# MSc. Econ: MATHEMATICAL STATISTICS, 1995 MAXIMUM-LIKELIHOOD ESTIMATION

Save this PDF as:

Size: px
Start display at page:

## Transcription

1 MAXIMUM-LIKELIHOOD ESTIMATION The General Theory of M-L Estimation In orer to erive an M-L estimator, we are boun to make an assumption about the functional form of the istribution which generates the ata However, the assumption can often be varie without affecting the form of the M-L estimator; an the general theory of maximum-likelihoo estimation can be evelope without reference to a specific istribution In fact, the M-L metho is of such generality that it provies a moel for most other methos of estimation For the other methos ten to generate estimators which can be epicte as approximations to the maximum-likelihoo estimators, if they are not actually ientical to the latter In orer to reveal the important characteristics of the likelihoo estimators, we shoul investigate the properties of the log-likelihoo function itself Consier the case where θ is the sole parameter of a log-likelihoo function log L(y; θ) wherein y =[y,,y T ] is a vector of sample elements In seeking to estimate the parameter, we regar θ as an argument of the function whilst the elements of y are consiere to be fixe However, in analysing the statistical properties of the function, we restore the ranom character to the sample elements The ranomness is conveye to the maximising value ˆθ which thereby acquires a istribution A funamental result is that, as the sample size increases, the likelihoo function ivie by the sample size tens to stabilise in the sense that it converges in probability, at every point in its omain, to a constant function In the process, the istribution of ˆθ becomes increasingly concentrate in the vicinity of the true parameter value θ 0 This accounts for the consistency of maximum-likelihoo estimation To emonstrate the convergence of the log-likelihoo function, we shall assume, as before, that the elements of y =[y,,y T ] form a ranom sample Then () L(y; θ) = an therefore T f(y t ; θ), t= (2) T log L(y; θ) = T T log f(y t ; θ) t= For any value of θ, this represents a sum of inepenently an ientically istribute ranom variables Therefore the law of large numbers can be applie

2 to show that MSc Econ: MATHEMATICAL STATISTICS, 995 (3) plim(t ) T log L(y; θ) =E log f(y t ; θ) } The next step is to emonstrate that Elog L(y; θ 0 )} Elog L(y; θ)}, which is to say that the expecte log-likelihoo function, to which the sample likelihoo function converges, is maximise by the true parameter value θ 0 The first erivative of log-likelihoo function is (4) = L(y; θ) L(y; θ) This is known as the score of the log-likelihoo function at θ Uner conitions which allow the erivative an the integral to commute, the erivative of the expectation is the expectation of the erivative Thus, from (4), (5) E log L(y; θ) } = y L(y; θ) } L(y; θ) L(y; θ 0 )y, where θ 0 is the true value of θ an L(y, θ 0 ) is the probability ensity function of y When θ = θ 0, the expression on the RHS simplifies in consequence of the cancellation of L(y, θ) in the enominator with L(y, θ 0 ) in the numerator Then we get L(y; θ 0 ) (6) y = L(y; θ 0 )y =0, y y where the final equality follows from the fact that the integral is unity, which implies that its erivative is zero Thus (7) E log L(y; θ 0 ) } } log L(y; θ0 ) = E =0; an this is a first-orer conition which inicates that the Elog L(y; θ)/t } is maximise at the true parameter value θ 0 Given that the log L(y; θ)/t converges to Elog L(y; θ)/t }, it follows, by some simple analytic arguments, that the maximising value of the former must converge to the maximising value of the latter: which is to say that ˆθ must converge to θ 0 Now let us ifferentiate (4) in respect to θ an take expectations Provie that the orer of these operations can be interchange, then (8) y L(y; θ)y = 2 2 L(y; θ)y =0, y 2

3 where the final equality follows in the same way as that of (6) The LHS can be expresse as 2 log L(y; θ) L(y; θ) (9) y 2 L(y; θ)y + y =0 y an, on substituting from (4) into the secon term, this becomes 2 } 2 log L(y; θ) y 2 L(y; θ)y + L(y; θ)y =0 (0) y Therefore, when θ = θ 0,weget } [ } ] 2 () E 2 log L(y; θ 0 ) log L(y; θ0 ) 2 = E =Φ This measure is know as Fisher s Information Since (7) inicates that the score log L(y; θ 0 )/ has an expecte value of zero, it follows that Fisher s Information represents the variance of the score at θ 0 Clearly, the information measure increases with the size of the sample To obtain a measure of the information about θ which is containe, on average, in a single observation, we may efine φ =Φ/T The importance of the information measure Φ is that its inverse provies an approximation to the variance of the maximum-likelihoo estimator which become increasingly accurate as the sample size increases Inee, this is the explanation of the terminology The famous Cramèr Rao theorem inicates that the inverse of the information measure provies a lower boun for the variance of any unbiase estimator of θ The fact that the asymptotic variance of the maximum-likelihoo estimator attains this boun, as we shall procee to show, is the proof of the estimator s efficiency The Asymptotic Distribution of the M-L Estimator The asymptotic istribution of the maximum-likelihoo estimator is establishe uner the assumption that the log-likelihoo function obeys certain regularity conitions Some of these conitions are not reaily explicable without a context Therefore, instea of itemising the conitions, we shall make an overall assumption which is appropriate to our own purposes but which is stronger than is strictly necessary We shall image that log L(y; θ) is an analytic function which can be represente by a Taylor-series expansion about the point θ 0 : (2) log L(θ) = log L(θ 0 )+ log L(θ 0) (θ θ 0 )+ 2 log L(θ 0 ) 2 2 (θ θ 0 ) 2 + 3! 3 log L(θ 0 ) 3 (θ θ 0 ) 3 + 3

4 In pursuing the asymptotic istribution of the maximum-likelihoo estimator, we can concentrate upon a quaratic approximation which is base the first three terms of this expansion The reason is that, as we have shown, the istribution of the estimator becomes increasingly concentrate in the vicinity of the true parameter value as the size of the sample increases Therefore the quaratic approximation becomes increasingly accurate for the range of values of θ which we are liable to consier It follows that, amongst the regularity conitions, there must be at least the provision that the erivatives of the function are finite-value up to the thir orer The quaratic approximation to the function, taken at the point θ 0,is (3) log L(θ) = log L(θ 0 )+ log L(θ 0) (θ θ 0 )+ 2 log L(θ 0 ) 2 2 (θ θ 0 ) 2 Its erivative with respect to θ is (4) log L(θ) = log L(θ 0) + 2 log L(θ 0 ) 2 (θ θ 0 ) By setting θ = ˆθ an by using the fact that log L(ˆθ)/ = 0, which follows from the efinition of the maximum-likelihoo estimator, we fin that (5) T (ˆθ θ0 )= T 2 } log L(θ 0 ) 2 T log L(θ 0 ) The argument which establishes the limiting istribution of T (ˆθ θ 0 ) has two strans First, the law of large numbers is invoke in to show that } (6) T 2 log L(y; θ 0 ) 2 = T t 2 log f(y t ; θ 0 ) 2 must converge to its expecte value which is the information measure φ =Φ/T Next, the central limit theorem is invoke to show that (7) log L(y; θ 0 ) T = T t log f(y t ; θ 0 ) has a limiting normal istribution which is N(0,φ) This result epens crucially on the fact that Φ = Tφ is the variance of log L(y; θ 0 )/ Thus the limiting istribution of the quantity T (ˆθ θ 0 ) is the normal N(0,φ ) istribution, since this is the istribution of φ times an N(0,φ) variable Within this argument, the evice of scaling ˆθ by T has the purpose of preventing the variance from vanishing, an the istribution from collapsing, 4

5 as the sample size increases inefinitely Having complete the argument, we can remove the scale factor; an the conclusion which is to be rawn is the following: (8) Let ˆθ be the maximum-likelihoo estimator obtaine by solving the equation log L(y, θ)/ = 0, an let θ 0 be the true value of the parameter Then ˆθ is istribute approximately accoring to the istribution N(θ 0, Φ ), where Φ is the inverse of Fisher s measure of information In establishing these results, we have consiere only the case where a single parameter is to estimate This has enable us to procee without the panoply of vectors an matrices Nevertheless, nothing essential has been omitte from our arguments In the case where θ is a vector of k elements, we efine the information matrix to be the matrix whose elements are the variances an covariances of the elements of the score vector Thus the generic element of the information matrix, in the ijth position, is (9) E } 2 log L(θ 0 ) log L(θ0 ) = E log L(θ } 0) θ i θ j θ i θ j 5

### MASSACHUSETTS INSTITUTE OF TECHNOLOGY 6.436J/15.085J Fall 2008 Lecture 14 10/27/2008 MOMENT GENERATING FUNCTIONS

MASSACHUSETTS INSTITUTE OF TECHNOLOGY 6.436J/15.085J Fall 2008 Lecture 14 10/27/2008 MOMENT GENERATING FUNCTIONS Contents 1. Moment generating functions 2. Sum of a ranom number of ranom variables 3. Transforms

### Math 230.01, Fall 2012: HW 1 Solutions

Math 3., Fall : HW Solutions Problem (p.9 #). Suppose a wor is picke at ranom from this sentence. Fin: a) the chance the wor has at least letters; SOLUTION: All wors are equally likely to be chosen. The

### CHAPTER 5 : CALCULUS

Dr Roger Ni (Queen Mary, University of Lonon) - 5. CHAPTER 5 : CALCULUS Differentiation Introuction to Differentiation Calculus is a branch of mathematics which concerns itself with change. Irrespective

### 10.2 Systems of Linear Equations: Matrices

SECTION 0.2 Systems of Linear Equations: Matrices 7 0.2 Systems of Linear Equations: Matrices OBJECTIVES Write the Augmente Matrix of a System of Linear Equations 2 Write the System from the Augmente Matrix

### Mathematics Review for Economists

Mathematics Review for Economists by John E. Floy University of Toronto May 9, 2013 This ocument presents a review of very basic mathematics for use by stuents who plan to stuy economics in grauate school

### Sensor Network Localization from Local Connectivity : Performance Analysis for the MDS-MAP Algorithm

Sensor Network Localization from Local Connectivity : Performance Analysis for the MDS-MAP Algorithm Sewoong Oh an Anrea Montanari Electrical Engineering an Statistics Department Stanfor University, Stanfor,

### On Adaboost and Optimal Betting Strategies

On Aaboost an Optimal Betting Strategies Pasquale Malacaria 1 an Fabrizio Smerali 1 1 School of Electronic Engineering an Computer Science, Queen Mary University of Lonon, Lonon, UK Abstract We explore

### Data Center Power System Reliability Beyond the 9 s: A Practical Approach

Data Center Power System Reliability Beyon the 9 s: A Practical Approach Bill Brown, P.E., Square D Critical Power Competency Center. Abstract Reliability has always been the focus of mission-critical

### Hull, Chapter 11 + Sections 17.1 and 17.2 Additional reference: John Cox and Mark Rubinstein, Options Markets, Chapter 5

Binomial Moel Hull, Chapter 11 + ections 17.1 an 17.2 Aitional reference: John Cox an Mark Rubinstein, Options Markets, Chapter 5 1. One-Perio Binomial Moel Creating synthetic options (replicating options)

### A Generalization of Sauer s Lemma to Classes of Large-Margin Functions

A Generalization of Sauer s Lemma to Classes of Large-Margin Functions Joel Ratsaby University College Lonon Gower Street, Lonon WC1E 6BT, Unite Kingom J.Ratsaby@cs.ucl.ac.uk, WWW home page: http://www.cs.ucl.ac.uk/staff/j.ratsaby/

### Exponential Functions: Differentiation and Integration. The Natural Exponential Function

46_54.q //4 :59 PM Page 5 5 CHAPTER 5 Logarithmic, Eponential, an Other Transcenental Functions Section 5.4 f () = e f() = ln The inverse function of the natural logarithmic function is the natural eponential

### Lagrangian and Hamiltonian Mechanics

Lagrangian an Hamiltonian Mechanics D.G. Simpson, Ph.D. Department of Physical Sciences an Engineering Prince George s Community College December 5, 007 Introuction In this course we have been stuying

### MODELLING OF TWO STRATEGIES IN INVENTORY CONTROL SYSTEM WITH RANDOM LEAD TIME AND DEMAND

art I. robobabilystic Moels Computer Moelling an New echnologies 27 Vol. No. 2-3 ransport an elecommunication Institute omonosova iga V-9 atvia MOEING OF WO AEGIE IN INVENOY CONO YEM WIH ANOM EA IME AN

### CURRENCY OPTION PRICING II

Jones Grauate School Rice University Masa Watanabe INTERNATIONAL FINANCE MGMT 657 Calibrating the Binomial Tree to Volatility Black-Scholes Moel for Currency Options Properties of the BS Moel Option Sensitivity

### Detecting Possibly Fraudulent or Error-Prone Survey Data Using Benford s Law

Detecting Possibly Frauulent or Error-Prone Survey Data Using Benfor s Law Davi Swanson, Moon Jung Cho, John Eltinge U.S. Bureau of Labor Statistics 2 Massachusetts Ave., NE, Room 3650, Washington, DC

### Sensitivity Analysis of Non-linear Performance with Probability Distortion

Preprints of the 19th Worl Congress The International Feeration of Automatic Control Cape Town, South Africa. August 24-29, 214 Sensitivity Analysis of Non-linear Performance with Probability Distortion

Calibration of the broa ban UV Raiometer Marian Morys an Daniel Berger Solar Light Co., Philaelphia, PA 19126 ABSTRACT Mounting concern about the ozone layer epletion an the potential ultraviolet exposure

### Differentiability of Exponential Functions

Differentiability of Exponential Functions Philip M. Anselone an John W. Lee Philip Anselone (panselone@actionnet.net) receive his Ph.D. from Oregon State in 1957. After a few years at Johns Hopkins an

### Digital barrier option contract with exponential random time

IMA Journal of Applie Mathematics Avance Access publishe June 9, IMA Journal of Applie Mathematics ) Page of 9 oi:.93/imamat/hxs3 Digital barrier option contract with exponential ranom time Doobae Jun

### Search Advertising Based Promotion Strategies for Online Retailers

Search Avertising Base Promotion Strategies for Online Retailers Amit Mehra The Inian School of Business yeraba, Inia Amit Mehra@isb.eu ABSTRACT Web site aresses of small on line retailers are often unknown

### 1 High-Dimensional Space

Contents High-Dimensional Space. Properties of High-Dimensional Space..................... 4. The High-Dimensional Sphere......................... 5.. The Sphere an the Cube in Higher Dimensions...........

### A SPATIAL UNIT LEVEL MODEL FOR SMALL AREA ESTIMATION

REVSTAT Statistical Journal Volume 9, Number 2, June 2011, 155 180 A SPATIAL UNIT LEVEL MODEL FOR SMALL AREA ESTIMATION Authors: Pero S. Coelho ISEGI Universiae Nova e Lisboa, Portugal Faculty of Economics,

### Web Appendices to Selling to Overcon dent Consumers

Web Appenices to Selling to Overcon ent Consumers Michael D. Grubb MIT Sloan School of Management Cambrige, MA 02142 mgrubbmit.eu www.mit.eu/~mgrubb May 2, 2008 B Option Pricing Intuition This appenix

### The most common model to support workforce management of telephone call centers is

Designing a Call Center with Impatient Customers O. Garnett A. Manelbaum M. Reiman Davison Faculty of Inustrial Engineering an Management, Technion, Haifa 32000, Israel Davison Faculty of Inustrial Engineering

### JON HOLTAN. if P&C Insurance Ltd., Oslo, Norway ABSTRACT

OPTIMAL INSURANCE COVERAGE UNDER BONUS-MALUS CONTRACTS BY JON HOLTAN if P&C Insurance Lt., Oslo, Norway ABSTRACT The paper analyses the questions: Shoul or shoul not an iniviual buy insurance? An if so,

### Lecture 17: Implicit differentiation

Lecture 7: Implicit ifferentiation Nathan Pflueger 8 October 203 Introuction Toay we iscuss a technique calle implicit ifferentiation, which provies a quicker an easier way to compute many erivatives we

### Inverse Trig Functions

Inverse Trig Functions c A Math Support Center Capsule February, 009 Introuction Just as trig functions arise in many applications, so o the inverse trig functions. What may be most surprising is that

### Optimal Control Of Production Inventory Systems With Deteriorating Items And Dynamic Costs

Applie Mathematics E-Notes, 8(2008), 194-202 c ISSN 1607-2510 Available free at mirror sites of http://www.math.nthu.eu.tw/ amen/ Optimal Control Of Prouction Inventory Systems With Deteriorating Items

### Risk Adjustment for Poker Players

Risk Ajustment for Poker Players William Chin DePaul University, Chicago, Illinois Marc Ingenoso Conger Asset Management LLC, Chicago, Illinois September, 2006 Introuction In this article we consier risk

### 2r 1. Definition (Degree Measure). Let G be a r-graph of order n and average degree d. Let S V (G). The degree measure µ(s) of S is defined by,

Theorem Simple Containers Theorem) Let G be a simple, r-graph of average egree an of orer n Let 0 < δ < If is large enough, then there exists a collection of sets C PV G)) satisfying: i) for every inepenent

### A Monte Carlo Simulation of Multivariate General

A Monte Carlo Simulation of Multivariate General Pareto Distribution an its Application 1 1 1 1 1 1 1 1 0 1 Luo Yao 1, Sui Danan, Wang Dongxiao,*, Zhou Zhenwei, He Weihong 1, Shi Hui 1 South China Sea

### Optimal Control Policy of a Production and Inventory System for multi-product in Segmented Market

RATIO MATHEMATICA 25 (2013), 29 46 ISSN:1592-7415 Optimal Control Policy of a Prouction an Inventory System for multi-prouct in Segmente Market Kuleep Chauhary, Yogener Singh, P. C. Jha Department of Operational

### Forecasting and Staffing Call Centers with Multiple Interdependent Uncertain Arrival Streams

Forecasting an Staffing Call Centers with Multiple Interepenent Uncertain Arrival Streams Han Ye Department of Statistics an Operations Research, University of North Carolina, Chapel Hill, NC 27599, hanye@email.unc.eu

### Game Theoretic Modeling of Cooperation among Service Providers in Mobile Cloud Computing Environments

2012 IEEE Wireless Communications an Networking Conference: Services, Applications, an Business Game Theoretic Moeling of Cooperation among Service Proviers in Mobile Clou Computing Environments Dusit

### 2012-2013 Enhanced Instructional Transition Guide Mathematics Algebra I Unit 08

01-013 Enhance Instructional Transition Guie Unit 08: Exponents an Polynomial Operations (18 ays) Possible Lesson 01 (4 ays) Possible Lesson 0 (7 ays) Possible Lesson 03 (7 ays) POSSIBLE LESSON 0 (7 ays)

### Mathematical Models of Therapeutical Actions Related to Tumour and Immune System Competition

Mathematical Moels of Therapeutical Actions Relate to Tumour an Immune System Competition Elena De Angelis (1 an Pierre-Emmanuel Jabin (2 (1 Dipartimento i Matematica, Politecnico i Torino Corso Duca egli

### y or f (x) to determine their nature.

Level C5 of challenge: D C5 Fining stationar points of cubic functions functions Mathematical goals Starting points Materials require Time neee To enable learners to: fin the stationar points of a cubic

### A New Pricing Model for Competitive Telecommunications Services Using Congestion Discounts

A New Pricing Moel for Competitive Telecommunications Services Using Congestion Discounts N. Keon an G. Ananalingam Department of Systems Engineering University of Pennsylvania Philaelphia, PA 19104-6315

### An intertemporal model of the real exchange rate, stock market, and international debt dynamics: policy simulations

This page may be remove to conceal the ientities of the authors An intertemporal moel of the real exchange rate, stock market, an international ebt ynamics: policy simulations Saziye Gazioglu an W. Davi

### The mean-field computation in a supermarket model with server multiple vacations

DOI.7/s66-3-7-5 The mean-fiel computation in a supermaret moel with server multiple vacations Quan-Lin Li Guirong Dai John C. S. Lui Yang Wang Receive: November / Accepte: 8 October 3 SpringerScienceBusinessMeiaNewYor3

### Factoring Dickson polynomials over finite fields

Factoring Dickson polynomials over finite fiels Manjul Bhargava Department of Mathematics, Princeton University. Princeton NJ 08544 manjul@math.princeton.eu Michael Zieve Department of Mathematics, University

### Which Networks Are Least Susceptible to Cascading Failures?

Which Networks Are Least Susceptible to Cascaing Failures? Larry Blume Davi Easley Jon Kleinberg Robert Kleinberg Éva Taros July 011 Abstract. The resilience of networks to various types of failures is

### Optimized Capacity Bounds for the Half-Duplex Gaussian MIMO Relay Channel

Optimize Capacity Bouns for the Half-Duplex Gaussian MIMO elay Channel Lennart Geres an Wolfgang Utschick International ITG Workshop on mart Antennas (WA) February 211 c 211 IEEE. Personal use of this

### (We assume that x 2 IR n with n > m f g are twice continuously ierentiable functions with Lipschitz secon erivatives. The Lagrangian function `(x y) i

An Analysis of Newton's Metho for Equivalent Karush{Kuhn{Tucker Systems Lus N. Vicente January 25, 999 Abstract In this paper we analyze the application of Newton's metho to the solution of systems of

### Lecture L25-3D Rigid Body Kinematics

J. Peraire, S. Winall 16.07 Dynamics Fall 2008 Version 2.0 Lecture L25-3D Rigi Boy Kinematics In this lecture, we consier the motion of a 3D rigi boy. We shall see that in the general three-imensional

### 4. Important theorems in quantum mechanics

TFY4215 Kjemisk fysikk og kvantemekanikk - Tillegg 4 1 TILLEGG 4 4. Important theorems in quantum mechanics Before attacking three-imensional potentials in the next chapter, we shall in chapter 4 of this

### IPA Derivatives for Make-to-Stock Production-Inventory Systems With Lost Sales

IPA Derivatives for Make-to-Stock Prouction-Inventory Systems With Lost Sales Yao Zhao Benjamin Melame Rutgers University Rutgers Business School Newark an New Brunswick Department of MSIS 94 Rockafeller

### Optimal Energy Commitments with Storage and Intermittent Supply

Submitte to Operations Research manuscript OPRE-2009-09-406 Optimal Energy Commitments with Storage an Intermittent Supply Jae Ho Kim Department of Electrical Engineering, Princeton University, Princeton,

### Ch 10. Arithmetic Average Options and Asian Opitons

Ch 10. Arithmetic Average Options an Asian Opitons I. Asian Option an the Analytic Pricing Formula II. Binomial Tree Moel to Price Average Options III. Combination of Arithmetic Average an Reset Options

### A New Evaluation Measure for Information Retrieval Systems

A New Evaluation Measure for Information Retrieval Systems Martin Mehlitz martin.mehlitz@ai-labor.e Christian Bauckhage Deutsche Telekom Laboratories christian.bauckhage@telekom.e Jérôme Kunegis jerome.kunegis@ai-labor.e

### Open World Face Recognition with Credibility and Confidence Measures

Open Worl Face Recognition with Creibility an Confience Measures Fayin Li an Harry Wechsler Department of Computer Science George Mason University Fairfax, VA 22030 {fli, wechsler}@cs.gmu.eu Abstract.

### ISSN: 2277-3754 ISO 9001:2008 Certified International Journal of Engineering and Innovative Technology (IJEIT) Volume 3, Issue 12, June 2014

ISSN: 77-754 ISO 900:008 Certifie International Journal of Engineering an Innovative echnology (IJEI) Volume, Issue, June 04 Manufacturing process with isruption uner Quaratic Deman for Deteriorating Inventory

### The one-year non-life insurance risk

The one-year non-life insurance risk Ohlsson, Esbjörn & Lauzeningks, Jan Abstract With few exceptions, the literature on non-life insurance reserve risk has been evote to the ultimo risk, the risk in the

### A Comparison of Performance Measures for Online Algorithms

A Comparison of Performance Measures for Online Algorithms Joan Boyar 1, Sany Irani 2, an Kim S. Larsen 1 1 Department of Mathematics an Computer Science, University of Southern Denmark, Campusvej 55,

### UCLA STAT 13 Introduction to Statistical Methods for the Life and Health Sciences. Chapter 9 Paired Data. Paired data. Paired data

UCLA STAT 3 Introuction to Statistical Methos for the Life an Health Sciences Instructor: Ivo Dinov, Asst. Prof. of Statistics an Neurology Chapter 9 Paire Data Teaching Assistants: Jacquelina Dacosta

### Web Appendices of Selling to Overcon dent Consumers

Web Appenices of Selling to Overcon ent Consumers Michael D. Grubb A Option Pricing Intuition This appenix provies aitional intuition base on option pricing for the result in Proposition 2. Consier the

### ThroughputScheduler: Learning to Schedule on Heterogeneous Hadoop Clusters

ThroughputScheuler: Learning to Scheule on Heterogeneous Haoop Clusters Shehar Gupta, Christian Fritz, Bob Price, Roger Hoover, an Johan e Kleer Palo Alto Research Center, Palo Alto, CA, USA {sgupta, cfritz,

### Example Optimization Problems selected from Section 4.7

Example Optimization Problems selecte from Section 4.7 19) We are aske to fin the points ( X, Y ) on the ellipse 4x 2 + y 2 = 4 that are farthest away from the point ( 1, 0 ) ; as it happens, this point

### Tracking Control of a Class of Hamiltonian Mechanical Systems with Disturbances

Proceeings of Australasian Conference on Robotics an Automation, -4 Dec 4, The University of Melbourne, Melbourne, Australia Tracking Control of a Class of Hamiltonian Mechanical Systems with Disturbances

### DEVELOPMENT OF A BRAKING MODEL FOR SPEED SUPERVISION SYSTEMS

DEVELOPMENT OF A BRAKING MODEL FOR SPEED SUPERVISION SYSTEMS Paolo Presciani*, Monica Malvezzi #, Giuseppe Luigi Bonacci +, Monica Balli + * FS Trenitalia Unità Tecnologie Materiale Rotabile Direzione

### Purpose of the Experiments. Principles and Error Analysis. ε 0 is the dielectric constant,ε 0. ε r. = 8.854 10 12 F/m is the permittivity of

Experiments with Parallel Plate Capacitors to Evaluate the Capacitance Calculation an Gauss Law in Electricity, an to Measure the Dielectric Constants of a Few Soli an Liqui Samples Table of Contents Purpose

### PROPERTIES OF MATRICES

PROPERIES OF MARICES ajoint... 4, 5 algebraic multiplicity... augmente matri... basis..., cofactor... 4 coorinate vector... 9 Cramer's rule... eterminant..., 5 iagonal matri... 6 iagonalizable... 8 imension...

### 20. Product rule, Quotient rule

20. Prouct rule, 20.1. Prouct rule Prouct rule, Prouct rule We have seen that the erivative of a sum is the sum of the erivatives: [f(x) + g(x)] = x x [f(x)] + x [(g(x)]. One might expect from this that

### Here the units used are radians and sin x = sin(x radians). Recall that sin x and cos x are defined and continuous everywhere and

Lecture 9 : Derivatives of Trigonometric Functions (Please review Trigonometry uner Algebra/Precalculus Review on the class webpage.) In this section we will look at the erivatives of the trigonometric

### Overview of Violations of the Basic Assumptions in the Classical Normal Linear Regression Model

Overview of Violations of the Basic Assumptions in the Classical Normal Linear Regression Model 1 September 004 A. Introduction and assumptions The classical normal linear regression model can be written

### New Hypothesis Testing-Based Methods for Fault Detection for Smart Grid Systems

New Hypothesis esting-base Methos for Fault Detection for Smart Gri Systems Qian He an Rick S. Blum ECE Department, Lehigh University 9 Memorial Drive West, Bethlehem, PA 85 {qih27; rblum}@lehigh.eu Abstract

### Comparative study regarding the methods of interpolation

Recent Avances in Geoesy an Geomatics Engineering Comparative stuy regaring the methos of interpolation PAUL DANIEL DUMITRU, MARIN PLOPEANU, DRAGOS BADEA Department of Geoesy an Photogrammetry Faculty

### Risk Management for Derivatives

Risk Management or Derivatives he Greeks are coming the Greeks are coming! Managing risk is important to a large number o iniviuals an institutions he most unamental aspect o business is a process where

### Strong solutions of the compressible nematic liquid crystal flow

Strong solutions of the compressible nematic liqui crystal flow Tao Huang Changyou Wang Huanyao Wen April 9, 11 Abstract We stuy strong solutions of the simplifie Ericksen-Leslie system moeling compressible

### Economics 241B Hypothesis Testing: Large Sample Inference

Economics 241B Hyothesis Testing: Large Samle Inference Statistical inference in large-samle theory is base on test statistics whose istributions are nown uner the truth of the null hyothesis. Derivation

### Modelling and Resolving Software Dependencies

June 15, 2005 Abstract Many Linux istributions an other moern operating systems feature the explicit eclaration of (often complex) epenency relationships between the pieces of software

### Chapter 11: Feedback and PID Control Theory

Chapter 11: Feeback an ID Control Theory Chapter 11: Feeback an ID Control Theory I. Introuction Feeback is a mechanism for regulating a physical system so that it maintains a certain state. Feeback works

### Scalar : Vector : Equal vectors : Negative vectors : Proper vector : Null Vector (Zero Vector): Parallel vectors : Antiparallel vectors :

ELEMENTS OF VECTOS 1 Scalar : physical quantity having only magnitue but not associate with any irection is calle a scalar eg: time, mass, istance, spee, work, energy, power, pressure, temperature, electric

### Firewall Design: Consistency, Completeness, and Compactness

C IS COS YS TE MS Firewall Design: Consistency, Completeness, an Compactness Mohame G. Goua an Xiang-Yang Alex Liu Department of Computer Sciences The University of Texas at Austin Austin, Texas 78712-1188,

### Answers to the Practice Problems for Test 2

Answers to the Practice Problems for Test 2 Davi Murphy. Fin f (x) if it is known that x [f(2x)] = x2. By the chain rule, x [f(2x)] = f (2x) 2, so 2f (2x) = x 2. Hence f (2x) = x 2 /2, but the lefthan

### CALCULATION INSTRUCTIONS

Energy Saving Guarantee Contract ppenix 8 CLCULTION INSTRUCTIONS Calculation Instructions for the Determination of the Energy Costs aseline, the nnual mounts of Savings an the Remuneration 1 asics ll prices

### Jitter effects on Analog to Digital and Digital to Analog Converters

Jitter effects on Analog to Digital an Digital to Analog Converters Jitter effects copyright 1999, 2000 Troisi Design Limite Jitter One of the significant problems in igital auio is clock jitter an its

### Safety Stock or Excess Capacity: Trade-offs under Supply Risk

Safety Stock or Excess Capacity: Trae-offs uner Supply Risk Aahaar Chaturvei Victor Martínez-e-Albéniz IESE Business School, University of Navarra Av. Pearson, 08034 Barcelona, Spain achaturvei@iese.eu

### Kater Pendulum. Introduction. It is well-known result that the period T of a simple pendulum is given by. T = 2π

Kater Penulum ntrouction t is well-known result that the perio of a simple penulum is given by π L g where L is the length. n principle, then, a penulum coul be use to measure g, the acceleration of gravity.

### Achieving quality audio testing for mobile phones

Test & Measurement Achieving quality auio testing for mobile phones The auio capabilities of a cellular hanset provie the funamental interface between the user an the raio transceiver. Just as RF testing

### A Case Study of Applying SOM in Market Segmentation of Automobile Insurance Customers

International Journal of Database Theory an Application, pp.25-36 http://x.oi.org/10.14257/ijta.2014.7.1.03 A Case Stuy of Applying SOM in Market Segmentation of Automobile Insurance Customers Vahi Golmah

### Introduction to Integration Part 1: Anti-Differentiation

Mathematics Learning Centre Introuction to Integration Part : Anti-Differentiation Mary Barnes c 999 University of Syney Contents For Reference. Table of erivatives......2 New notation.... 2 Introuction

### Unbalanced Power Flow Analysis in a Micro Grid

International Journal of Emerging Technology an Avance Engineering Unbalance Power Flow Analysis in a Micro Gri Thai Hau Vo 1, Mingyu Liao 2, Tianhui Liu 3, Anushree 4, Jayashri Ravishankar 5, Toan Phung

### Patch Complexity, Finite Pixel Correlations and Optimal Denoising

Patch Complexity, Finite Pixel Correlations an Optimal Denoising Anat Levin Boaz Naler Freo Duran William T. Freeman Weizmann Institute MIT CSAIL Abstract. Image restoration tasks are ill-pose problems,

### Pythagorean Triples Over Gaussian Integers

International Journal of Algebra, Vol. 6, 01, no., 55-64 Pythagorean Triples Over Gaussian Integers Cheranoot Somboonkulavui 1 Department of Mathematics, Faculty of Science Chulalongkorn University Bangkok

### The Quick Calculus Tutorial

The Quick Calculus Tutorial This text is a quick introuction into Calculus ieas an techniques. It is esigne to help you if you take the Calculus base course Physics 211 at the same time with Calculus I,

### COPULA INFERENCE FOR MULTIPLE LIVES ANALYSIS PRELIMINARIES * Yogo Purwono

COPULA INFERENCE FOR MULTIPLE LIVES ANALYSIS PRELIMINARIES * Yogo Purwono Department of Management Faculty of Economics, University of Inonesia Depok, INDONESIA Abstract. Copula moels are becoming increasingly

### i=1 In practice, the natural logarithm of the likelihood function, called the log-likelihood function and denoted by

Statistics 580 Maximum Likelihood Estimation Introduction Let y (y 1, y 2,..., y n be a vector of iid, random variables from one of a family of distributions on R n and indexed by a p-dimensional parameter

### A Data Placement Strategy in Scientific Cloud Workflows

A Data Placement Strategy in Scientific Clou Workflows Dong Yuan, Yun Yang, Xiao Liu, Jinjun Chen Faculty of Information an Communication Technologies, Swinburne University of Technology Hawthorn, Melbourne,

### Cost Efficient Datacenter Selection for Cloud Services

Cost Efficient Datacenter Selection for Clou Services Hong u, Baochun Li henryxu, bli@eecg.toronto.eu Department of Electrical an Computer Engineering University of Toronto Abstract Many clou services

### Learning using Large Datasets

Learning using Large Datasets Léon Bottou a an Olivier Bousquet b a NEC Laboratories America, Princeton, NJ08540, USA b Google Zürich, 8002 Zürich, Switzerlan Abstract. This contribution evelops a theoretical

### On the Multivariate t Distribution

Technical report from Automatic Control at Linköpings universitet On the Multivariate t Distribution Michael Roth Division of Automatic Control E-mail: roth@isy.liu.se 7th April 03 Report no.: LiTH-ISY-R-3059

### Di usion on Social Networks. Current Version: June 6, 2006 Appeared in: Économie Publique, Numéro 16, pp 3-16, 2005/1.

Di usion on Social Networks Matthew O. Jackson y Caltech Leeat Yariv z Caltech Current Version: June 6, 2006 Appeare in: Économie Publique, Numéro 16, pp 3-16, 2005/1. Abstract. We analyze a moel of i

### Minimizing Makespan in Flow Shop Scheduling Using a Network Approach

Minimizing Makespan in Flow Shop Scheuling Using a Network Approach Amin Sahraeian Department of Inustrial Engineering, Payame Noor University, Asaluyeh, Iran 1 Introuction Prouction systems can be ivie

### The Impact of Forecasting Methods on Bullwhip Effect in Supply Chain Management

The Imact of Forecasting Methos on Bullwhi Effect in Suly Chain Management HX Sun, YT Ren Deartment of Inustrial an Systems Engineering, National University of Singaore, Singaore Schoo of Mechanical an

### Compact Routing on Internet-Like Graphs

Compact Routing on Internet-Like Graphs Dmitri Krioukov Email: ima@krioukov.net Kevin Fall Intel Research, Berkeley Email: kfall@intel-research.net Xiaowei Yang Massachusetts Institute of Technology Email:

### Definition of the spin current: The angular spin current and its physical consequences

Definition of the spin current: The angular spin current an its physical consequences Qing-feng Sun 1, * an X. C. Xie 2,3 1 Beijing National Lab for Conense Matter Physics an Institute of Physics, Chinese

### An Introduction to Event-triggered and Self-triggered Control

An Introuction to Event-triggere an Self-triggere Control W.P.M.H. Heemels K.H. Johansson P. Tabuaa Abstract Recent evelopments in computer an communication technologies have le to a new type of large-scale