA and Mezbahur
More informationDivide and Conquer Kernel Ridge Regression. A Distributed Algorithm with Minimax Optimal Rates
: A Distributed Algorithm with Minimax Optimal Rates Yuchen Zhang, John C. Duchi, Martin Wainwright (UC Berkeley;http://arxiv.org/pdf/1305.509; Apr 9, 014) Gatsby Unit, Tea Talk June 10, 014 Outline Motivation.
More informationOpen Access A Stat istical Model for Wind Power Forecast Error Based on Kernel Density
Send Orders for Reprints to reprints@benthamscience.ae The Open Electrical & Electronic Engineering Journal, 2014, 8, 501-507 501 Open Access A Stat istical Model for Wind Power Forecast Error Based on
More informationNonparametric Heteroscedastic Transformation Regression Models for Skewed Data, with an Application to Health Care Costs
Nonparametric Heteroscedastic Transformation Regression Models for Skewed Data, with an Application to Health Care Costs Xiao-Hua Zhou, Huazhen Lin, Eric Johnson Journal of Royal Statistical Society Series
More informationDensity Estimation. We are concerned more here with the non-parametric case (see Roger Barlow s lectures for parametric statistics)
Density Estimation Density Estimation: Deals with the problem of estimating probability density functions (PDFs) based on some data sampled from the PDF. May use assumed forms of the distribution, parameterized
More informationIntroduction. Linear Regression. coefficient estimates for the wage equation: E(Y X) = X 1 β X d β d = X β
Introduction - Introduction -2 Introduction Linear Regression E(Y X) = X β +...+X d β d = X β Example: Wage equation Y = log wages, X = schooling (measured in years), labor market experience (measured
More informationSpatially Smoothed Kernel Density Estimation via Generalized Empirical Likelihood
Spatially Smoothed Kernel Density Estimation via Generalized Empirical Likelihood Kuangyu Wen & Ximing Wu Texas A&M University Info-Metrics Institute Conference: Recent Innovations in Info-Metrics October
More informationTest for Discontinuities in Nonparametric Regression
Communications of the Korean Statistical Society Vol. 15, No. 5, 2008, pp. 709 717 Test for Discontinuities in Nonparametric Regression Dongryeon Park 1) Abstract The difference of two one-sided kernel
More informationPositive data kernel density estimation via the logkde package for R
Positive data kernel density estimation via the logkde package for R Andrew T. Jones 1, Hien D. Nguyen 2, and Geoffrey J. McLachlan 1 which is constructed from the sample { i } n i=1. Here, K (x) is a
More informationMaximum Smoothed Likelihood for Multivariate Nonparametric Mixtures
Maximum Smoothed Likelihood for Multivariate Nonparametric Mixtures David Hunter Pennsylvania State University, USA Joint work with: Tom Hettmansperger, Hoben Thomas, Didier Chauveau, Pierre Vandekerkhove,
More informationConfidence intervals for kernel density estimation
Stata User Group - 9th UK meeting - 19/20 May 2003 Confidence intervals for kernel density estimation Carlo Fiorio c.fiorio@lse.ac.uk London School of Economics and STICERD Stata User Group - 9th UK meeting
More informationSTAT 830 Non-parametric Inference Basics
STAT 830 Non-parametric Inference Basics Richard Lockhart Simon Fraser University STAT 801=830 Fall 2012 Richard Lockhart (Simon Fraser University)STAT 830 Non-parametric Inference Basics STAT 801=830
More informationSection 4.3: Continuous Data Histograms
Section 4.3: Continuous Data Histograms Discrete-Event Simulation: A First Course c 2006 Pearson Ed., Inc. 0-13-142917-5 Discrete-Event Simulation: A First Course Section 4.3: Continuous Data Histograms
More informationContinuous Probability Distributions. Uniform Distribution
Continuous Probability Distributions Uniform Distribution Important Terms & Concepts Learned Probability Mass Function (PMF) Cumulative Distribution Function (CDF) Complementary Cumulative Distribution
More informationOn the Inverse Gaussian Kernel Estimator of the Hazard Rate Function
On the Inverse Gaussian Kernel Estimator of the Hazard Rate Function May 6, 206 The Islamic University of Gaza Deanery of Higher Studies Faculty of Science Department of Mathematics On the Inverse Gaussian
More informationDiscussion Paper No. 28
Discussion Paper No. 28 Asymptotic Property of Wrapped Cauchy Kernel Density Estimation on the Circle Yasuhito Tsuruta Masahiko Sagae Asymptotic Property of Wrapped Cauchy Kernel Density Estimation on
More informationLearning Objectives for Stat 225
Learning Objectives for Stat 225 08/20/12 Introduction to Probability: Get some general ideas about probability, and learn how to use sample space to compute the probability of a specific event. Set Theory:
More informationMore on Estimation. Maximum Likelihood Estimation.
More on Estimation. In the previous chapter we looked at the properties of estimators and the criteria we could use to choose between types of estimators. Here we examine more closely some very popular
More informationAsymptotically Optimal Regression Trees
Working Paper 208:2 Department of Economics School of Economics and Management Asymptotically Optimal Regression Trees Erik Mohlin May 208 Asymptotically Optimal Regression Trees Erik Mohlin Lund University
More informationFuzzy histograms and density estimation
Fuzzy histograms and density estimation Kevin LOQUIN 1 and Olivier STRAUSS LIRMM - 161 rue Ada - 3439 Montpellier cedex 5 - France 1 Kevin.Loquin@lirmm.fr Olivier.Strauss@lirmm.fr The probability density
More informationBickel Rosenblatt test
University of Latvia 28.05.2011. A classical Let X 1,..., X n be i.i.d. random variables with a continuous probability density function f. Consider a simple hypothesis H 0 : f = f 0 with a significance
More informationIntroduction to Regression
Introduction to Regression Chad M. Schafer cschafer@stat.cmu.edu Carnegie Mellon University Introduction to Regression p. 1/100 Outline General Concepts of Regression, Bias-Variance Tradeoff Linear Regression
More informationData-Based Choice of Histogram Bin Width. M. P. Wand. Australian Graduate School of Management. University of New South Wales.
Data-Based Choice of Histogram Bin Width M. P. Wand Australian Graduate School of Management University of New South Wales 13th May, 199 Abstract The most important parameter of a histogram is the bin
More informationESTIMATORS IN THE CONTEXT OF ACTUARIAL LOSS MODEL A COMPARISON OF TWO NONPARAMETRIC DENSITY MENGJUE TANG A THESIS MATHEMATICS AND STATISTICS
A COMPARISON OF TWO NONPARAMETRIC DENSITY ESTIMATORS IN THE CONTEXT OF ACTUARIAL LOSS MODEL MENGJUE TANG A THESIS IN THE DEPARTMENT OF MATHEMATICS AND STATISTICS PRESENTED IN PARTIAL FULFILLMENT OF THE
More informationExploiting k-nearest Neighbor Information with Many Data
Exploiting k-nearest Neighbor Information with Many Data 2017 TEST TECHNOLOGY WORKSHOP 2017. 10. 24 (Tue.) Yung-Kyun Noh Robotics Lab., Contents Nonparametric methods for estimating density functions Nearest
More informationComputer Emulation With Density Estimation
Computer Emulation With Density Estimation Jake Coleman, Robert Wolpert May 8, 2017 Jake Coleman, Robert Wolpert Emulation and Density Estimation May 8, 2017 1 / 17 Computer Emulation Motivation Expensive
More informationDensity Estimation (II)
Density Estimation (II) Yesterday Overview & Issues Histogram Kernel estimators Ideogram Today Further development of optimization Estimating variance and bias Adaptive kernels Multivariate kernel estimation
More informationOn variable bandwidth kernel density estimation
JSM 04 - Section on Nonparametric Statistics On variable bandwidth kernel density estimation Janet Nakarmi Hailin Sang Abstract In this paper we study the ideal variable bandwidth kernel estimator introduced
More informationStatistics - Lecture One. Outline. Charlotte Wickham 1. Basic ideas about estimation
Statistics - Lecture One Charlotte Wickham wickham@stat.berkeley.edu http://www.stat.berkeley.edu/~wickham/ Outline 1. Basic ideas about estimation 2. Method of Moments 3. Maximum Likelihood 4. Confidence
More informationStatistics 135 Fall 2007 Midterm Exam
Name: Student ID Number: Statistics 135 Fall 007 Midterm Exam Ignore the finite population correction in all relevant problems. The exam is closed book, but some possibly useful facts about probability
More informationThe Priestley-Chao Estimator - Bias, Variance and Mean-Square Error
The Priestley-Chao Estimator - Bias, Variance and Mean-Square Error Bias, variance and mse properties In the previous section we saw that the eact mean and variance of the Pristley-Chao estimator ˆm()
More informationMath 494: Mathematical Statistics
Math 494: Mathematical Statistics Instructor: Jimin Ding jmding@wustl.edu Department of Mathematics Washington University in St. Louis Class materials are available on course website (www.math.wustl.edu/
More informationPattern Recognition and Machine Learning. Bishop Chapter 2: Probability Distributions
Pattern Recognition and Machine Learning Chapter 2: Probability Distributions Cécile Amblard Alex Kläser Jakob Verbeek October 11, 27 Probability Distributions: General Density Estimation: given a finite
More informationLocal Polynomial Regression
VI Local Polynomial Regression (1) Global polynomial regression We observe random pairs (X 1, Y 1 ),, (X n, Y n ) where (X 1, Y 1 ),, (X n, Y n ) iid (X, Y ). We want to estimate m(x) = E(Y X = x) based
More informationNonparametric Methods
Nonparametric Methods Franco Peracchi University of Rome Tor Vergata and EIEF January 2011 Contents 1 Density estimators 2 1.1 Empirical densities......................... 4 1.2 The kernel method.........................
More informationIntensity Analysis of Spatial Point Patterns Geog 210C Introduction to Spatial Data Analysis
Intensity Analysis of Spatial Point Patterns Geog 210C Introduction to Spatial Data Analysis Chris Funk Lecture 4 Spatial Point Patterns Definition Set of point locations with recorded events" within study
More informationPractice Problems Section Problems
Practice Problems Section 4-4-3 4-4 4-5 4-6 4-7 4-8 4-10 Supplemental Problems 4-1 to 4-9 4-13, 14, 15, 17, 19, 0 4-3, 34, 36, 38 4-47, 49, 5, 54, 55 4-59, 60, 63 4-66, 68, 69, 70, 74 4-79, 81, 84 4-85,
More informationNonparametric estimation using wavelet methods. Dominique Picard. Laboratoire Probabilités et Modèles Aléatoires Université Paris VII
Nonparametric estimation using wavelet methods Dominique Picard Laboratoire Probabilités et Modèles Aléatoires Université Paris VII http ://www.proba.jussieu.fr/mathdoc/preprints/index.html 1 Nonparametric
More informationIntelligent Data Analysis. Principal Component Analysis. School of Computer Science University of Birmingham
Intelligent Data Analysis Principal Component Analysis Peter Tiňo School of Computer Science University of Birmingham Discovering low-dimensional spatial layout in higher dimensional spaces - 1-D/3-D example
More informationAdditive Models: Extensions and Related Models.
Additive Models: Extensions and Related Models. Enno Mammen Byeong U. Park Melanie Schienle August 8, 202 Abstract We give an overview over smooth backfitting type estimators in additive models. Moreover
More informationIntroduction to Curve Estimation
Introduction to Curve Estimation Density 0.000 0.002 0.004 0.006 700 800 900 1000 1100 1200 1300 Wilcoxon score Michael E. Tarter & Micheal D. Lock Model-Free Curve Estimation Monographs on Statistics
More informationDESIGN-ADAPTIVE MINIMAX LOCAL LINEAR REGRESSION FOR LONGITUDINAL/CLUSTERED DATA
Statistica Sinica 18(2008), 515-534 DESIGN-ADAPTIVE MINIMAX LOCAL LINEAR REGRESSION FOR LONGITUDINAL/CLUSTERED DATA Kani Chen 1, Jianqing Fan 2 and Zhezhen Jin 3 1 Hong Kong University of Science and Technology,
More informationSTAT 6350 Analysis of Lifetime Data. Probability Plotting
STAT 6350 Analysis of Lifetime Data Probability Plotting Purpose of Probability Plots Probability plots are an important tool for analyzing data and have been particular popular in the analysis of life
More informationPreface. 1 Nonparametric Density Estimation and Testing. 1.1 Introduction. 1.2 Univariate Density Estimation
Preface Nonparametric econometrics has become one of the most important sub-fields in modern econometrics. The primary goal of this lecture note is to introduce various nonparametric and semiparametric
More informationStatistica Sinica Preprint No: SS
Statistica Sinica Preprint No: SS-017-0013 Title A Bootstrap Method for Constructing Pointwise and Uniform Confidence Bands for Conditional Quantile Functions Manuscript ID SS-017-0013 URL http://wwwstatsinicaedutw/statistica/
More information