Computing MLE Bias Empirically
|
|
- Imogene Osborne
- 5 years ago
- Views:
Transcription
1 Computng MLE Bas Emprcally Kar Wa Lm Australan atonal Unversty January 3, 27 Abstract Ths note studes the bas arses from the MLE estmate of the rate parameter and the mean parameter of an exponental dstrbuton. Motvaton Although maxmum lkelhood estmaton (MLE) methods provde estmates that are useful, the estmates themselves are not guaranteed to be unbased. evertheless, MLE methods are stll hghly regarded n practce due to several of ther propertes, notably, the estmates are consstent and asymptotcally normal (Casella and Berger, 22; Panchenko, 26). The most popular example that llustrates the bas of the MLE methods s the MLE estmate of the varance parameter σ 2 of a normal dstrbuton (µ, σ 2 ), we refer the readers to Lang (22) for detals. Another example that s of nterest s that of an exponental dstrbuton. In ths case, the MLE estmate of the rate parameter λ of an exponental dstrbuton Exp(λ) s based, however, the MLE estmate for the mean parameter µ = /λ s unbased. Thus, the exponental dstrbuton makes a good case study for understandng the MLE bas. In ths note, we attempt to quantfy the bas of the MLE estmates emprcally through smulatons. For ths purpose, we wll use the exponental dstrbuton as example. 2 MLE for Exponental Dstrbuton In ths secton, we provde a bref dervaton of the MLE estmate of the rate parameter λ and the mean parameter µ of an exponental dstrbuton. We note that MLE estmates are values that maxmse the lkelhood (probablty densty functon) or loglkelhood of the observed data. Let {x } be..d. random varables that are exponentally dstrbuted, wrtten as The lkelhood functon assocated wth X = {x } can be wrtten as wth the followng log lkelhood: x Exp(λ). () p(x) = λ e λ x, (2) = L(λ, X) = log λ λ x. (3) =
2 Solvng for the MLE estmate ˆλ = arg mn λ L(λ, X) gves us ˆλ = (4) where s the mean of {x }: = x (5) = We note that by usng the nvarant property of the MLE, the MLE estmate ˆµ s smply 2. Bas of the MLE Estmates ˆµ =. (6) An estmate s unbased f the expectaton of the estmate equals to ts true value. We frst note that the MLE estmate ˆµ s unbased, evdenced by E[ˆµ] = E[] = For the MLE estmate ˆλ, ts expectaton s [ ] E[ˆλ] = E E[x ] = µ. (7) = E[] = λ. (8) The nequalty follows from Jensen s nequalty wth the convex functon f(x) = /x. To quantfy the bas of ˆλ, we can derve the bas B(ˆλ) drectly usng the propertes of gamma dstrbuton. Frst, we note that the exponental dstrbuton s a specal case of the gamma dstrbuton wth shape parameter, that s, x Gamma(, λ). (9) Snce the summaton of gamma random varables s also gamma dstrbuted (see Proposton 3., Taylor, 29), we have x Gamma(, λ). () = otce that the nverse of a gamma random varable s nverse-gamma dstrbuted (Wkpeda, 26): = x InvGamma(, λ), () wth expectaton [ ] E = x = λ, for >. (2) Wth ths establshed, we can derve the bas of the MLE estmate ˆλ as follows: [ ] [ ] B(ˆλ) = E λ = E = x λ = λ λ = λ. (3) We can see that the bas approaches zero as ncreases. 2
3 3 Emprcal Bas for Exponental Dstrbuton In ths secton, we perform experments to evaluate the bas of the MLE estmates emprcally through Monte Carlo method. The emprcally bas of an estmate ˆλ can be computed as ˆB(ˆλ) = M M ˆλ (j) λ, (4) j= where ˆλ (j) s the MLE estmate for λ n the j-th smulaton experment. To be precse, notng that x (j) ˆλ (j) = s the -th sample of the j-th smulaton experment., (5) = x(j) In the followng, we compute ˆB(ˆλ) by varyng from to and settng M =. For smplcty, we let λ = µ =. The experments are performed usng Matlab. Fgure below plots the mean of the MLE estmates ˆλ aganst the number of samples used n the experments. The mean ˆλ s computed as follows: ˆλ = M M ˆλ (j), (6) j= wth varyng. From the followng plot, we can see that the mean of the MLE estmates devate from ts true value. However, the devaton decreases as the number of samples ncreases..5.4 MLE Estmates of λ vs Ground Truth MLE estmate Ground truth.3 λ ext, we plot the emprcal bas of the MLE estmator ˆλ (blue) and ts theoretcal counterpart (red) vs the number of samples n the fgure below. Here, we can see that the emprcal bas fluctuates around ts theoretcal values. Further, we also supermpose the emprcal bas of the MLE estmate ˆµ (green) for llustratve purpose. Snce the MLE estmator for µ s unbased, the emprcal bas fluctuates around (yellow). 3
4 -3 5 Emprcal Bas vs Theoretcal Bas of the MLE Estmator Bas of estmator µ Bas of estmator λ Theoretcal bas for µ Theoretcal bas for λ Bas We note that the fluctuatons come from two ndependent sources. Frstly, the fluctuaton s due to varablty wthn the MLE estmates, that s, V[ˆλ]. The second source of varablty comes from the Monte Carlo smulatons, whch approaches zero only when M. To llustrate, n the fgure below, we compare the standard errors of ˆB(ˆλ) aganst ther theoretcal values, gven as [ ] [ V[ˆλ] = V = V = x ] = 2 λ 2 ( ) 2, for > 2. (7) ( 2) Ths corresponds to the frst source of varablty. The varablty from the Monte Carlo smulaton s n fact smaller, whch s gven by V[ ˆB(ˆλ)] = M V[ˆλ] = 2 λ 2 M( ) 2, for > 2. (8) ( 2) Emprcal Standard Error vs Theoretcal Standard Devaton of the MLE Estmator.7 Emprcal standard error Theoretcal standard devaton.6.5 Standard Error We note that the standard errors computed for the estmator ˆµ have smlar values as those of ˆλ (thus not plotted). Ths s because ther varances are very close to one another (due to λ = µ) for large : V[ˆµ] = V [ ] = V [ = x ] = λ 2 = µ2. (9) 4
5 3. Bas Correcton To recap, the expected value of the MLE estmator ˆλ s λ. We can smply multply a correcton factor to the MLE estmator to elmnate the bas, ths gves us an unbased estmator ˆλ = = = x. (2) In the followng graph, we compute the emprcal bas of the adjusted estmator. 5 x Emprcal Bas vs Theoretcal Bas of the Corrected Estmator 3 Bas of estmator λ * 4 Theoretcal bas for λ * 3 2 Bas As expected, the MLE estmates centred around, verfyng that the estmator ˆλ s unbased. We note that the varance of the estmator s V[ˆλ ] = ( )2 λ 2 2 V[ˆλ] =, for > 2, (2) ( 2) whch s slghtly lower than that of the MLE estmator ˆλ, ths corrected estmator s thus better n terms of both bas and varablty. References Casella, G. and Berger, R. L. (22). Statstcal Inference, volume 2. Duxbury Pacfc Grove, CA. Lang, D. (22). Maxmum lkelhood estmator for varance s based: Proof. Retreved from Panchenko, D. (26). Lecture 3 Propertes of MLE: consstency, asymptotc normalty. Fsher nformaton. Retreved from statstcs-for-applcatons-fall-26/lecture-notes/ lecture3.pdf. Taylor, J. (29). Lecture 2: Sums of ndependent random varables. Retreved from jtaylor/teachng/fall2/stp42/ lectures/lecture2.pdf. Wkpeda (26). Inverse-gamma dstrbuton. Retreved from org/wk/inverse-gamma_dstrbuton. 5
Econ107 Applied Econometrics Topic 3: Classical Model (Studenmund, Chapter 4)
I. Classcal Assumptons Econ7 Appled Econometrcs Topc 3: Classcal Model (Studenmund, Chapter 4) We have defned OLS and studed some algebrac propertes of OLS. In ths topc we wll study statstcal propertes
More informationMATH 829: Introduction to Data Mining and Analysis The EM algorithm (part 2)
1/16 MATH 829: Introducton to Data Mnng and Analyss The EM algorthm (part 2) Domnque Gullot Departments of Mathematcal Scences Unversty of Delaware Aprl 20, 2016 Recall 2/16 We are gven ndependent observatons
More informationANSWERS. Problem 1. and the moment generating function (mgf) by. defined for any real t. Use this to show that E( U) var( U)
Econ 413 Exam 13 H ANSWERS Settet er nndelt 9 deloppgaver, A,B,C, som alle anbefales å telle lkt for å gøre det ltt lettere å stå. Svar er gtt . Unfortunately, there s a prntng error n the hnt of
More informationParametric fractional imputation for missing data analysis. Jae Kwang Kim Survey Working Group Seminar March 29, 2010
Parametrc fractonal mputaton for mssng data analyss Jae Kwang Km Survey Workng Group Semnar March 29, 2010 1 Outlne Introducton Proposed method Fractonal mputaton Approxmaton Varance estmaton Multple mputaton
More informationMASSACHUSETTS INSTITUTE OF TECHNOLOGY 6.265/15.070J Fall 2013 Lecture 12 10/21/2013. Martingale Concentration Inequalities and Applications
MASSACHUSETTS INSTITUTE OF TECHNOLOGY 6.65/15.070J Fall 013 Lecture 1 10/1/013 Martngale Concentraton Inequaltes and Applcatons Content. 1. Exponental concentraton for martngales wth bounded ncrements.
More informationJoint Statistical Meetings - Biopharmaceutical Section
Iteratve Ch-Square Test for Equvalence of Multple Treatment Groups Te-Hua Ng*, U.S. Food and Drug Admnstraton 1401 Rockvlle Pke, #200S, HFM-217, Rockvlle, MD 20852-1448 Key Words: Equvalence Testng; Actve
More informationLecture 4: September 12
36-755: Advanced Statstcal Theory Fall 016 Lecture 4: September 1 Lecturer: Alessandro Rnaldo Scrbe: Xao Hu Ta Note: LaTeX template courtesy of UC Berkeley EECS dept. Dsclamer: These notes have not been
More informationWhich estimator of the dispersion parameter for the Gamma family generalized linear models is to be chosen?
STATISTICS Dalarna Unversty D-level Master s Thess 007 Whch estmator of the dsperson parameter for the Gamma famly generalzed lnear models s to be chosen? Submtted by: Juan Du Regstraton Number: 8096-T084
More informationAsymptotic Efficiencies of the MLE Based on Bivariate Record Values from Bivariate Normal Distribution
JIRSS 013 Vol. 1, No., pp 35-5 Downloaded from jrss.rstat.r at 1:45 +0430 on Monday September 17th 018 Asymptotc ffcences of the ML Based on Bvarate Record Values from Bvarate Normal Dstrbuton Morteza
More informationA nonparametric two-sample wald test of equality of variances
Unversty of Wollongong Research Onlne Centre for Statstcal & Survey Methodology Workng Paper Seres Faculty of Engneerng and Informaton Scences 0 A nonparametrc two-sample wald test of equalty of varances
More informationLecture 4 Hypothesis Testing
Lecture 4 Hypothess Testng We may wsh to test pror hypotheses about the coeffcents we estmate. We can use the estmates to test whether the data rejects our hypothess. An example mght be that we wsh to
More informationU-Pb Geochronology Practical: Background
U-Pb Geochronology Practcal: Background Basc Concepts: accuracy: measure of the dfference between an expermental measurement and the true value precson: measure of the reproducblty of the expermental result
More informationAn (almost) unbiased estimator for the S-Gini index
An (almost unbased estmator for the S-Gn ndex Thomas Demuynck February 25, 2009 Abstract Ths note provdes an unbased estmator for the absolute S-Gn and an almost unbased estmator for the relatve S-Gn for
More informationF71SM1 STATISTICAL METHODS TUTORIAL ON 7 ESTIMATION SOLUTIONS
F7SM STATISTICAL METHODS TUTORIAL ON 7 ESTIMATION SOLUTIONS RJG. (a) E[X] = 0 xf(x)dx = θ 0 y e y dy = θγ(3) = θ [or note X gamma(, /θ) wth mean θ (from Yellow Book)] Settng X = θ MME θ = X/ (b) L(θ) =
More informationSupplementary material: Margin based PU Learning. Matrix Concentration Inequalities
Supplementary materal: Margn based PU Learnng We gve the complete proofs of Theorem and n Secton We frst ntroduce the well-known concentraton nequalty, so the covarance estmator can be bounded Then we
More informationSTATS 306B: Unsupervised Learning Spring Lecture 10 April 30
STATS 306B: Unsupervsed Learnng Sprng 2014 Lecture 10 Aprl 30 Lecturer: Lester Mackey Scrbe: Joey Arthur, Rakesh Achanta 10.1 Factor Analyss 10.1.1 Recap Recall the factor analyss (FA) model for lnear
More informationLogistic Regression. CAP 5610: Machine Learning Instructor: Guo-Jun QI
Logstc Regresson CAP 561: achne Learnng Instructor: Guo-Jun QI Bayes Classfer: A Generatve model odel the posteror dstrbuton P(Y X) Estmate class-condtonal dstrbuton P(X Y) for each Y Estmate pror dstrbuton
More informationParameters Estimation of the Modified Weibull Distribution Based on Type I Censored Samples
Appled Mathematcal Scences, Vol. 5, 011, no. 59, 899-917 Parameters Estmaton of the Modfed Webull Dstrbuton Based on Type I Censored Samples Soufane Gasm École Supereure des Scences et Technques de Tuns
More informationGoodness of fit and Wilks theorem
DRAFT 0.0 Glen Cowan 3 June, 2013 Goodness of ft and Wlks theorem Suppose we model data y wth a lkelhood L(µ) that depends on a set of N parameters µ = (µ 1,...,µ N ). Defne the statstc t µ ln L(µ) L(ˆµ),
More information/ n ) are compared. The logic is: if the two
STAT C141, Sprng 2005 Lecture 13 Two sample tests One sample tests: examples of goodness of ft tests, where we are testng whether our data supports predctons. Two sample tests: called as tests of ndependence
More informationP exp(tx) = 1 + t 2k M 2k. k N
1. Subgaussan tals Defnton. Say that a random varable X has a subgaussan dstrbuton wth scale factor σ< f P exp(tx) exp(σ 2 t 2 /2) for all real t. For example, f X s dstrbuted N(,σ 2 ) then t s subgaussan.
More informationj) = 1 (note sigma notation) ii. Continuous random variable (e.g. Normal distribution) 1. density function: f ( x) 0 and f ( x) dx = 1
Random varables Measure of central tendences and varablty (means and varances) Jont densty functons and ndependence Measures of assocaton (covarance and correlaton) Interestng result Condtonal dstrbutons
More informationEfficient nonresponse weighting adjustment using estimated response probability
Effcent nonresponse weghtng adjustment usng estmated response probablty Jae Kwang Km Department of Appled Statstcs, Yonse Unversty, Seoul, 120-749, KOREA Key Words: Regresson estmator, Propensty score,
More informationDurban Watson for Testing the Lack-of-Fit of Polynomial Regression Models without Replications
Durban Watson for Testng the Lack-of-Ft of Polynomal Regresson Models wthout Replcatons Ruba A. Alyaf, Maha A. Omar, Abdullah A. Al-Shha ralyaf@ksu.edu.sa, maomar@ksu.edu.sa, aalshha@ksu.edu.sa Department
More informationSee Book Chapter 11 2 nd Edition (Chapter 10 1 st Edition)
Count Data Models See Book Chapter 11 2 nd Edton (Chapter 10 1 st Edton) Count data consst of non-negatve nteger values Examples: number of drver route changes per week, the number of trp departure changes
More information4 Analysis of Variance (ANOVA) 5 ANOVA. 5.1 Introduction. 5.2 Fixed Effects ANOVA
4 Analyss of Varance (ANOVA) 5 ANOVA 51 Introducton ANOVA ANOVA s a way to estmate and test the means of multple populatons We wll start wth one-way ANOVA If the populatons ncluded n the study are selected
More informationEstimation: Part 2. Chapter GREG estimation
Chapter 9 Estmaton: Part 2 9. GREG estmaton In Chapter 8, we have seen that the regresson estmator s an effcent estmator when there s a lnear relatonshp between y and x. In ths chapter, we generalzed the
More informationStat260: Bayesian Modeling and Inference Lecture Date: February 22, Reference Priors
Stat60: Bayesan Modelng and Inference Lecture Date: February, 00 Reference Prors Lecturer: Mchael I. Jordan Scrbe: Steven Troxler and Wayne Lee In ths lecture, we assume that θ R; n hgher-dmensons, reference
More informationConvergence of random processes
DS-GA 12 Lecture notes 6 Fall 216 Convergence of random processes 1 Introducton In these notes we study convergence of dscrete random processes. Ths allows to characterze phenomena such as the law of large
More informationENG 8801/ Special Topics in Computer Engineering: Pattern Recognition. Memorial University of Newfoundland Pattern Recognition
EG 880/988 - Specal opcs n Computer Engneerng: Pattern Recognton Memoral Unversty of ewfoundland Pattern Recognton Lecture 7 May 3, 006 http://wwwengrmunca/~charlesr Offce Hours: uesdays hursdays 8:30-9:30
More informationLecture 3. Ax x i a i. i i
18.409 The Behavor of Algorthms n Practce 2/14/2 Lecturer: Dan Spelman Lecture 3 Scrbe: Arvnd Sankar 1 Largest sngular value In order to bound the condton number, we need an upper bound on the largest
More informationLinear Regression Analysis: Terminology and Notation
ECON 35* -- Secton : Basc Concepts of Regresson Analyss (Page ) Lnear Regresson Analyss: Termnology and Notaton Consder the generc verson of the smple (two-varable) lnear regresson model. It s represented
More informationLecture 10 Support Vector Machines II
Lecture 10 Support Vector Machnes II 22 February 2016 Taylor B. Arnold Yale Statstcs STAT 365/665 1/28 Notes: Problem 3 s posted and due ths upcomng Frday There was an early bug n the fake-test data; fxed
More informationComposite Hypotheses testing
Composte ypotheses testng In many hypothess testng problems there are many possble dstrbutons that can occur under each of the hypotheses. The output of the source s a set of parameters (ponts n a parameter
More informationprinceton univ. F 13 cos 521: Advanced Algorithm Design Lecture 3: Large deviations bounds and applications Lecturer: Sanjeev Arora
prnceton unv. F 13 cos 521: Advanced Algorthm Desgn Lecture 3: Large devatons bounds and applcatons Lecturer: Sanjeev Arora Scrbe: Today s topc s devaton bounds: what s the probablty that a random varable
More informationECONOMICS 351*-A Mid-Term Exam -- Fall Term 2000 Page 1 of 13 pages. QUEEN'S UNIVERSITY AT KINGSTON Department of Economics
ECOOMICS 35*-A Md-Term Exam -- Fall Term 000 Page of 3 pages QUEE'S UIVERSITY AT KIGSTO Department of Economcs ECOOMICS 35* - Secton A Introductory Econometrcs Fall Term 000 MID-TERM EAM ASWERS MG Abbott
More informationEcon Statistical Properties of the OLS estimator. Sanjaya DeSilva
Econ 39 - Statstcal Propertes of the OLS estmator Sanjaya DeSlva September, 008 1 Overvew Recall that the true regresson model s Y = β 0 + β 1 X + u (1) Applyng the OLS method to a sample of data, we estmate
More informationNow we relax this assumption and allow that the error variance depends on the independent variables, i.e., heteroskedasticity
ECON 48 / WH Hong Heteroskedastcty. Consequences of Heteroskedastcty for OLS Assumpton MLR. 5: Homoskedastcty var ( u x ) = σ Now we relax ths assumpton and allow that the error varance depends on the
More informationParametric fractional imputation for missing data analysis
Secton on Survey Research Methods JSM 2008 Parametrc fractonal mputaton for mssng data analyss Jae Kwang Km Wayne Fuller Abstract Under a parametrc model for mssng data, the EM algorthm s a popular tool
More informationMaximum Likelihood Estimation of Binary Dependent Variables Models: Probit and Logit. 1. General Formulation of Binary Dependent Variables Models
ECO 452 -- OE 4: Probt and Logt Models ECO 452 -- OE 4 Maxmum Lkelhood Estmaton of Bnary Dependent Varables Models: Probt and Logt hs note demonstrates how to formulate bnary dependent varables models
More informationECONOMETRICS II (ECO 2401S) University of Toronto. Department of Economics. Winter 2017 Instructor: Victor Aguirregabiria
ECOOMETRICS II ECO 40S Unversty of Toronto Department of Economcs Wnter 07 Instructor: Vctor Agurregabra SOLUTIO TO FIAL EXAM Tuesday, Aprl 8, 07 From :00pm-5:00pm 3 hours ISTRUCTIOS: - Ths s a closed-book
More informationRockefeller College University at Albany
Rockefeller College Unverst at Alban PAD 705 Handout: Maxmum Lkelhood Estmaton Orgnal b Davd A. Wse John F. Kenned School of Government, Harvard Unverst Modfcatons b R. Karl Rethemeer Up to ths pont n
More informationNotes on Frequency Estimation in Data Streams
Notes on Frequency Estmaton n Data Streams In (one of) the data streamng model(s), the data s a sequence of arrvals a 1, a 2,..., a m of the form a j = (, v) where s the dentty of the tem and belongs to
More informationLOGIT ANALYSIS. A.K. VASISHT Indian Agricultural Statistics Research Institute, Library Avenue, New Delhi
LOGIT ANALYSIS A.K. VASISHT Indan Agrcultural Statstcs Research Insttute, Lbrary Avenue, New Delh-0 02 amtvassht@asr.res.n. Introducton In dummy regresson varable models, t s assumed mplctly that the dependent
More informationEstimation of the Mean of Truncated Exponential Distribution
Journal of Mathematcs and Statstcs 4 (4): 84-88, 008 ISSN 549-644 008 Scence Publcatons Estmaton of the Mean of Truncated Exponental Dstrbuton Fars Muslm Al-Athar Department of Mathematcs, Faculty of Scence,
More informationInterval Estimation in the Classical Normal Linear Regression Model. 1. Introduction
ECONOMICS 35* -- NOTE 7 ECON 35* -- NOTE 7 Interval Estmaton n the Classcal Normal Lnear Regresson Model Ths note outlnes the basc elements of nterval estmaton n the Classcal Normal Lnear Regresson Model
More informationThe EM Algorithm (Dempster, Laird, Rubin 1977) The missing data or incomplete data setting: ODL(φ;Y ) = [Y;φ] = [Y X,φ][X φ] = X
The EM Algorthm (Dempster, Lard, Rubn 1977 The mssng data or ncomplete data settng: An Observed Data Lkelhood (ODL that s a mxture or ntegral of Complete Data Lkelhoods (CDL. (1a ODL(;Y = [Y;] = [Y,][
More informationLecture 6 More on Complete Randomized Block Design (RBD)
Lecture 6 More on Complete Randomzed Block Desgn (RBD) Multple test Multple test The multple comparsons or multple testng problem occurs when one consders a set of statstcal nferences smultaneously. For
More information3.1 Expectation of Functions of Several Random Variables. )' be a k-dimensional discrete or continuous random vector, with joint PMF p (, E X E X1 E X
Statstcs 1: Probablty Theory II 37 3 EPECTATION OF SEVERAL RANDOM VARIABLES As n Probablty Theory I, the nterest n most stuatons les not on the actual dstrbuton of a random vector, but rather on a number
More informationLecture 3: Probability Distributions
Lecture 3: Probablty Dstrbutons Random Varables Let us begn by defnng a sample space as a set of outcomes from an experment. We denote ths by S. A random varable s a functon whch maps outcomes nto the
More informationPopulation Design in Nonlinear Mixed Effects Multiple Response Models: extension of PFIM and evaluation by simulation with NONMEM and MONOLIX
Populaton Desgn n Nonlnear Mxed Effects Multple Response Models: extenson of PFIM and evaluaton by smulaton wth NONMEM and MONOLIX May 4th 007 Carolne Bazzol, Sylve Retout, France Mentré Inserm U738 Unversty
More informationEconomics 130. Lecture 4 Simple Linear Regression Continued
Economcs 130 Lecture 4 Contnued Readngs for Week 4 Text, Chapter and 3. We contnue wth addressng our second ssue + add n how we evaluate these relatonshps: Where do we get data to do ths analyss? How do
More informationUncertainty as the Overlap of Alternate Conditional Distributions
Uncertanty as the Overlap of Alternate Condtonal Dstrbutons Olena Babak and Clayton V. Deutsch Centre for Computatonal Geostatstcs Department of Cvl & Envronmental Engneerng Unversty of Alberta An mportant
More informationLecture 12: Classification
Lecture : Classfcaton g Dscrmnant functons g The optmal Bayes classfer g Quadratc classfers g Eucldean and Mahalanobs metrcs g K Nearest Neghbor Classfers Intellgent Sensor Systems Rcardo Guterrez-Osuna
More informationEffective plots to assess bias and precision in method comparison studies
Effectve plots to assess bas and precson n method comparson studes Bern, November, 016 Patrck Taffé, PhD Insttute of Socal and Preventve Medcne () Unversty of Lausanne, Swtzerland Patrck.Taffe@chuv.ch
More informationFirst Year Examination Department of Statistics, University of Florida
Frst Year Examnaton Department of Statstcs, Unversty of Florda May 7, 010, 8:00 am - 1:00 noon Instructons: 1. You have four hours to answer questons n ths examnaton.. You must show your work to receve
More informationMaximum Likelihood Estimation
Maxmum Lkelhood Estmaton INFO-2301: Quanttatve Reasonng 2 Mchael Paul and Jordan Boyd-Graber MARCH 7, 2017 INFO-2301: Quanttatve Reasonng 2 Paul and Boyd-Graber Maxmum Lkelhood Estmaton 1 of 9 Why MLE?
More informationDr. Shalabh Department of Mathematics and Statistics Indian Institute of Technology Kanpur
Analyss of Varance and Desgn of Experment-I MODULE VIII LECTURE - 34 ANALYSIS OF VARIANCE IN RANDOM-EFFECTS MODEL AND MIXED-EFFECTS EFFECTS MODEL Dr Shalabh Department of Mathematcs and Statstcs Indan
More informationExpected Value and Variance
MATH 38 Expected Value and Varance Dr. Neal, WKU We now shall dscuss how to fnd the average and standard devaton of a random varable X. Expected Value Defnton. The expected value (or average value, or
More information3.1 ML and Empirical Distribution
67577 Intro. to Machne Learnng Fall semester, 2008/9 Lecture 3: Maxmum Lkelhood/ Maxmum Entropy Dualty Lecturer: Amnon Shashua Scrbe: Amnon Shashua 1 In the prevous lecture we defned the prncple of Maxmum
More informationNon-Mixture Cure Model for Interval Censored Data: Simulation Study ABSTRACT
Malaysan Journal of Mathematcal Scences 8(S): 37-44 (2014) Specal Issue: Internatonal Conference on Mathematcal Scences and Statstcs 2013 (ICMSS2013) MALAYSIAN JOURNAL OF MATHEMATICAL SCIENCES Journal
More informationDr. Shalabh Department of Mathematics and Statistics Indian Institute of Technology Kanpur
Analyss of Varance and Desgn of Experment-I MODULE VII LECTURE - 3 ANALYSIS OF COVARIANCE Dr Shalabh Department of Mathematcs and Statstcs Indan Insttute of Technology Kanpur Any scentfc experment s performed
More informationSTAT 3008 Applied Regression Analysis
STAT 3008 Appled Regresson Analyss Tutoral : Smple Lnear Regresson LAI Chun He Department of Statstcs, The Chnese Unversty of Hong Kong 1 Model Assumpton To quantfy the relatonshp between two factors,
More informationASYMPTOTIC PROPERTIES OF ESTIMATES FOR THE PARAMETERS IN THE LOGISTIC REGRESSION MODEL
Asymptotc Asan-Afrcan Propertes Journal of Estmates Economcs for and the Econometrcs, Parameters n Vol. the Logstc, No., Regresson 20: 65-74 Model 65 ASYMPTOTIC PROPERTIES OF ESTIMATES FOR THE PARAMETERS
More informationDevelopment Pattern and Prediction Error for the Stochastic Bornhuetter-Ferguson Claims Reserving Method
Development Pattern and Predcton Error for the Stochastc Bornhuetter-Ferguson Clams Reservng Method Annna Saluz, Alos Gsler, Maro V. Wüthrch ETH Zurch ASTIN Colloquum Madrd, June 2011 Overvew 1 Notaton
More informationStatistics and Probability Theory in Civil, Surveying and Environmental Engineering
Statstcs and Probablty Theory n Cvl, Surveyng and Envronmental Engneerng Pro. Dr. Mchael Havbro Faber ETH Zurch, Swtzerland Contents o Todays Lecture Overvew o Uncertanty Modelng Random Varables - propertes
More informationRandomness and Computation
Randomness and Computaton or, Randomzed Algorthms Mary Cryan School of Informatcs Unversty of Ednburgh RC 208/9) Lecture 0 slde Balls n Bns m balls, n bns, and balls thrown unformly at random nto bns usually
More informationTesting for seasonal unit roots in heterogeneous panels
Testng for seasonal unt roots n heterogeneous panels Jesus Otero * Facultad de Economía Unversdad del Rosaro, Colomba Jeremy Smth Department of Economcs Unversty of arwck Monca Gulett Aston Busness School
More informationConjugacy and the Exponential Family
CS281B/Stat241B: Advanced Topcs n Learnng & Decson Makng Conjugacy and the Exponental Famly Lecturer: Mchael I. Jordan Scrbes: Bran Mlch 1 Conjugacy In the prevous lecture, we saw conjugate prors for the
More informationA Comparative Study for Estimation Parameters in Panel Data Model
A Comparatve Study for Estmaton Parameters n Panel Data Model Ahmed H. Youssef and Mohamed R. Abonazel hs paper examnes the panel data models when the regresson coeffcents are fxed random and mxed and
More informationComputation of Higher Order Moments from Two Multinomial Overdispersion Likelihood Models
Computaton of Hgher Order Moments from Two Multnomal Overdsperson Lkelhood Models BY J. T. NEWCOMER, N. K. NEERCHAL Department of Mathematcs and Statstcs, Unversty of Maryland, Baltmore County, Baltmore,
More informationTHE ROYAL STATISTICAL SOCIETY 2006 EXAMINATIONS SOLUTIONS HIGHER CERTIFICATE
THE ROYAL STATISTICAL SOCIETY 6 EXAMINATIONS SOLUTIONS HIGHER CERTIFICATE PAPER I STATISTICAL THEORY The Socety provdes these solutons to assst canddates preparng for the eamnatons n future years and for
More informationSimulation and Probability Distribution
CHAPTER Probablty, Statstcs, and Relablty for Engneers and Scentsts Second Edton PROBABILIT DISTRIBUTION FOR CONTINUOUS RANDOM VARIABLES A. J. Clark School of Engneerng Department of Cvl and Envronmental
More informationCS 2750 Machine Learning. Lecture 5. Density estimation. CS 2750 Machine Learning. Announcements
CS 750 Machne Learnng Lecture 5 Densty estmaton Mlos Hauskrecht mlos@cs.ptt.edu 539 Sennott Square CS 750 Machne Learnng Announcements Homework Due on Wednesday before the class Reports: hand n before
More informationMATH 281A: Homework #6
MATH 28A: Homework #6 Jongha Ryu Due date: November 8, 206 Problem. (Problem 2..2. Soluton. If X,..., X n Bern(p, then T = X s a complete suffcent statstc. Our target s g(p = p, and the nave guess suggested
More informationExplaining the Stein Paradox
Explanng the Sten Paradox Kwong Hu Yung 1999/06/10 Abstract Ths report offers several ratonale for the Sten paradox. Sectons 1 and defnes the multvarate normal mean estmaton problem and ntroduces Sten
More informationExcess Error, Approximation Error, and Estimation Error
E0 370 Statstcal Learnng Theory Lecture 10 Sep 15, 011 Excess Error, Approxaton Error, and Estaton Error Lecturer: Shvan Agarwal Scrbe: Shvan Agarwal 1 Introducton So far, we have consdered the fnte saple
More informationDifferentiating Gaussian Processes
Dfferentatng Gaussan Processes Andrew McHutchon Aprl 17, 013 1 Frst Order Dervatve of the Posteror Mean The posteror mean of a GP s gven by, f = x, X KX, X 1 y x, X α 1 Only the x, X term depends on the
More informationAperture Photometry Uncertainties assuming Priors and Correlated Noise
Aperture Photometry Uncertantes assumng Prors and Correlated Nose F. Masc, verson.0, 10/06/009 1. Summary We derve a general formula for the nose varance n the flux of a source estmated from aperture photometry
More informationThermodynamics and statistical mechanics in materials modelling II
Course MP3 Lecture 8/11/006 (JAE) Course MP3 Lecture 8/11/006 Thermodynamcs and statstcal mechancs n materals modellng II A bref résumé of the physcal concepts used n materals modellng Dr James Ellott.1
More informationOn Quantifying the Accuracy of Maximum Likelihood Estimation of Participant Reliability in Social Sensing
On Quantfyng the Accuracy of Maxmum Lkelhood Estmaton of Partcpant Relablty n Socal Sensng Dong Wang, Tarek Abdelzaher Department of Computer Scence Unversty of Illnos Urbana, IL 61801 dwang24@llnos.edu,
More informationLecture 17 : Stochastic Processes II
: Stochastc Processes II 1 Contnuous-tme stochastc process So far we have studed dscrete-tme stochastc processes. We studed the concept of Makov chans and martngales, tme seres analyss, and regresson analyss
More information18.1 Introduction and Recap
CS787: Advanced Algorthms Scrbe: Pryananda Shenoy and Shjn Kong Lecturer: Shuch Chawla Topc: Streamng Algorthmscontnued) Date: 0/26/2007 We contnue talng about streamng algorthms n ths lecture, ncludng
More informationExam. Econometrics - Exam 1
Econometrcs - Exam 1 Exam Problem 1: (15 ponts) Suppose that the classcal regresson model apples but that the true value of the constant s zero. In order to answer the followng questons assume just one
More informationStatistical inference for generalized Pareto distribution based on progressive Type-II censored data with random removals
Internatonal Journal of Scentfc World, 2 1) 2014) 1-9 c Scence Publshng Corporaton www.scencepubco.com/ndex.php/ijsw do: 10.14419/jsw.v21.1780 Research Paper Statstcal nference for generalzed Pareto dstrbuton
More informationUNIVERSITY OF TORONTO Faculty of Arts and Science. December 2005 Examinations STA437H1F/STA1005HF. Duration - 3 hours
UNIVERSITY OF TORONTO Faculty of Arts and Scence December 005 Examnatons STA47HF/STA005HF Duraton - hours AIDS ALLOWED: (to be suppled by the student) Non-programmable calculator One handwrtten 8.5'' x
More informationLOW BIAS INTEGRATED PATH ESTIMATORS. James M. Calvin
Proceedngs of the 007 Wnter Smulaton Conference S G Henderson, B Bller, M-H Hseh, J Shortle, J D Tew, and R R Barton, eds LOW BIAS INTEGRATED PATH ESTIMATORS James M Calvn Department of Computer Scence
More information4.1. Lecture 4: Fitting distributions: goodness of fit. Goodness of fit: the underlying principle
Lecture 4: Fttng dstrbutons: goodness of ft Goodness of ft Testng goodness of ft Testng normalty An mportant note on testng normalty! L4.1 Goodness of ft measures the extent to whch some emprcal dstrbuton
More informationCIS526: Machine Learning Lecture 3 (Sept 16, 2003) Linear Regression. Preparation help: Xiaoying Huang. x 1 θ 1 output... θ M x M
CIS56: achne Learnng Lecture 3 (Sept 6, 003) Preparaton help: Xaoyng Huang Lnear Regresson Lnear regresson can be represented by a functonal form: f(; θ) = θ 0 0 +θ + + θ = θ = 0 ote: 0 s a dummy attrbute
More informationProblem of Estimation. Ordinary Least Squares (OLS) Ordinary Least Squares Method. Basic Econometrics in Transportation. Bivariate Regression Analysis
1/60 Problem of Estmaton Basc Econometrcs n Transportaton Bvarate Regresson Analyss Amr Samm Cvl Engneerng Department Sharf Unversty of Technology Ordnary Least Squares (OLS) Maxmum Lkelhood (ML) Generally,
More informationCopyright 2017 by Taylor Enterprises, Inc., All Rights Reserved. Adjusted Control Limits for U Charts. Dr. Wayne A. Taylor
Taylor Enterprses, Inc. Adjusted Control Lmts for U Charts Copyrght 207 by Taylor Enterprses, Inc., All Rghts Reserved. Adjusted Control Lmts for U Charts Dr. Wayne A. Taylor Abstract: U charts are used
More informationprinceton univ. F 17 cos 521: Advanced Algorithm Design Lecture 7: LP Duality Lecturer: Matt Weinberg
prnceton unv. F 17 cos 521: Advanced Algorthm Desgn Lecture 7: LP Dualty Lecturer: Matt Wenberg Scrbe: LP Dualty s an extremely useful tool for analyzng structural propertes of lnear programs. Whle there
More informationVapnik-Chervonenkis theory
Vapnk-Chervonenks theory Rs Kondor June 13, 2008 For the purposes of ths lecture, we restrct ourselves to the bnary supervsed batch learnng settng. We assume that we have an nput space X, and an unknown
More informationInference from Data Partitions
Inference from Data Parttons Rcardo Bórquez and Melvn Hnch May 18, 2010 1 Introducton Consder a statonary process X = {X t, t = 1, 2,..., T } defned on (Ω, F, {F t }, P ) where F denotes the Borel sets,
More informationSimulated Power of the Discrete Cramér-von Mises Goodness-of-Fit Tests
Smulated of the Cramér-von Mses Goodness-of-Ft Tests Steele, M., Chaselng, J. and 3 Hurst, C. School of Mathematcal and Physcal Scences, James Cook Unversty, Australan School of Envronmental Studes, Grffth
More informationThe Order Relation and Trace Inequalities for. Hermitian Operators
Internatonal Mathematcal Forum, Vol 3, 08, no, 507-57 HIKARI Ltd, wwwm-hkarcom https://doorg/0988/mf088055 The Order Relaton and Trace Inequaltes for Hermtan Operators Y Huang School of Informaton Scence
More informationThe Gaussian classifier. Nuno Vasconcelos ECE Department, UCSD
he Gaussan classfer Nuno Vasconcelos ECE Department, UCSD Bayesan decson theory recall that we have state of the world X observatons g decson functon L[g,y] loss of predctng y wth g Bayes decson rule s
More informationsince [1-( 0+ 1x1i+ 2x2 i)] [ 0+ 1x1i+ assumed to be a reasonable approximation
Econ 388 R. Butler 204 revsons Lecture 4 Dummy Dependent Varables I. Lnear Probablty Model: the Regresson model wth a dummy varables as the dependent varable assumpton, mplcaton regular multple regresson
More informationCourse 395: Machine Learning - Lectures
Course 395: Machne Learnng - Lectures Lecture 1-2: Concept Learnng (M. Pantc Lecture 3-4: Decson Trees & CC Intro (M. Pantc Lecture 5-6: Artfcal Neural Networks (S.Zaferou Lecture 7-8: Instance ased Learnng
More informationChapter 11: Simple Linear Regression and Correlation
Chapter 11: Smple Lnear Regresson and Correlaton 11-1 Emprcal Models 11-2 Smple Lnear Regresson 11-3 Propertes of the Least Squares Estmators 11-4 Hypothess Test n Smple Lnear Regresson 11-4.1 Use of t-tests
More information