Asymptotic standard errors of MLE
|
|
- Randall Copeland
- 5 years ago
- Views:
Transcription
1 Asymptotic standard errors of MLE Suppose, in the previous example of Carbon and Nitrogen in soil data, that we get the parameter estimates For maximum likelihood estimation, we can use Hessian matrix of the loglikelihood function to get the asymptotic standard errors of the maximum likelihood estimates Hessian matrix is the matrix of the the second-order partial derivatives of a function The observed information matrix is the negative of the Hessian matrix of the loglikelihood function evaluated at the maximum likelihood estimators Mikyoung Jun (Texas A&M) stat647 lecture 11 October 8, / 9
2 Asymptotic standard errors of MLE It is known in statistics theory that maximum likelihood estimators are asymptotically normal with the mean being the true parameter values and the covariance matrix being the inverse of the observed information matrix In particular, the square root of the diagonal entries of the inverse of the observed information matrix are asymptotic standard errors of the parameter estimates Mikyoung Jun (Texas A&M) stat647 lecture 11 October 8, / 9
3 Asymptotic standard errors of MLE In practice, how do we get the asymptotic standard errors of the parameter estimates? Let us start by the Carbon-Nitrogen example Suppose we are doing maximum likelihood estimation to estimate the parameters In R, you can ask nlm or optim functions to return Hessian matrix Once you get the Hessian matrix, you need to be careful because of the parameter transformation For Carbon-Nitrogen data, we get ˆα = 0.040(0.021), ˆβ = 50.73(33.10), ˆν = 0.52(0.25), ˆδ = 0.029(0.0038), ˆβ 0 = 3.12(0.14), ˆβ 1 = 0.80(0.019) Mikyoung Jun (Texas A&M) stat647 lecture 11 October 8, / 9
4 Spatial autoregressive models In time series, autoregressive models express the data at time t as a linear combination of the values in the past For example, if Z(t) is a time series of interest, AR(p) model is in the form p Z(t) = c + φ i Z(t i) + ɛ(t) i=1 We can do similar things with spatial data We will see SAR models and CAR models today Mikyoung Jun (Texas A&M) stat647 lecture 11 October 8, / 9
5 Simultaneous Autoregressive (SAR) Model Consider a spatial regression problem with Gaussian data If B is a matrix of spatial dependence parameters with b i i = 0, Z(s) = m(s)β + e(s) e(s) = B e(s) + v The residuals v i, i = 1,, n, have mean zero and a diagonal covariance matrix Mikyoung Jun (Texas A&M) stat647 lecture 11 October 8, / 9
6 Simultaneous Autoregressive (SAR) Model We can also express this as (I B)(Z(s) m(s)β) = v Then the covariance matrix of Z(s), Σ SAR = (I B) 1 Σ v (I B T ) 1 if Σ v is the diagonal covariance matrix of v The covariance structure of Z(s) is completely determined by B and Σ v Mikyoung Jun (Texas A&M) stat647 lecture 11 October 8, / 9
7 Simultaneous Autoregressive (SAR) Model In practice, we may need to model B using a parametric model for b ij We may let B = ρw where W is a proximity matrix that consists of 0 s and 1 s Under this setting, we can easily see how SAR model reduces to a spatial model with uncorrelated errors Mikyoung Jun (Texas A&M) stat647 lecture 11 October 8, / 9
8 Markov property In time series, if the conditional distribution of Z(t + 1) given Z(s), s = 1,, t is the same as that of Z(t + 1) given Z(t), we say the process has Markov property We can extend this to spatial data We may say a spatial random field Z(s) is a Markov random field if Z(s i ) only depends on its neighbors N i Mikyoung Jun (Texas A&M) stat647 lecture 11 October 8, / 9
9 Conditional Autoregressive (CAR) Model This model uses the concept of Markov property We consider f (Z(s i ) Z(s) i ) where Z(s) i denotes the vector of all the data except Z(s i ) Specifically we assume each of the conditional distributions is Gaussian and we let E(Z(s i ) Z(s) i ) = m(s i )β + n c ij (Z(s j ) m(s i )β) j=1 Var(Z(s i ) Z(s) i ) = σ 2 i, i = 1,, n Here c ij is nonzero only if s i N i and c ii = 0 Mikyoung Jun (Texas A&M) stat647 lecture 11 October 8, / 9
10 Conditional Autoregressive (CAR) Model Given conditional distributions, it is not easy to construct joint distributions to do estimation and inference SAR model is only defined for multivariate Gaussian distribution while CAR model may not Mikyoung Jun (Texas A&M) stat647 lecture 11 October 8, / 9
What s for today. Introduction to Space-time models. c Mikyoung Jun (Texas A&M) Stat647 Lecture 14 October 16, / 19
What s for today Introduction to Space-time models c Mikyoung Jun (Texas A&M) Stat647 Lecture 14 October 16, 2012 1 / 19 Space-time Data So far we looked at the data that vary over space Now we add another
More informationSimple example of analysis on spatial-temporal data set
Simple example of analysis on spatial-temporal data set I used the ground level ozone data in North Carolina (from Suhasini Subba Rao s website) The original data consists of 920 days of data over 72 locations
More informationWhat s for today. Continue to discuss about nonstationary models Moving windows Convolution model Weighted stationary model
What s for today Continue to discuss about nonstationary models Moving windows Convolution model Weighted stationary model c Mikyoung Jun (Texas A&M) Stat647 Lecture 11 October 2, 2012 1 / 23 Nonstationary
More informationLecture 7 Autoregressive Processes in Space
Lecture 7 Autoregressive Processes in Space Dennis Sun Stanford University Stats 253 July 8, 2015 1 Last Time 2 Autoregressive Processes in Space 3 Estimating Parameters 4 Testing for Spatial Autocorrelation
More informationWhat s for today. Random Fields Autocovariance Stationarity, Isotropy. c Mikyoung Jun (Texas A&M) stat647 Lecture 2 August 30, / 13
What s for today Random Fields Autocovariance Stationarity, Isotropy c Mikyoung Jun (Texas A&M) stat647 Lecture 2 August 30, 2012 1 / 13 Stochastic Process and Random Fields A stochastic process is a family
More informationMS&E 226: Small Data. Lecture 11: Maximum likelihood (v2) Ramesh Johari
MS&E 226: Small Data Lecture 11: Maximum likelihood (v2) Ramesh Johari ramesh.johari@stanford.edu 1 / 18 The likelihood function 2 / 18 Estimating the parameter This lecture develops the methodology behind
More informationSpatial inference. Spatial inference. Accounting for spatial correlation. Multivariate normal distributions
Spatial inference I will start with a simple model, using species diversity data Strong spatial dependence, Î = 0.79 what is the mean diversity? How precise is our estimate? Sampling discussion: The 64
More informationWhat s for today. All about Variogram Nugget effect. Mikyoung Jun (Texas A&M) stat647 lecture 4 September 6, / 17
What s for today All about Variogram Nugget effect Mikyoung Jun (Texas A&M) stat647 lecture 4 September 6, 2012 1 / 17 What is the variogram? Let us consider a stationary (or isotropic) random field Z
More informationAreal Unit Data Regular or Irregular Grids or Lattices Large Point-referenced Datasets
Areal Unit Data Regular or Irregular Grids or Lattices Large Point-referenced Datasets Is there spatial pattern? Chapter 3: Basics of Areal Data Models p. 1/18 Areal Unit Data Regular or Irregular Grids
More informationAn Introduction to Spatial Statistics. Chunfeng Huang Department of Statistics, Indiana University
An Introduction to Spatial Statistics Chunfeng Huang Department of Statistics, Indiana University Microwave Sounding Unit (MSU) Anomalies (Monthly): 1979-2006. Iron Ore (Cressie, 1986) Raw percent data
More informationLattice Data. Tonglin Zhang. Spatial Statistics for Point and Lattice Data (Part III)
Title: Spatial Statistics for Point Processes and Lattice Data (Part III) Lattice Data Tonglin Zhang Outline Description Research Problems Global Clustering and Local Clusters Permutation Test Spatial
More informationLecture 16 Solving GLMs via IRWLS
Lecture 16 Solving GLMs via IRWLS 09 November 2015 Taylor B. Arnold Yale Statistics STAT 312/612 Notes problem set 5 posted; due next class problem set 6, November 18th Goals for today fixed PCA example
More informationAreal data models. Spatial smoothers. Brook s Lemma and Gibbs distribution. CAR models Gaussian case Non-Gaussian case
Areal data models Spatial smoothers Brook s Lemma and Gibbs distribution CAR models Gaussian case Non-Gaussian case SAR models Gaussian case Non-Gaussian case CAR vs. SAR STAR models Inference for areal
More informationGauge Plots. Gauge Plots JAPANESE BEETLE DATA MAXIMUM LIKELIHOOD FOR SPATIALLY CORRELATED DISCRETE DATA JAPANESE BEETLE DATA
JAPANESE BEETLE DATA 6 MAXIMUM LIKELIHOOD FOR SPATIALLY CORRELATED DISCRETE DATA Gauge Plots TuscaroraLisa Central Madsen Fairways, 996 January 9, 7 Grubs Adult Activity Grub Counts 6 8 Organic Matter
More informationProblems. Suppose both models are fitted to the same data. Show that SS Res, A SS Res, B
Simple Linear Regression 35 Problems 1 Consider a set of data (x i, y i ), i =1, 2,,n, and the following two regression models: y i = β 0 + β 1 x i + ε, (i =1, 2,,n), Model A y i = γ 0 + γ 1 x i + γ 2
More informationNonparametric Bayesian Methods (Gaussian Processes)
[70240413 Statistical Machine Learning, Spring, 2015] Nonparametric Bayesian Methods (Gaussian Processes) Jun Zhu dcszj@mail.tsinghua.edu.cn http://bigml.cs.tsinghua.edu.cn/~jun State Key Lab of Intelligent
More informationCS281 Section 4: Factor Analysis and PCA
CS81 Section 4: Factor Analysis and PCA Scott Linderman At this point we have seen a variety of machine learning models, with a particular emphasis on models for supervised learning. In particular, we
More informationPeter Hoff Linear and multilinear models April 3, GLS for multivariate regression 5. 3 Covariance estimation for the GLM 8
Contents 1 Linear model 1 2 GLS for multivariate regression 5 3 Covariance estimation for the GLM 8 4 Testing the GLH 11 A reference for some of this material can be found somewhere. 1 Linear model Recall
More informationSaddlepoint-Based Bootstrap Inference in Dependent Data Settings
Saddlepoint-Based Bootstrap Inference in Dependent Data Settings Alex Trindade Dept. of Mathematics & Statistics, Texas Tech University Rob Paige, Missouri University of Science and Technology Indika Wickramasinghe,
More informationStat 5101 Lecture Notes
Stat 5101 Lecture Notes Charles J. Geyer Copyright 1998, 1999, 2000, 2001 by Charles J. Geyer May 7, 2001 ii Stat 5101 (Geyer) Course Notes Contents 1 Random Variables and Change of Variables 1 1.1 Random
More informationLinear models. Rasmus Waagepetersen Department of Mathematics Aalborg University Denmark. October 5, 2016
Linear models Rasmus Waagepetersen Department of Mathematics Aalborg University Denmark October 5, 2016 1 / 16 Outline for today linear models least squares estimation orthogonal projections estimation
More informationMatrices and Multivariate Statistics - II
Matrices and Multivariate Statistics - II Richard Mott November 2011 Multivariate Random Variables Consider a set of dependent random variables z = (z 1,..., z n ) E(z i ) = µ i cov(z i, z j ) = σ ij =
More informationSTAT5044: Regression and Anova
STAT5044: Regression and Anova Inyoung Kim 1 / 15 Outline 1 Fitting GLMs 2 / 15 Fitting GLMS We study how to find the maxlimum likelihood estimator ˆβ of GLM parameters The likelihood equaions are usually
More informationWeighted Least Squares
Weighted Least Squares The standard linear model assumes that Var(ε i ) = σ 2 for i = 1,..., n. As we have seen, however, there are instances where Var(Y X = x i ) = Var(ε i ) = σ2 w i. Here w 1,..., w
More informationDe-mystifying random effects models
De-mystifying random effects models Peter J Diggle Lecture 4, Leahurst, October 2012 Linear regression input variable x factor, covariate, explanatory variable,... output variable y response, end-point,
More informationOutline. Overview of Issues. Spatial Regression. Luc Anselin
Spatial Regression Luc Anselin University of Illinois, Urbana-Champaign http://www.spacestat.com Outline Overview of Issues Spatial Regression Specifications Space-Time Models Spatial Latent Variable Models
More information1 Mixed effect models and longitudinal data analysis
1 Mixed effect models and longitudinal data analysis Mixed effects models provide a flexible approach to any situation where data have a grouping structure which introduces some kind of correlation between
More informationFor more information about how to cite these materials visit
Author(s): Kerby Shedden, Ph.D., 2010 License: Unless otherwise noted, this material is made available under the terms of the Creative Commons Attribution Share Alike 3.0 License: http://creativecommons.org/licenses/by-sa/3.0/
More information11 : Gaussian Graphic Models and Ising Models
10-708: Probabilistic Graphical Models 10-708, Spring 2017 11 : Gaussian Graphic Models and Ising Models Lecturer: Bryon Aragam Scribes: Chao-Ming Yen 1 Introduction Different from previous maximum likelihood
More informationDecomposable and Directed Graphical Gaussian Models
Decomposable Decomposable and Directed Graphical Gaussian Models Graphical Models and Inference, Lecture 13, Michaelmas Term 2009 November 26, 2009 Decomposable Definition Basic properties Wishart density
More informationNon-stationary Cross-Covariance Models for Multivariate Processes on a Globe
Scandinavian Journal of Statistics, Vol. 38: 726 747, 2011 doi: 10.1111/j.1467-9469.2011.00751.x Published by Blackwell Publishing Ltd. Non-stationary Cross-Covariance Models for Multivariate Processes
More informationRegression, Ridge Regression, Lasso
Regression, Ridge Regression, Lasso Fabio G. Cozman - fgcozman@usp.br October 2, 2018 A general definition Regression studies the relationship between a response variable Y and covariates X 1,..., X n.
More informationMassachusetts Institute of Technology Department of Electrical Engineering and Computer Science Algorithms For Inference Fall 2014
Massachusetts Institute of Technology Department of Electrical Engineering and Computer Science 6.438 Algorithms For Inference Fall 2014 Problem Set 3 Issued: Thursday, September 25, 2014 Due: Thursday,
More informationNonstationary cross-covariance models for multivariate processes on a globe
Nonstationary cross-covariance models for multivariate processes on a globe Mikyoung Jun 1 April 15, 2011 Abstract: In geophysical and environmental problems, it is common to have multiple variables of
More informationLecture 3 September 1
STAT 383C: Statistical Modeling I Fall 2016 Lecture 3 September 1 Lecturer: Purnamrita Sarkar Scribe: Giorgio Paulon, Carlos Zanini Disclaimer: These scribe notes have been slightly proofread and may have
More informationMarkov random fields. The Markov property
Markov random fields The Markov property Discrete time: (X k X k!1,x k!2,... = (X k X k!1 A time symmetric version: (X k! X!k = (X k X k!1,x k+1 A more general version: Let A be a set of indices >k, B
More information[y i α βx i ] 2 (2) Q = i=1
Least squares fits This section has no probability in it. There are no random variables. We are given n points (x i, y i ) and want to find the equation of the line that best fits them. We take the equation
More information10. Time series regression and forecasting
10. Time series regression and forecasting Key feature of this section: Analysis of data on a single entity observed at multiple points in time (time series data) Typical research questions: What is the
More informationSTA442/2101: Assignment 5
STA442/2101: Assignment 5 Craig Burkett Quiz on: Oct 23 rd, 2015 The questions are practice for the quiz next week, and are not to be handed in. I would like you to bring in all of the code you used to
More informationECE521 week 3: 23/26 January 2017
ECE521 week 3: 23/26 January 2017 Outline Probabilistic interpretation of linear regression - Maximum likelihood estimation (MLE) - Maximum a posteriori (MAP) estimation Bias-variance trade-off Linear
More informationUCSD ECE153 Handout #30 Prof. Young-Han Kim Thursday, May 15, Homework Set #6 Due: Thursday, May 22, 2011
UCSD ECE153 Handout #30 Prof. Young-Han Kim Thursday, May 15, 2014 Homework Set #6 Due: Thursday, May 22, 2011 1. Linear estimator. Consider a channel with the observation Y = XZ, where the signal X and
More informationStat260: Bayesian Modeling and Inference Lecture Date: February 10th, Jeffreys priors. exp 1 ) p 2
Stat260: Bayesian Modeling and Inference Lecture Date: February 10th, 2010 Jeffreys priors Lecturer: Michael I. Jordan Scribe: Timothy Hunter 1 Priors for the multivariate Gaussian Consider a multivariate
More informationRegression Models - Introduction
Regression Models - Introduction In regression models there are two types of variables that are studied: A dependent variable, Y, also called response variable. It is modeled as random. An independent
More informationEconometrics. Week 4. Fall Institute of Economic Studies Faculty of Social Sciences Charles University in Prague
Econometrics Week 4 Institute of Economic Studies Faculty of Social Sciences Charles University in Prague Fall 2012 1 / 23 Recommended Reading For the today Serial correlation and heteroskedasticity in
More informationSTAT5044: Regression and Anova. Inyoung Kim
STAT5044: Regression and Anova Inyoung Kim 2 / 47 Outline 1 Regression 2 Simple Linear regression 3 Basic concepts in regression 4 How to estimate unknown parameters 5 Properties of Least Squares Estimators:
More informationMS&E 226: Small Data
MS&E 226: Small Data Lecture 12: Frequentist properties of estimators (v4) Ramesh Johari ramesh.johari@stanford.edu 1 / 39 Frequentist inference 2 / 39 Thinking like a frequentist Suppose that for some
More informationSTAT Financial Time Series
STAT 6104 - Financial Time Series Chapter 4 - Estimation in the time Domain Chun Yip Yau (CUHK) STAT 6104:Financial Time Series 1 / 46 Agenda 1 Introduction 2 Moment Estimates 3 Autoregressive Models (AR
More informationMATH 829: Introduction to Data Mining and Analysis Graphical Models I
MATH 829: Introduction to Data Mining and Analysis Graphical Models I Dominique Guillot Departments of Mathematical Sciences University of Delaware May 2, 2016 1/12 Independence and conditional independence:
More informationSTAT 135 Lab 13 (Review) Linear Regression, Multivariate Random Variables, Prediction, Logistic Regression and the δ-method.
STAT 135 Lab 13 (Review) Linear Regression, Multivariate Random Variables, Prediction, Logistic Regression and the δ-method. Rebecca Barter May 5, 2015 Linear Regression Review Linear Regression Review
More informationGaussian Processes 1. Schedule
1 Schedule 17 Jan: Gaussian processes (Jo Eidsvik) 24 Jan: Hands-on project on Gaussian processes (Team effort, work in groups) 31 Jan: Latent Gaussian models and INLA (Jo Eidsvik) 7 Feb: Hands-on project
More informationLinear Methods for Prediction
Chapter 5 Linear Methods for Prediction 5.1 Introduction We now revisit the classification problem and focus on linear methods. Since our prediction Ĝ(x) will always take values in the discrete set G we
More informationX t = a t + r t, (7.1)
Chapter 7 State Space Models 71 Introduction State Space models, developed over the past 10 20 years, are alternative models for time series They include both the ARIMA models of Chapters 3 6 and the Classical
More informationBayesian spatial hierarchical modeling for temperature extremes
Bayesian spatial hierarchical modeling for temperature extremes Indriati Bisono Dr. Andrew Robinson Dr. Aloke Phatak Mathematics and Statistics Department The University of Melbourne Maths, Informatics
More informationLecture 34: Properties of the LSE
Lecture 34: Properties of the LSE The following results explain why the LSE is popular. Gauss-Markov Theorem Assume a general linear model previously described: Y = Xβ + E with assumption A2, i.e., Var(E
More informationSTAT 730 Chapter 4: Estimation
STAT 730 Chapter 4: Estimation Timothy Hanson Department of Statistics, University of South Carolina Stat 730: Multivariate Analysis 1 / 23 The likelihood We have iid data, at least initially. Each datum
More informationMultivariate Bayesian Linear Regression MLAI Lecture 11
Multivariate Bayesian Linear Regression MLAI Lecture 11 Neil D. Lawrence Department of Computer Science Sheffield University 21st October 2012 Outline Univariate Bayesian Linear Regression Multivariate
More informationivporbit:an R package to estimate the probit model with continuous endogenous regressors
MPRA Munich Personal RePEc Archive ivporbit:an R package to estimate the probit model with continuous endogenous regressors Taha Zaghdoudi University of Jendouba, Faculty of Law Economics and Management
More informationSTAT 100C: Linear models
STAT 100C: Linear models Arash A. Amini June 9, 2018 1 / 56 Table of Contents Multiple linear regression Linear model setup Estimation of β Geometric interpretation Estimation of σ 2 Hat matrix Gram matrix
More informationChapter 3 - Estimation by direct maximization of the likelihood
Chapter 3 - Estimation by direct maximization of the likelihood 02433 - Hidden Markov Models Martin Wæver Pedersen, Henrik Madsen Course week 3 MWP, compiled June 7, 2011 Recall: Recursive scheme for the
More informationLecture 6: Hypothesis Testing
Lecture 6: Hypothesis Testing Mauricio Sarrias Universidad Católica del Norte November 6, 2017 1 Moran s I Statistic Mandatory Reading Moran s I based on Cliff and Ord (1972) Kelijan and Prucha (2001)
More informationσ(a) = a N (x; 0, 1 2 ) dx. σ(a) = Φ(a) =
Until now we have always worked with likelihoods and prior distributions that were conjugate to each other, allowing the computation of the posterior distribution to be done in closed form. Unfortunately,
More informationLECTURE 2 LINEAR REGRESSION MODEL AND OLS
SEPTEMBER 29, 2014 LECTURE 2 LINEAR REGRESSION MODEL AND OLS Definitions A common question in econometrics is to study the effect of one group of variables X i, usually called the regressors, on another
More informationWeighted Least Squares
Weighted Least Squares The standard linear model assumes that Var(ε i ) = σ 2 for i = 1,..., n. As we have seen, however, there are instances where Var(Y X = x i ) = Var(ε i ) = σ2 w i. Here w 1,..., w
More informationAsymptotic Multivariate Kriging Using Estimated Parameters with Bayesian Prediction Methods for Non-linear Predictands
Asymptotic Multivariate Kriging Using Estimated Parameters with Bayesian Prediction Methods for Non-linear Predictands Elizabeth C. Mannshardt-Shamseldin Advisor: Richard L. Smith Duke University Department
More informationA time series is called strictly stationary if the joint distribution of every collection (Y t
5 Time series A time series is a set of observations recorded over time. You can think for example at the GDP of a country over the years (or quarters) or the hourly measurements of temperature over a
More informationMIT Spring 2015
Regression Analysis MIT 18.472 Dr. Kempthorne Spring 2015 1 Outline Regression Analysis 1 Regression Analysis 2 Multiple Linear Regression: Setup Data Set n cases i = 1, 2,..., n 1 Response (dependent)
More informationLecture Notes 5 Convergence and Limit Theorems. Convergence with Probability 1. Convergence in Mean Square. Convergence in Probability, WLLN
Lecture Notes 5 Convergence and Limit Theorems Motivation Convergence with Probability Convergence in Mean Square Convergence in Probability, WLLN Convergence in Distribution, CLT EE 278: Convergence and
More informationWhat s for today. More on Binomial distribution Poisson distribution. c Mikyoung Jun (Texas A&M) stat211 lecture 7 February 8, / 16
What s for today More on Binomial distribution Poisson distribution c Mikyoung Jun (Texas A&M) stat211 lecture 7 February 8, 2011 1 / 16 Review: Binomial distribution Question: among the following, what
More informationStatistics for analyzing and modeling precipitation isotope ratios in IsoMAP
Statistics for analyzing and modeling precipitation isotope ratios in IsoMAP The IsoMAP uses the multiple linear regression and geostatistical methods to analyze isotope data Suppose the response variable
More informationEconomics 536 Lecture 7. Introduction to Specification Testing in Dynamic Econometric Models
University of Illinois Fall 2016 Department of Economics Roger Koenker Economics 536 Lecture 7 Introduction to Specification Testing in Dynamic Econometric Models In this lecture I want to briefly describe
More informationGeneralized Linear Models. Kurt Hornik
Generalized Linear Models Kurt Hornik Motivation Assuming normality, the linear model y = Xβ + e has y = β + ε, ε N(0, σ 2 ) such that y N(μ, σ 2 ), E(y ) = μ = β. Various generalizations, including general
More informationBIO5312 Biostatistics Lecture 13: Maximum Likelihood Estimation
BIO5312 Biostatistics Lecture 13: Maximum Likelihood Estimation Yujin Chung November 29th, 2016 Fall 2016 Yujin Chung Lec13: MLE Fall 2016 1/24 Previous Parametric tests Mean comparisons (normality assumption)
More informationMultivariate Regression Analysis
Matrices and vectors The model from the sample is: Y = Xβ +u with n individuals, l response variable, k regressors Y is a n 1 vector or a n l matrix with the notation Y T = (y 1,y 2,...,y n ) 1 x 11 x
More informationChapter 12: An introduction to Time Series Analysis. Chapter 12: An introduction to Time Series Analysis
Chapter 12: An introduction to Time Series Analysis Introduction In this chapter, we will discuss forecasting with single-series (univariate) Box-Jenkins models. The common name of the models is Auto-Regressive
More informationRegression diagnostics
Regression diagnostics Kerby Shedden Department of Statistics, University of Michigan November 5, 018 1 / 6 Motivation When working with a linear model with design matrix X, the conventional linear model
More informationMean square continuity
Mean square continuity Suppose Z is a random field on R d We say Z is mean square continuous at s if lim E{Z(x) x s Z(s)}2 = 0 If Z is stationary, Z is mean square continuous at s if and only if K is continuous
More informationProblem Selected Scores
Statistics Ph.D. Qualifying Exam: Part II November 20, 2010 Student Name: 1. Answer 8 out of 12 problems. Mark the problems you selected in the following table. Problem 1 2 3 4 5 6 7 8 9 10 11 12 Selected
More informationLecture 20: Linear model, the LSE, and UMVUE
Lecture 20: Linear model, the LSE, and UMVUE Linear Models One of the most useful statistical models is X i = β τ Z i + ε i, i = 1,...,n, where X i is the ith observation and is often called the ith response;
More informationLinear Regression. Junhui Qian. October 27, 2014
Linear Regression Junhui Qian October 27, 2014 Outline The Model Estimation Ordinary Least Square Method of Moments Maximum Likelihood Estimation Properties of OLS Estimator Unbiasedness Consistency Efficiency
More informationRegression Models - Introduction
Regression Models - Introduction In regression models, two types of variables that are studied: A dependent variable, Y, also called response variable. It is modeled as random. An independent variable,
More informationCross-covariance Functions for Tangent Vector Fields on the Sphere
Cross-covariance Functions for Tangent Vector Fields on the Sphere Minjie Fan 1 Tomoko Matsuo 2 1 Department of Statistics University of California, Davis 2 Cooperative Institute for Research in Environmental
More informationHeteroskedasticity. Part VII. Heteroskedasticity
Part VII Heteroskedasticity As of Oct 15, 2015 1 Heteroskedasticity Consequences Heteroskedasticity-robust inference Testing for Heteroskedasticity Weighted Least Squares (WLS) Feasible generalized Least
More informationGLS and FGLS. Econ 671. Purdue University. Justin L. Tobias (Purdue) GLS and FGLS 1 / 22
GLS and FGLS Econ 671 Purdue University Justin L. Tobias (Purdue) GLS and FGLS 1 / 22 In this lecture we continue to discuss properties associated with the GLS estimator. In addition we discuss the practical
More informationOpen Problems in Mixed Models
xxiii Determining how to deal with a not positive definite covariance matrix of random effects, D during maximum likelihood estimation algorithms. Several strategies are discussed in Section 2.15. For
More informationSpatial Process Estimates as Smoothers: A Review
Spatial Process Estimates as Smoothers: A Review Soutir Bandyopadhyay 1 Basic Model The observational model considered here has the form Y i = f(x i ) + ɛ i, for 1 i n. (1.1) where Y i is the observed
More informationVariations. ECE 6540, Lecture 10 Maximum Likelihood Estimation
Variations ECE 6540, Lecture 10 Last Time BLUE (Best Linear Unbiased Estimator) Formulation Advantages Disadvantages 2 The BLUE A simplification Assume the estimator is a linear system For a single parameter
More informationSpatio-Temporal Modelling of Credit Default Data
1/20 Spatio-Temporal Modelling of Credit Default Data Sathyanarayan Anand Advisor: Prof. Robert Stine The Wharton School, University of Pennsylvania April 29, 2011 2/20 Outline 1 Background 2 Conditional
More informationProperties of the least squares estimates
Properties of the least squares estimates 2019-01-18 Warmup Let a and b be scalar constants, and X be a scalar random variable. Fill in the blanks E ax + b) = Var ax + b) = Goal Recall that the least squares
More informationLecture 18. Models for areal data. Colin Rundel 03/22/2017
Lecture 18 Models for areal data Colin Rundel 03/22/2017 1 areal / lattice data 2 Example - NC SIDS SID79 3 EDA - Moran s I If we have observations at n spatial locations s 1,... s n ) I = n i=1 n n j=1
More informationBayesian Linear Regression
Bayesian Linear Regression Sudipto Banerjee 1 Biostatistics, School of Public Health, University of Minnesota, Minneapolis, Minnesota, U.S.A. September 15, 2010 1 Linear regression models: a Bayesian perspective
More informationParameter estimation: ACVF of AR processes
Parameter estimation: ACVF of AR processes Yule-Walker s for AR processes: a method of moments, i.e. µ = x and choose parameters so that γ(h) = ˆγ(h) (for h small ). 12 novembre 2013 1 / 8 Parameter estimation:
More informationAssociation studies and regression
Association studies and regression CM226: Machine Learning for Bioinformatics. Fall 2016 Sriram Sankararaman Acknowledgments: Fei Sha, Ameet Talwalkar Association studies and regression 1 / 104 Administration
More informationSpecification Errors, Measurement Errors, Confounding
Specification Errors, Measurement Errors, Confounding Kerby Shedden Department of Statistics, University of Michigan October 10, 2018 1 / 32 An unobserved covariate Suppose we have a data generating model
More informationTechnical Vignette 5: Understanding intrinsic Gaussian Markov random field spatial models, including intrinsic conditional autoregressive models
Technical Vignette 5: Understanding intrinsic Gaussian Markov random field spatial models, including intrinsic conditional autoregressive models Christopher Paciorek, Department of Statistics, University
More informationSummer School in Statistics for Astronomers V June 1 - June 6, Regression. Mosuk Chow Statistics Department Penn State University.
Summer School in Statistics for Astronomers V June 1 - June 6, 2009 Regression Mosuk Chow Statistics Department Penn State University. Adapted from notes prepared by RL Karandikar Mean and variance Recall
More informationProbabilistic Graphical Models
Probabilistic Graphical Models Lecture 11 CRFs, Exponential Family CS/CNS/EE 155 Andreas Krause Announcements Homework 2 due today Project milestones due next Monday (Nov 9) About half the work should
More informationMCMC algorithms for fitting Bayesian models
MCMC algorithms for fitting Bayesian models p. 1/1 MCMC algorithms for fitting Bayesian models Sudipto Banerjee sudiptob@biostat.umn.edu University of Minnesota MCMC algorithms for fitting Bayesian models
More informationSTAT 518 Intro Student Presentation
STAT 518 Intro Student Presentation Wen Wei Loh April 11, 2013 Title of paper Radford M. Neal [1999] Bayesian Statistics, 6: 475-501, 1999 What the paper is about Regression and Classification Flexible
More informationCovariance and Correlation
Covariance and Correlation ST 370 The probability distribution of a random variable gives complete information about its behavior, but its mean and variance are useful summaries. Similarly, the joint probability
More informationCox s proportional hazards model and Cox s partial likelihood
Cox s proportional hazards model and Cox s partial likelihood Rasmus Waagepetersen October 12, 2018 1 / 27 Non-parametric vs. parametric Suppose we want to estimate unknown function, e.g. survival function.
More information