b) The mean of the fitted (predicted) values of Y is equal to the mean of the Y values: c) The residuals of the regression line sum up to zero: = ei

Save this PDF as:
 WORD  PNG  TXT  JPG

Size: px
Start display at page:

Download "b) The mean of the fitted (predicted) values of Y is equal to the mean of the Y values: c) The residuals of the regression line sum up to zero: = ei"

Transcription

1 Mathematcal Propertes of the Least Squares Regresson The least squares regresson lne obeys certan mathematcal propertes whch are useful to know n practce. The followng propertes can be establshed algebracally: a The least squares regresson lne passes through the pont of sample means of Y and X. Ths can be easly seen from (4.9 whch can be rewrtten as follows, Y = b +b.x (4. b The mean of the ftted (predcted values of Y s equal to the mean of the Y values: Let Yˆ = b+b. X then we have - Y ˆ = ( b+b.x = ( Y - b.x +b. X =Y - b.x +b.x = Y n n (4.3 c The resduals of the regresson lne sum up to zero: e = ( Y - Yˆ =Y - Y ˆ = (4.4 n n d The resduals e are uncorrelated wth the X values: e X = e X - e X snce e = = e ( X - X = ( Y - Y.( X - X - b. ( X - X snce b =Y - b.x = snce b = ( Y - Y.( X - X / ( X - X e The resduals e are uncorrelated wth the ftted values Y. Ths property follows logcally from the prevous one snce each ftted value of Y s lnear functon of the correspondng X value. f The least squares regresson splts the varaton n the Y varable nto two components - the explaned varaton due to the varaton n X and the resdual varaton: where, TSS = RSS + ESS (4.5

2 TSS = (Y - Y ESS = (Yˆ - Yˆ RSS = e = (Y - Yˆ (4.6 TSS s the total varaton observed n the dependent varable Y. It s called the total sum of squares. ESS, the Explaned Sum of Squares, s the varaton of the predcted values (b +b.x. Ths s the varaton n Y accounted for by the varaton n the explanatory varable X. What s left s the RSS, the Resdual Sum of Squares. The reason why the ESS and RSS neatly add up to the TSS s that the resduals are uncorrelated wth the ftted Y values and, hence, there s no term wth the sum of covarances. Ths last property suggests a useful way to measure the goodness of ft of the estmated sample regresson. Ths s done as follows, R = ESS/TSS (4.7 where R, called R-square, s the coeffcent of determnaton. It gves us the proporton of the total sum of squares of the dependent varable explaned by the varaton n the explanatory varable. In fact, the R equals the square of the lnear correlaton coeffcent between the observed and the predcted values of the dependent varable Y, computed as follows, r = Cov (Y,Y ˆ V( Y.V( Yˆ = (Y - Y.( Yˆ - Yˆ (Y - Y. (Yˆ - Yˆ (4.7a A correlaton coeffcent measures the degree of lnear assocaton between two varables. Note, however, that f the underlyng relaton between the varables s non-lnear, the correlaton coeffcent may perform poorly, notwthstandng that fact that a strong non-lnear assocaton exsts between two varables.

3 Statstcal Propertes of LS Lnear Regresson We brefly revew the man ponts wthout much further elaboraton, apart from a few specfc ponts whch concern regresson only. We shall merely remnd you of the results of formal dervatons wthout botherng about proofs whch can be found n most ntroductory texts on statstcs or econometrcs. Standard Errors Gven the assumptons of the classcal lnear regresson model, the varances of the least squares estmators are gven by, var( b = σ (4.9 n σ var( b = ( X - X (4. Furthermore, an unbased estmator of σ s gven by s as follows: s = (Y - b - b X n - (4. where s s called the standard error of regresson snce σ s the varance of the error term whch measures the devaton of ndvduals ponts from the regresson lne. Replacng σ by s n (4.9 and (4., we get unbased estmates of the varances of b and b. Obvously, the estmated standard errors are the square roots of these varances. The total sum of squares of X, ( X - X whch features n the denomnator of the varances of the ntercept and slope coeffcents s a measure of the total varaton n the X values. Thus, other thngs beng equal, the hgher the varaton n the X values, the lower wll be the varances of the estmators, whch mples that hgher wll be the precson n estmaton. In other words, the range of observed X plays a crucal role n the relablty of

4 the estmates. Thnk about ths. It would ndeed be dffcult to measure the response of Y on X f X hardly vares at all. The greater the range over whch X vares, the easer t s to capture ts mpact on the varaton n Y. Samplng Dstrbutons To construct the confdence ntervals and to perform tests of hypotheses we need the probablty dstrbuton of the errors whch mples that we use the normalty assumpton of the error terms. Under ths assumpton, the least squares estmators b and b each follow a normal dstrbuton. However, snce we generally do not know the varance of the error term, we cannot make use of the normal dstrbuton drectly. Instead, we use the t-dstrbuton defned as follows n the case of b, b - β = _ t se( b t ( n - (4. where se(b, the standard error of b, s gven by, se( b = s [ ( X - X ] (4.3 usng (4. and (4.. The statstc, t (n-, denotes the Student's t-dstrbuton wth (n- degrees of freedom. The reason why we now have only (n- degrees of freedom s that, n smple regresson, we use the sample data to estmate coeffcents: the slope and the ntercept of the lne. In the case of the sample mean, n contrast, we only estmated one parameter (the mean tself from the sample. Smlarly, for b, we get, b - β = _ t se( b t ( n - (4.4 where se(b, the standard error of b, s gven by, se( X b = s + (4.5 n ( X - X

5 usng (4.9 and (4.. Confdence Intervals for the Parameters ß and ß The confdence lmts for ß and ß wth (-α per cent confdence co-effcent (say, 95 per cent, n whch case α=.5 are gven by, + _ t n -,.se( b (4.6 b + _ t n -,.se( b (4.7 b respectvely, where t(n-,α/ s the (-α/ percentle of a t-dstrbuton wth (n- degrees of freedom, and se(b and se(b are gven by (4.3 and (4.5 respectvely. Confdence Interval for the Condtonal Mean of Y At tmes, we may be nterested to construct a confdence nterval for the condtonal mean. For example, after fttng a regresson of household savngs on ncome, we may want to construct a confdence nterval for average savngs gven the level of ncome n order to assess the savngs potental of a certan type of households. Suppose, µ = β +.X (4.8 β.e. µ s the condtonal mean of Y gven X=X. The pont estmate of µ s gven by, +b. X b whle ts (-α per cent confdence nterval can be obtaned as follows, µ + _ t n -,.se( µ (4.9 where, ( X - X se( µ = s + (4.3 n ( X - X

6 Confdence Interval for the Predcted Y Values There are other occasons where we mght be nterested n the uncertanty n predcton on the bass of the estmated regresson. For example, when estmatng a regresson of paddy yeld (physcal output per unt area on annual ranfall, we may want to predct next year's yeld gven the antcpated ranfall. In ths case, our nterest s not to obtan a confdence nterval of the condtonal mean of the yeld.e. the mean yeld at a gven level of ranfall. Rather, we want to fnd a confdence nterval for the yeld (Y tself, gven the ranfall (X? Obvously, n ths case, = β + β. X + ε = µ + ε Y where µ s gven by (4.8. The (-α per cent confdence nterval for the Y gven X=X s then obtaned as follows, where, + _ t n -,.se(y (4.3 Y ( X - X se( Y = s + + (4.3 n ( X - X In ths case, therefore, the standard error of Y s larger than that of µ snce the latter corresponds to the condtonal mean of the yeld for a gven level of ranfall, whle the former corresponds to the predcted value of the yeld. In both cases, (4.3 and (4.3, the confdence ntervals wll be larger, the farther the X value s away from ts mean n the sample. Standard Error of a Resdual Fnally, the resduals e are the estmators of errors ε (see (4.7 and (4.8. The standard error of e s obtaned as follows, ( X - X se ( e = s - h where h = + n ( X - X (4.33

7 where s s gven by (4.. Note that whle the standard devaton of the error term s assumed to be homoscedastc, equaton 4.33 shows that the resduals of the regresson lne are heteroscedastc n nature. The standard error of each resdual depends on the value of h. The statstc h s called the hat statstc: h wll be larger, the greater the dstance of X from ts mean. A value of X whch s far away from ts mean (for example, an outler n the unvarate analyss of X wll produce a large hat statstc whch, as we shall see n secton 4.7, can exert undue nfluence on the locaton of a regresson lne. A data pont wth a large hat statstc s sad to exert leverage on the least squares regresson lne, the mportance of whch wll be shown n secton 4.7.

Causal, Explanatory Forecasting. Analysis. Regression Analysis. Simple Linear Regression. Which is Independent? Forecasting

Causal, Explanatory Forecasting. Analysis. Regression Analysis. Simple Linear Regression. Which is Independent? Forecasting Causal, Explanatory Forecastng Assumes cause-and-effect relatonshp between system nputs and ts output Forecastng wth Regresson Analyss Rchard S. Barr Inputs System Cause + Effect Relatonshp The job of

More information

The covariance is the two variable analog to the variance. The formula for the covariance between two variables is

The covariance is the two variable analog to the variance. The formula for the covariance between two variables is Regresson Lectures So far we have talked only about statstcs that descrbe one varable. What we are gong to be dscussng for much of the remander of the course s relatonshps between two or more varables.

More information

THE METHOD OF LEAST SQUARES THE METHOD OF LEAST SQUARES

THE METHOD OF LEAST SQUARES THE METHOD OF LEAST SQUARES The goal: to measure (determne) an unknown quantty x (the value of a RV X) Realsaton: n results: y 1, y 2,..., y j,..., y n, (the measured values of Y 1, Y 2,..., Y j,..., Y n ) every result s encumbered

More information

Economic Interpretation of Regression. Theory and Applications

Economic Interpretation of Regression. Theory and Applications Economc Interpretaton of Regresson Theor and Applcatons Classcal and Baesan Econometrc Methods Applcaton of mathematcal statstcs to economc data for emprcal support Economc theor postulates a qualtatve

More information

SIMPLE LINEAR CORRELATION

SIMPLE LINEAR CORRELATION SIMPLE LINEAR CORRELATION Smple lnear correlaton s a measure of the degree to whch two varables vary together, or a measure of the ntensty of the assocaton between two varables. Correlaton often s abused.

More information

x f(x) 1 0.25 1 0.75 x 1 0 1 1 0.04 0.01 0.20 1 0.12 0.03 0.60

x f(x) 1 0.25 1 0.75 x 1 0 1 1 0.04 0.01 0.20 1 0.12 0.03 0.60 BIVARIATE DISTRIBUTIONS Let be a varable that assumes the values { 1,,..., n }. Then, a functon that epresses the relatve frequenc of these values s called a unvarate frequenc functon. It must be true

More information

The Analysis of Covariance. ERSH 8310 Keppel and Wickens Chapter 15

The Analysis of Covariance. ERSH 8310 Keppel and Wickens Chapter 15 The Analyss of Covarance ERSH 830 Keppel and Wckens Chapter 5 Today s Class Intal Consderatons Covarance and Lnear Regresson The Lnear Regresson Equaton TheAnalyss of Covarance Assumptons Underlyng the

More information

CHAPTER 14 MORE ABOUT REGRESSION

CHAPTER 14 MORE ABOUT REGRESSION CHAPTER 14 MORE ABOUT REGRESSION We learned n Chapter 5 that often a straght lne descrbes the pattern of a relatonshp between two quanttatve varables. For nstance, n Example 5.1 we explored the relatonshp

More information

Linear Regression Analysis for STARDEX

Linear Regression Analysis for STARDEX Lnear Regresson Analss for STARDEX Malcolm Halock, Clmatc Research Unt The followng document s an overvew of lnear regresson methods for reference b members of STARDEX. Whle t ams to cover the most common

More information

Study on CET4 Marks in China s Graded English Teaching

Study on CET4 Marks in China s Graded English Teaching Study on CET4 Marks n Chna s Graded Englsh Teachng CHE We College of Foregn Studes, Shandong Insttute of Busness and Technology, P.R.Chna, 264005 Abstract: Ths paper deploys Logt model, and decomposes

More information

The Analysis of Outliers in Statistical Data

The Analysis of Outliers in Statistical Data THALES Project No. xxxx The Analyss of Outlers n Statstcal Data Research Team Chrysses Caron, Assocate Professor (P.I.) Vaslk Karot, Doctoral canddate Polychrons Economou, Chrstna Perrakou, Postgraduate

More information

PSYCHOLOGICAL RESEARCH (PYC 304-C) Lecture 12

PSYCHOLOGICAL RESEARCH (PYC 304-C) Lecture 12 14 The Ch-squared dstrbuton PSYCHOLOGICAL RESEARCH (PYC 304-C) Lecture 1 If a normal varable X, havng mean µ and varance σ, s standardsed, the new varable Z has a mean 0 and varance 1. When ths standardsed

More information

Calibration and Linear Regression Analysis: A Self-Guided Tutorial

Calibration and Linear Regression Analysis: A Self-Guided Tutorial Calbraton and Lnear Regresson Analyss: A Self-Guded Tutoral Part The Calbraton Curve, Correlaton Coeffcent and Confdence Lmts CHM314 Instrumental Analyss Department of Chemstry, Unversty of Toronto Dr.

More information

Inequality and The Accounting Period. Quentin Wodon and Shlomo Yitzhaki. World Bank and Hebrew University. September 2001.

Inequality and The Accounting Period. Quentin Wodon and Shlomo Yitzhaki. World Bank and Hebrew University. September 2001. Inequalty and The Accountng Perod Quentn Wodon and Shlomo Ytzha World Ban and Hebrew Unversty September Abstract Income nequalty typcally declnes wth the length of tme taen nto account for measurement.

More information

1. Measuring association using correlation and regression

1. Measuring association using correlation and regression How to measure assocaton I: Correlaton. 1. Measurng assocaton usng correlaton and regresson We often would lke to know how one varable, such as a mother's weght, s related to another varable, such as a

More information

Regression Models for a Binary Response Using EXCEL and JMP

Regression Models for a Binary Response Using EXCEL and JMP SEMATECH 997 Statstcal Methods Symposum Austn Regresson Models for a Bnary Response Usng EXCEL and JMP Davd C. Trndade, Ph.D. STAT-TECH Consultng and Tranng n Appled Statstcs San Jose, CA Topcs Practcal

More information

MULTIPLE LINEAR REGRESSION IN MINITAB

MULTIPLE LINEAR REGRESSION IN MINITAB MULTIPLE LINEAR REGRESSION IN MINITAB Ths document shows a complcated Mntab multple regresson. It ncludes descrptons of the Mntab commands, and the Mntab output s heavly annotated. Comments n { } are used

More information

CHAPTER 5 RELATIONSHIPS BETWEEN QUANTITATIVE VARIABLES

CHAPTER 5 RELATIONSHIPS BETWEEN QUANTITATIVE VARIABLES CHAPTER 5 RELATIONSHIPS BETWEEN QUANTITATIVE VARIABLES In ths chapter, we wll learn how to descrbe the relatonshp between two quanttatve varables. Remember (from Chapter 2) that the terms quanttatve varable

More information

benefit is 2, paid if the policyholder dies within the year, and probability of death within the year is ).

benefit is 2, paid if the policyholder dies within the year, and probability of death within the year is ). REVIEW OF RISK MANAGEMENT CONCEPTS LOSS DISTRIBUTIONS AND INSURANCE Loss and nsurance: When someone s subject to the rsk of ncurrng a fnancal loss, the loss s generally modeled usng a random varable or

More information

Time Series Analysis in Studies of AGN Variability. Bradley M. Peterson The Ohio State University

Time Series Analysis in Studies of AGN Variability. Bradley M. Peterson The Ohio State University Tme Seres Analyss n Studes of AGN Varablty Bradley M. Peterson The Oho State Unversty 1 Lnear Correlaton Degree to whch two parameters are lnearly correlated can be expressed n terms of the lnear correlaton

More information

PRACTICE 1: MUTUAL FUNDS EVALUATION USING MATLAB.

PRACTICE 1: MUTUAL FUNDS EVALUATION USING MATLAB. PRACTICE 1: MUTUAL FUNDS EVALUATION USING MATLAB. INDEX 1. Load data usng the Edtor wndow and m-fle 2. Learnng to save results from the Edtor wndow. 3. Computng the Sharpe Rato 4. Obtanng the Treynor Rato

More information

Latent Class Regression. Statistics for Psychosocial Research II: Structural Models December 4 and 6, 2006

Latent Class Regression. Statistics for Psychosocial Research II: Structural Models December 4 and 6, 2006 Latent Class Regresson Statstcs for Psychosocal Research II: Structural Models December 4 and 6, 2006 Latent Class Regresson (LCR) What s t and when do we use t? Recall the standard latent class model

More information

The OC Curve of Attribute Acceptance Plans

The OC Curve of Attribute Acceptance Plans The OC Curve of Attrbute Acceptance Plans The Operatng Characterstc (OC) curve descrbes the probablty of acceptng a lot as a functon of the lot s qualty. Fgure 1 shows a typcal OC Curve. 10 8 6 4 1 3 4

More information

Examples of Multiple Linear Regression Models

Examples of Multiple Linear Regression Models ECON *: Examples of Multple Regresson Models Examples of Multple Lnear Regresson Models Data: Stata tutoral data set n text fle autoraw or autotxt Sample data: A cross-sectonal sample of 7 cars sold n

More information

8.5 UNITARY AND HERMITIAN MATRICES. The conjugate transpose of a complex matrix A, denoted by A*, is given by

8.5 UNITARY AND HERMITIAN MATRICES. The conjugate transpose of a complex matrix A, denoted by A*, is given by 6 CHAPTER 8 COMPLEX VECTOR SPACES 5. Fnd the kernel of the lnear transformaton gven n Exercse 5. In Exercses 55 and 56, fnd the mage of v, for the ndcated composton, where and are gven by the followng

More information

Lecture 9: Logit/Probit. Prof. Sharyn O Halloran Sustainable Development U9611 Econometrics II

Lecture 9: Logit/Probit. Prof. Sharyn O Halloran Sustainable Development U9611 Econometrics II Lecture 9: Logt/Probt Prof. Sharyn O Halloran Sustanable Development U96 Econometrcs II Revew of Lnear Estmaton So far, we know how to handle lnear estmaton models of the type: Y = β 0 + β *X + β 2 *X

More information

DI Fund Sufficiency Evaluation Methodological Recommendations and DIA Russia Practice

DI Fund Sufficiency Evaluation Methodological Recommendations and DIA Russia Practice DI Fund Suffcency Evaluaton Methodologcal Recommendatons and DIA Russa Practce Andre G. Melnkov Deputy General Drector DIA Russa THE DEPOSIT INSURANCE CONFERENCE IN THE MENA REGION AMMAN-JORDAN, 18 20

More information

Can Auto Liability Insurance Purchases Signal Risk Attitude?

Can Auto Liability Insurance Purchases Signal Risk Attitude? Internatonal Journal of Busness and Economcs, 2011, Vol. 10, No. 2, 159-164 Can Auto Lablty Insurance Purchases Sgnal Rsk Atttude? Chu-Shu L Department of Internatonal Busness, Asa Unversty, Tawan Sheng-Chang

More information

Communication Networks II Contents

Communication Networks II Contents 8 / 1 -- Communcaton Networs II (Görg) -- www.comnets.un-bremen.de Communcaton Networs II Contents 1 Fundamentals of probablty theory 2 Traffc n communcaton networs 3 Stochastc & Marovan Processes (SP

More information

2.4 Bivariate distributions

2.4 Bivariate distributions page 28 2.4 Bvarate dstrbutons 2.4.1 Defntons Let X and Y be dscrete r.v.s defned on the same probablty space (S, F, P). Instead of treatng them separately, t s often necessary to thnk of them actng together

More information

L10: Linear discriminants analysis

L10: Linear discriminants analysis L0: Lnear dscrmnants analyss Lnear dscrmnant analyss, two classes Lnear dscrmnant analyss, C classes LDA vs. PCA Lmtatons of LDA Varants of LDA Other dmensonalty reducton methods CSCE 666 Pattern Analyss

More information

Analysis of Covariance

Analysis of Covariance Chapter 551 Analyss of Covarance Introducton A common tas n research s to compare the averages of two or more populatons (groups). We mght want to compare the ncome level of two regons, the ntrogen content

More information

Formula of Total Probability, Bayes Rule, and Applications

Formula of Total Probability, Bayes Rule, and Applications 1 Formula of Total Probablty, Bayes Rule, and Applcatons Recall that for any event A, the par of events A and A has an ntersecton that s empty, whereas the unon A A represents the total populaton of nterest.

More information

H 1 : at least one is not zero

H 1 : at least one is not zero Chapter 6 More Multple Regresson Model The F-test Jont Hypothess Tests Consder the lnear regresson equaton: () y = β + βx + βx + β4x4 + e for =,,..., N The t-statstc gve a test of sgnfcance of an ndvdual

More information

Statistical Methods to Develop Rating Models

Statistical Methods to Develop Rating Models Statstcal Methods to Develop Ratng Models [Evelyn Hayden and Danel Porath, Österrechsche Natonalbank and Unversty of Appled Scences at Manz] Source: The Basel II Rsk Parameters Estmaton, Valdaton, and

More information

Recurrence. 1 Definitions and main statements

Recurrence. 1 Definitions and main statements Recurrence 1 Defntons and man statements Let X n, n = 0, 1, 2,... be a MC wth the state space S = (1, 2,...), transton probabltes p j = P {X n+1 = j X n = }, and the transton matrx P = (p j ),j S def.

More information

Least Squares Fitting of Data

Least Squares Fitting of Data Least Squares Fttng of Data Davd Eberly Geoetrc Tools, LLC http://www.geoetrctools.co/ Copyrght c 1998-2016. All Rghts Reserved. Created: July 15, 1999 Last Modfed: January 5, 2015 Contents 1 Lnear Fttng

More information

Forecasting the Direction and Strength of Stock Market Movement

Forecasting the Direction and Strength of Stock Market Movement Forecastng the Drecton and Strength of Stock Market Movement Jngwe Chen Mng Chen Nan Ye cjngwe@stanford.edu mchen5@stanford.edu nanye@stanford.edu Abstract - Stock market s one of the most complcated systems

More information

The Application of Fractional Brownian Motion in Option Pricing

The Application of Fractional Brownian Motion in Option Pricing Vol. 0, No. (05), pp. 73-8 http://dx.do.org/0.457/jmue.05.0..6 The Applcaton of Fractonal Brownan Moton n Opton Prcng Qng-xn Zhou School of Basc Scence,arbn Unversty of Commerce,arbn zhouqngxn98@6.com

More information

Answer: A). There is a flatter IS curve in the high MPC economy. Original LM LM after increase in M. IS curve for low MPC economy

Answer: A). There is a flatter IS curve in the high MPC economy. Original LM LM after increase in M. IS curve for low MPC economy 4.02 Quz Solutons Fall 2004 Multple-Choce Questons (30/00 ponts) Please, crcle the correct answer for each of the followng 0 multple-choce questons. For each queston, only one of the answers s correct.

More information

Luby s Alg. for Maximal Independent Sets using Pairwise Independence

Luby s Alg. for Maximal Independent Sets using Pairwise Independence Lecture Notes for Randomzed Algorthms Luby s Alg. for Maxmal Independent Sets usng Parwse Independence Last Updated by Erc Vgoda on February, 006 8. Maxmal Independent Sets For a graph G = (V, E), an ndependent

More information

GRAVITY DATA VALIDATION AND OUTLIER DETECTION USING L 1 -NORM

GRAVITY DATA VALIDATION AND OUTLIER DETECTION USING L 1 -NORM GRAVITY DATA VALIDATION AND OUTLIER DETECTION USING L 1 -NORM BARRIOT Jean-Perre, SARRAILH Mchel BGI/CNES 18.av.E.Beln 31401 TOULOUSE Cedex 4 (France) Emal: jean-perre.barrot@cnes.fr 1/Introducton The

More information

Analysis of Premium Liabilities for Australian Lines of Business

Analysis of Premium Liabilities for Australian Lines of Business Summary of Analyss of Premum Labltes for Australan Lnes of Busness Emly Tao Honours Research Paper, The Unversty of Melbourne Emly Tao Acknowledgements I am grateful to the Australan Prudental Regulaton

More information

ErrorPropagation.nb 1. Error Propagation

ErrorPropagation.nb 1. Error Propagation ErrorPropagaton.nb Error Propagaton Suppose that we make observatons of a quantty x that s subject to random fluctuatons or measurement errors. Our best estmate of the true value for ths quantty s then

More information

Portfolio Loss Distribution

Portfolio Loss Distribution Portfolo Loss Dstrbuton Rsky assets n loan ortfolo hghly llqud assets hold-to-maturty n the bank s balance sheet Outstandngs The orton of the bank asset that has already been extended to borrowers. Commtment

More information

Control Charts for Means (Simulation)

Control Charts for Means (Simulation) Chapter 290 Control Charts for Means (Smulaton) Introducton Ths procedure allows you to study the run length dstrbuton of Shewhart (Xbar), Cusum, FIR Cusum, and EWMA process control charts for means usng

More information

n + d + q = 24 and.05n +.1d +.25q = 2 { n + d + q = 24 (3) n + 2d + 5q = 40 (2)

n + d + q = 24 and.05n +.1d +.25q = 2 { n + d + q = 24 (3) n + 2d + 5q = 40 (2) MATH 16T Exam 1 : Part I (In-Class) Solutons 1. (0 pts) A pggy bank contans 4 cons, all of whch are nckels (5 ), dmes (10 ) or quarters (5 ). The pggy bank also contans a con of each denomnaton. The total

More information

THE DISTRIBUTION OF LOAN PORTFOLIO VALUE * Oldrich Alfons Vasicek

THE DISTRIBUTION OF LOAN PORTFOLIO VALUE * Oldrich Alfons Vasicek HE DISRIBUION OF LOAN PORFOLIO VALUE * Oldrch Alfons Vascek he amount of captal necessary to support a portfolo of debt securtes depends on the probablty dstrbuton of the portfolo loss. Consder a portfolo

More information

STATISTICAL DATA ANALYSIS IN EXCEL

STATISTICAL DATA ANALYSIS IN EXCEL Mcroarray Center STATISTICAL DATA ANALYSIS IN EXCEL Lecture 6 Some Advanced Topcs Dr. Petr Nazarov 14-01-013 petr.nazarov@crp-sante.lu Statstcal data analyss n Ecel. 6. Some advanced topcs Correcton for

More information

Exhaustive Regression. An Exploration of Regression-Based Data Mining Techniques Using Super Computation

Exhaustive Regression. An Exploration of Regression-Based Data Mining Techniques Using Super Computation Exhaustve Regresson An Exploraton of Regresson-Based Data Mnng Technques Usng Super Computaton Antony Daves, Ph.D. Assocate Professor of Economcs Duquesne Unversty Pttsburgh, PA 58 Research Fellow The

More information

Marginal Benefit Incidence Analysis Using a Single Cross-section of Data. Mohamed Ihsan Ajwad and Quentin Wodon 1. World Bank.

Marginal Benefit Incidence Analysis Using a Single Cross-section of Data. Mohamed Ihsan Ajwad and Quentin Wodon 1. World Bank. Margnal Beneft Incdence Analyss Usng a Sngle Cross-secton of Data Mohamed Ihsan Ajwad and uentn Wodon World Bank August 200 Abstract In a recent paper, Lanjouw and Ravallon proposed an attractve and smple

More information

QUANTUM MECHANICS, BRAS AND KETS

QUANTUM MECHANICS, BRAS AND KETS PH575 SPRING QUANTUM MECHANICS, BRAS AND KETS The followng summares the man relatons and defntons from quantum mechancs that we wll be usng. State of a phscal sstem: The state of a phscal sstem s represented

More information

1 De nitions and Censoring

1 De nitions and Censoring De ntons and Censorng. Survval Analyss We begn by consderng smple analyses but we wll lead up to and take a look at regresson on explanatory factors., as n lnear regresson part A. The mportant d erence

More information

Psych 5741 (Carey): 8/22/97 Parametric Statistics - 1

Psych 5741 (Carey): 8/22/97 Parametric Statistics - 1 Psych 5741 (Carey): 8//97 Parametrc Statstcs - 1 1 Parametrc Statstcs: Tradtonal Approach 11 Denton o parametrc statstcs: Parametrc statstcs assume that the varable(s) o nterest n the populaton(s) o nterest

More information

v a 1 b 1 i, a 2 b 2 i,..., a n b n i.

v a 1 b 1 i, a 2 b 2 i,..., a n b n i. SECTION 8.4 COMPLEX VECTOR SPACES AND INNER PRODUCTS 455 8.4 COMPLEX VECTOR SPACES AND INNER PRODUCTS All the vector spaces we have studed thus far n the text are real vector spaces snce the scalars are

More information

Estimation of Dispersion Parameters in GLMs with and without Random Effects

Estimation of Dispersion Parameters in GLMs with and without Random Effects Mathematcal Statstcs Stockholm Unversty Estmaton of Dsperson Parameters n GLMs wth and wthout Random Effects Meng Ruoyan Examensarbete 2004:5 Postal address: Mathematcal Statstcs Dept. of Mathematcs Stockholm

More information

Variance estimation for the instrumental variables approach to measurement error in generalized linear models

Variance estimation for the instrumental variables approach to measurement error in generalized linear models he Stata Journal (2003) 3, Number 4, pp. 342 350 Varance estmaton for the nstrumental varables approach to measurement error n generalzed lnear models James W. Hardn Arnold School of Publc Health Unversty

More information

Underwriting Risk. Glenn Meyers. Insurance Services Office, Inc.

Underwriting Risk. Glenn Meyers. Insurance Services Office, Inc. Underwrtng Rsk By Glenn Meyers Insurance Servces Offce, Inc. Abstract In a compettve nsurance market, nsurers have lmted nfluence on the premum charged for an nsurance contract. hey must decde whether

More information

An Evaluation of the Extended Logistic, Simple Logistic, and Gompertz Models for Forecasting Short Lifecycle Products and Services

An Evaluation of the Extended Logistic, Simple Logistic, and Gompertz Models for Forecasting Short Lifecycle Products and Services An Evaluaton of the Extended Logstc, Smple Logstc, and Gompertz Models for Forecastng Short Lfecycle Products and Servces Charles V. Trappey a,1, Hsn-yng Wu b a Professor (Management Scence), Natonal Chao

More information

Testing GOF & Estimating Overdispersion

Testing GOF & Estimating Overdispersion Testng GOF & Estmatng Overdsperson Your Most General Model Needs to Ft the Dataset It s mportant that the most general (complcated) model n your canddate model lst fts the data well. Ths model s a benchmark

More information

Support Vector Machines

Support Vector Machines Support Vector Machnes Max Wellng Department of Computer Scence Unversty of Toronto 10 Kng s College Road Toronto, M5S 3G5 Canada wellng@cs.toronto.edu Abstract Ths s a note to explan support vector machnes.

More information

Lecture 3: Force of Interest, Real Interest Rate, Annuity

Lecture 3: Force of Interest, Real Interest Rate, Annuity Lecture 3: Force of Interest, Real Interest Rate, Annuty Goals: Study contnuous compoundng and force of nterest Dscuss real nterest rate Learn annuty-mmedate, and ts present value Study annuty-due, and

More information

Calculation of Sampling Weights

Calculation of Sampling Weights Perre Foy Statstcs Canada 4 Calculaton of Samplng Weghts 4.1 OVERVIEW The basc sample desgn used n TIMSS Populatons 1 and 2 was a two-stage stratfed cluster desgn. 1 The frst stage conssted of a sample

More information

Simon Acomb NAG Financial Mathematics Day

Simon Acomb NAG Financial Mathematics Day 1 Why People Who Prce Dervatves Are Interested In Correlaton mon Acomb NAG Fnancal Mathematcs Day Correlaton Rsk What Is Correlaton No lnear relatonshp between ponts Co-movement between the ponts Postve

More information

A random variable is a variable whose value depends on the outcome of a random event/experiment.

A random variable is a variable whose value depends on the outcome of a random event/experiment. Random varables and Probablty dstrbutons A random varable s a varable whose value depends on the outcome of a random event/experment. For example, the score on the roll of a de, the heght of a randomly

More information

Lecture 14: Implementing CAPM

Lecture 14: Implementing CAPM Lecture 14: Implementng CAPM Queston: So, how do I apply the CAPM? Current readng: Brealey and Myers, Chapter 9 Reader, Chapter 15 M. Spegel and R. Stanton, 2000 1 Key Results So Far All nvestors should

More information

greatest common divisor

greatest common divisor 4. GCD 1 The greatest common dvsor of two ntegers a and b (not both zero) s the largest nteger whch s a common factor of both a and b. We denote ths number by gcd(a, b), or smply (a, b) when there s no

More information

BERNSTEIN POLYNOMIALS

BERNSTEIN POLYNOMIALS On-Lne Geometrc Modelng Notes BERNSTEIN POLYNOMIALS Kenneth I. Joy Vsualzaton and Graphcs Research Group Department of Computer Scence Unversty of Calforna, Davs Overvew Polynomals are ncredbly useful

More information

START Selected Topics in Assurance

START Selected Topics in Assurance START Selected Topcs n Assurance Related Technologes Table of Contents Introducton Some Statstcal Bacground Fttng Normal and Lognormal Dstrbuton Fttng an Exponental Dstrbuton Fttng a Webull Dstrbuton A

More information

Properties of Indoor Received Signal Strength for WLAN Location Fingerprinting

Properties of Indoor Received Signal Strength for WLAN Location Fingerprinting Propertes of Indoor Receved Sgnal Strength for WLAN Locaton Fngerprntng Kamol Kaemarungs and Prashant Krshnamurthy Telecommuncatons Program, School of Informaton Scences, Unversty of Pttsburgh E-mal: kakst2,prashk@ptt.edu

More information

Chapter 15: Debt and Taxes

Chapter 15: Debt and Taxes Chapter 15: Debt and Taxes-1 Chapter 15: Debt and Taxes I. Basc Ideas 1. Corporate Taxes => nterest expense s tax deductble => as debt ncreases, corporate taxes fall => ncentve to fund the frm wth debt

More information

Fuzzy Regression and the Term Structure of Interest Rates Revisited

Fuzzy Regression and the Term Structure of Interest Rates Revisited Fuzzy Regresson and the Term Structure of Interest Rates Revsted Arnold F. Shapro Penn State Unversty Smeal College of Busness, Unversty Park, PA 68, USA Phone: -84-865-396, Fax: -84-865-684, E-mal: afs@psu.edu

More information

Institute of Informatics, Faculty of Business and Management, Brno University of Technology,Czech Republic

Institute of Informatics, Faculty of Business and Management, Brno University of Technology,Czech Republic Lagrange Multplers as Quanttatve Indcators n Economcs Ivan Mezník Insttute of Informatcs, Faculty of Busness and Management, Brno Unversty of TechnologCzech Republc Abstract The quanttatve role of Lagrange

More information

CHOLESTEROL REFERENCE METHOD LABORATORY NETWORK. Sample Stability Protocol

CHOLESTEROL REFERENCE METHOD LABORATORY NETWORK. Sample Stability Protocol CHOLESTEROL REFERENCE METHOD LABORATORY NETWORK Sample Stablty Protocol Background The Cholesterol Reference Method Laboratory Network (CRMLN) developed certfcaton protocols for total cholesterol, HDL

More information

Quantization Effects in Digital Filters

Quantization Effects in Digital Filters Quantzaton Effects n Dgtal Flters Dstrbuton of Truncaton Errors In two's complement representaton an exact number would have nfntely many bts (n general). When we lmt the number of bts to some fnte value

More information

Passive Filters. References: Barbow (pp 265-275), Hayes & Horowitz (pp 32-60), Rizzoni (Chap. 6)

Passive Filters. References: Barbow (pp 265-275), Hayes & Horowitz (pp 32-60), Rizzoni (Chap. 6) Passve Flters eferences: Barbow (pp 6575), Hayes & Horowtz (pp 360), zzon (Chap. 6) Frequencyselectve or flter crcuts pass to the output only those nput sgnals that are n a desred range of frequences (called

More information

ESTIMATING THE MARKET VALUE OF FRANKING CREDITS: EMPIRICAL EVIDENCE FROM AUSTRALIA

ESTIMATING THE MARKET VALUE OF FRANKING CREDITS: EMPIRICAL EVIDENCE FROM AUSTRALIA ESTIMATING THE MARKET VALUE OF FRANKING CREDITS: EMPIRICAL EVIDENCE FROM AUSTRALIA Duc Vo Beauden Gellard Stefan Mero Economc Regulaton Authorty 469 Wellngton Street, Perth, WA 6000, Australa Phone: (08)

More information

Credit Limit Optimization (CLO) for Credit Cards

Credit Limit Optimization (CLO) for Credit Cards Credt Lmt Optmzaton (CLO) for Credt Cards Vay S. Desa CSCC IX, Ednburgh September 8, 2005 Copyrght 2003, SAS Insttute Inc. All rghts reserved. SAS Propretary Agenda Background Tradtonal approaches to credt

More information

Part 1: quick summary 5. Part 2: understanding the basics of ANOVA 8

Part 1: quick summary 5. Part 2: understanding the basics of ANOVA 8 Statstcs Rudolf N. Cardnal Graduate-level statstcs for psychology and neuroscence NOV n practce, and complex NOV desgns Verson of May 4 Part : quck summary 5. Overvew of ths document 5. Background knowledge

More information

A statistical approach to determine Microbiologically Influenced Corrosion (MIC) Rates of underground gas pipelines.

A statistical approach to determine Microbiologically Influenced Corrosion (MIC) Rates of underground gas pipelines. A statstcal approach to determne Mcrobologcally Influenced Corroson (MIC) Rates of underground gas ppelnes. by Lech A. Grzelak A thess submtted to the Delft Unversty of Technology n conformty wth the requrements

More information

SPEE Recommended Evaluation Practice #6 Definition of Decline Curve Parameters Background:

SPEE Recommended Evaluation Practice #6 Definition of Decline Curve Parameters Background: SPEE Recommended Evaluaton Practce #6 efnton of eclne Curve Parameters Background: The producton hstores of ol and gas wells can be analyzed to estmate reserves and future ol and gas producton rates and

More information

INTRODUCTION TO MONTE CARLO SIMULATION. Samik Raychaudhuri

INTRODUCTION TO MONTE CARLO SIMULATION. Samik Raychaudhuri Proceedngs of the 2008 Wnter Smulaton Conference S. J. Mason, R. R. Hll, L. Mönch, O. Rose, T. Jefferson, J. W. Fowler eds. INTRODUCTION TO MONTE CARLO SIMULATION Samk Raychaudhur Oracle Crystal Ball Global

More information

Although ordinary least-squares (OLS) regression

Although ordinary least-squares (OLS) regression egresson through the Orgn Blackwell Oxford, TEST 0141-98X 003 5 31000 Orgnal Joseph Teachng G. UK Artcle Publshng Esenhauer through Statstcs the Ltd Trust Orgn 001 KEYWODS: Teachng; egresson; Analyss of

More information

Diagnostic Tests of Cross Section Independence for Nonlinear Panel Data Models

Diagnostic Tests of Cross Section Independence for Nonlinear Panel Data Models DISCUSSION PAPER SERIES IZA DP No. 2756 Dagnostc ests of Cross Secton Independence for Nonlnear Panel Data Models Cheng Hsao M. Hashem Pesaran Andreas Pck Aprl 2007 Forschungsnsttut zur Zukunft der Arbet

More information

The Magnetic Field. Concepts and Principles. Moving Charges. Permanent Magnets

The Magnetic Field. Concepts and Principles. Moving Charges. Permanent Magnets . The Magnetc Feld Concepts and Prncples Movng Charges All charged partcles create electrc felds, and these felds can be detected by other charged partcles resultng n electrc force. However, a completely

More information

Joe Pimbley, unpublished, 2005. Yield Curve Calculations

Joe Pimbley, unpublished, 2005. Yield Curve Calculations Joe Pmbley, unpublshed, 005. Yeld Curve Calculatons Background: Everythng s dscount factors Yeld curve calculatons nclude valuaton of forward rate agreements (FRAs), swaps, nterest rate optons, and forward

More information

Simple Interest Loans (Section 5.1) :

Simple Interest Loans (Section 5.1) : Chapter 5 Fnance The frst part of ths revew wll explan the dfferent nterest and nvestment equatons you learned n secton 5.1 through 5.4 of your textbook and go through several examples. The second part

More information

INVESTIGATION OF VEHICULAR USERS FAIRNESS IN CDMA-HDR NETWORKS

INVESTIGATION OF VEHICULAR USERS FAIRNESS IN CDMA-HDR NETWORKS 21 22 September 2007, BULGARIA 119 Proceedngs of the Internatonal Conference on Informaton Technologes (InfoTech-2007) 21 st 22 nd September 2007, Bulgara vol. 2 INVESTIGATION OF VEHICULAR USERS FAIRNESS

More information

We are now ready to answer the question: What are the possible cardinalities for finite fields?

We are now ready to answer the question: What are the possible cardinalities for finite fields? Chapter 3 Fnte felds We have seen, n the prevous chapters, some examples of fnte felds. For example, the resdue class rng Z/pZ (when p s a prme) forms a feld wth p elements whch may be dentfed wth the

More information

CLUSTER SAMPLING DR. SHALABH DEPARTMENT OF MATHEMATICS AND STATISTICS INDIAN INSTITUTE OF TECHNOLOGY KANPUR 1

CLUSTER SAMPLING DR. SHALABH DEPARTMENT OF MATHEMATICS AND STATISTICS INDIAN INSTITUTE OF TECHNOLOGY KANPUR 1 amplng Theory MODULE IX LECTURE - 30 CLUTER AMPLIG DR HALABH DEPARTMET OF MATHEMATIC AD TATITIC IDIA ITITUTE OF TECHOLOGY KAPUR It s one of the asc assumptons n any samplng procedure that the populaton

More information

Texas Instruments 30X IIS Calculator

Texas Instruments 30X IIS Calculator Texas Instruments 30X IIS Calculator Keystrokes for the TI-30X IIS are shown for a few topcs n whch keystrokes are unque. Start by readng the Quk Start secton. Then, before begnnng a specfc unt of the

More information

Control Charts with Supplementary Runs Rules for Monitoring Bivariate Processes

Control Charts with Supplementary Runs Rules for Monitoring Bivariate Processes Control Charts wth Supplementary Runs Rules for Montorng varate Processes Marcela. G. Machado *, ntono F.. Costa * * Producton Department, Sao Paulo State Unversty, Campus of Guaratnguetá, 56-4 Guaratnguetá,

More information

FORCED CONVECTION HEAT TRANSFER IN A DOUBLE PIPE HEAT EXCHANGER

FORCED CONVECTION HEAT TRANSFER IN A DOUBLE PIPE HEAT EXCHANGER FORCED CONVECION HEA RANSFER IN A DOUBLE PIPE HEA EXCHANGER Dr. J. Mchael Doster Department of Nuclear Engneerng Box 7909 North Carolna State Unversty Ralegh, NC 27695-7909 Introducton he convectve heat

More information

1 Example 1: Axis-aligned rectangles

1 Example 1: Axis-aligned rectangles COS 511: Theoretcal Machne Learnng Lecturer: Rob Schapre Lecture # 6 Scrbe: Aaron Schld February 21, 2013 Last class, we dscussed an analogue for Occam s Razor for nfnte hypothess spaces that, n conjuncton

More information

An empirical study for credit card approvals in the Greek banking sector

An empirical study for credit card approvals in the Greek banking sector An emprcal study for credt card approvals n the Greek bankng sector Mara Mavr George Ioannou Bergamo, Italy 17-21 May 2004 Management Scences Laboratory Department of Management Scence & Technology Athens

More information

An Empirical Study of Search Engine Advertising Effectiveness

An Empirical Study of Search Engine Advertising Effectiveness An Emprcal Study of Search Engne Advertsng Effectveness Sanjog Msra, Smon School of Busness Unversty of Rochester Edeal Pnker, Smon School of Busness Unversty of Rochester Alan Rmm-Kaufman, Rmm-Kaufman

More information

Multilevel Analysis (ver. 1.0)

Multilevel Analysis (ver. 1.0) Multlevel Analyss (ver. 1.0) Oscar Torres-Reyna Data Consultant otorres@prnceton.edu http://dss.prnceton.edu/tranng/ Motvaton Use multlevel model whenever your data s grouped (or nested) n more than one

More information

New bounds in Balog-Szemerédi-Gowers theorem

New bounds in Balog-Szemerédi-Gowers theorem New bounds n Balog-Szemeréd-Gowers theorem By Tomasz Schoen Abstract We prove, n partcular, that every fnte subset A of an abelan group wth the addtve energy κ A 3 contans a set A such that A κ A and A

More information

total A A reag total A A r eag

total A A reag total A A r eag hapter 5 Standardzng nalytcal Methods hapter Overvew 5 nalytcal Standards 5B albratng the Sgnal (S total ) 5 Determnng the Senstvty (k ) 5D Lnear Regresson and albraton urves 5E ompensatng for the Reagent

More information

Ring structure of splines on triangulations

Ring structure of splines on triangulations www.oeaw.ac.at Rng structure of splnes on trangulatons N. Vllamzar RICAM-Report 2014-48 www.rcam.oeaw.ac.at RING STRUCTURE OF SPLINES ON TRIANGULATIONS NELLY VILLAMIZAR Introducton For a trangulated regon

More information