An Extreme Value Theory Approach for Analyzing the Extreme Risk of the Gold Prices

 Gloria Cameron
 3 months ago
 Views:
Transcription
1 A Extreme Value Theory Approach for Aalyzig the Extreme Risk of the Gold Prices JiahBag Jag Abstract Fiacial markets frequetly experiece extreme movemets i the egative side. Accurate computatio of value at risk ad expected shortfall are the mai tasks of the risk maagers or portfolio maagers. I this paper, gold prices (i US dollars) have bee examied to illustrate the mai idea of extreme value theory ad discuss the tail behaviour. GEV ad GPD are used to compute VaR ad ES. The results show that GPD model with threshold is a better choice. Keywords: value at risk, expected shortfall, geeralized extreme value distributio, geeralized Pareto distributio
2 Itroductio Risk maagers ad Portfolio maagers cocer extreme egative side movemets i the fiacial markets. A log list of research has posted o this topic. Ramaza ad Faruk (2006) examie the dyamics of extreme values of overight borrowig rates i a iterbak moey market. Geeralized Pareto distributio has bee picked for its well fittig. Feradez (2005) usig extreme value theory to the Uited States, Europe, Asia, ad Lati America fiacial markets for computig value at risk (VaR). Oe of his fidigs is, o average, EVT provides the most accurate estimate of VaR. Byström (2005) applied extreme value theory to the case of extremely large electricity price chages ad declared a good fit with geeralized Pareto distributio (GPD). Bali (2003) determies the type of asymptotic distributio for the extreme chages i U.S. Treasury yields. I his paper, the thitailed Gumbel ad expoetial distributio are worse tha the fattailed Fréchet ad Pareto distributios. Neftci (2000) foud that the extreme distributio theory fit well for the extreme evets i fiacial markets. Geçay ad Selçuk (2004) ivestigate the extreme value theory to geerate VaR estimates ad study the tail forecasts of daily returs for stress testig. Maroh (2005) studies the tail idex i the case of geeralized order statistics, ad declares the asymptotic properties of the Fréchet distributio. Brooks, Clare, Dalle Molle ad Persad, G., (2005) apply a umber of differet extreme value models for computig the value at risk of three LIFFE futures cotracts. Based upo the log list of applicatios of extreme value theory ad theory basis, this paper will first discuss the geeralized extreme value methods ad the fit the model with gold prices (i US dollars) to compute the chace of ew record geerated i the ext period. Differet models are also bee fitted with gold data. The results show that GPD model with threshold is a better choice. 2. Methodology 2.1 Geeralized Extreme Value (GEV) Distributio Extreme movemets i gold prices ca be measured by the daily returs of the gold prices per troy ouce (reported o Lodo Gold Market). Losses or egative returs are the mai cocers i the field of fiacial risk maagemet, for examples stock market crashes. For simplicity, extreme movemets i the left tail of the distributio ca be characterized by the positive umbers of the right high quitiles. Let Xi of the ith daily retur of the gold prices betwee day i ad day i 1. Defie where Pi ad i 1 X (l P l P ) = i i i 1 be the egative P are the gold prices of day i ad day i 1. Suppose that X1, X 2,..., X be iid radom variables with a ukow cumulative distributio fuctio (CDF) F( x) = Pr( X x). Extreme values are defied as maxima (or miimum) of the i idepedetly ad idetically distributed radom variables X1, X 2,..., X. Let X ( ) the maximum egative side movemets i the daily gold prices returs, that is X X X X = ( ) max( 1, 2,..., ). Sice the extreme movemets are the focus of this study, the exact distributio of X ( ) ca be writte as be
3 A Extreme Value Theory Approach for Aalyzig The Extreme Risk of the Gold Prices 99 Pr( X a) Pr( X a, X a,..., X a) = ( ) 1 2 I practice the paret distributio F = i = 1 F ( a) = F ( a ) (1) is usually ukow or ot precisely kow. The empirical estimatio of the distributio F ( a) is poor i this case. Fisher ad Tippet (1928) derived the asymptotic distributio of F ( a ). Suppose µ ( ) ad σ ( ) sequeces of real umber locatio ad scale measures of the maximum statistic X ( ). The the stadardized maximum statistic are Z ( X µ ) / σ * ( ) = ( ) ( ) ( ) Coverges to z = ( x µ ) / σ which has oe of three forms of odegeerate distributio families such as H ( z ) e x p { e x p [ z ] }, z = < < H z z z ( ) 1 / ξ = e x p { }, > 0 = 0, e l s e H z z z ( ) 1 / ξ = e x p { [ ] }, < 0 = 1, e l s e (2) These forms go uder the ames of Gumbel, Fréchet, ad Weibull respectively. Here µ ad σ are the mea retur ad volatility of the extreme values x ad ξ is the shape parameter or called 1/ ξ the tail idex of the extreme statistic distributio. With ξ = 0, ξ > 0, ξ < 0 represet Gumbel, Fréchet, ad Weibull types of tail behavior respectively. ad short tails respectively. I fact Gumbel, Fréchet, ad Weibull types ca be fit for expoetial, log, Embrechts ad Mikosch (1997) proposed a geeralized extreme value (GEV) distributio which icluded those three types ad ca be used for the case statioary GARCH processes. GEV distributio has the followig form H ( x; µ, σ ) = exp{ exp[ ( x µ ) / σ ]}, x ; ξ = 0 ξ 2.2 Parameters Estimatio for GEV 1/ ξ = exp{ [1 + ξ ( µ ) / σ ], 1 + ξ ( µ ) / σ > 0; ξ 0 x Suppose that block maxima B1, B2,..., Bk are idepedet variables from a GEV distributio, the loglikelihood fuctio for the GEV, uder the case of ξ 0, ca be give x (3)
4 as l L k l (1 1 / ) l {1 ( B ) / } = σ + ξ + ξ µ σ i i = 1 k {1 ( B ) / } + ξ µ σ i i = 1 For the Gumbel type GEV form, the loglikelihood fuctio ca be writte as k = σ i µ σ i µ σ i 1 i 1 = = k l L k l ( B ) / e x p { ( B ) / } (5) As Smith (1985) declared that, for ξ > 0.5, the maximum likelihood estimators, for ξ, µ, ad σ, satisfy the regular coditios ad therefore havig asymptotic ad cosistet properties. The umber of blocks, k ad the block size form a crucial tradeoff betwee variace ad bias of parameters estimatio. 1 / k ξ (4) POT ad GPD Fittig models with more data is better tha less, so peaks over thresholds (POT) method utilizes data over a specified threshold. Defie the excess distributio as F ( x ) = P r ( X h < x X > h ) h = F ( x + h ) F ( h ) 1 F ( h ) where h is the threshold ad F is a ukow distributio such that the CDF of the maxima will coverge to a GEV type distributio. (6) For large value of threshold h, there exists a fuctio τ ( h) > 0 such that the excess distributio of equatio (6) will approximated by the geeralized Pareto distributio (GPD) with the followig form H ( x ) = 1 e x p ( x / τ ( h )), ξ = 0 ξ, τ ( h ) 1 / ξ = 1 (1 + ξ / τ ( )), ξ 0 where x > 0 for the case of ξ 0, ad 0 x τ ( h) / ξ for the case of ξ < 0. Defie x1, x2,..., xk as the extreme values which are positive values after subtractig threshold h. For large value of h, x1, x2,..., xk is a radom sample from a GPD, so the ukow parameters ξ ad τ ( h) loglikelihood fuctio. VaR ad ES derived as x ca be estimated with maximum likelihood estimatio o GPD Based o equatio (6) ad GPD distributio, the ukow distributio F F( y) (1 F( h)) H ( x) F( h) = ξ, τ ( h) + (8) h (7) ca be
5 A Extreme Value Theory Approach for Aalyzig The Extreme Risk of the Gold Prices 101 where y = h + x. F( h) ca be estimated with oparametric empirical estimator F( h) = ( k) / where k is the umber of extreme values exceed the threshold h. Therefore the estimator of (8) is where ξ ad τ ( h) F( y) (1 F( h)) H ( x; ξ, τ ( h)) F( h) are mle of GPD loglikelihood. shortfall ca be computed usig (9). First, defie F( VaRq ) distributio fuctio up to q th quatile VaR q. Next, give that as From (10), GPD distributio. VaRq = + (9) 1 VaR q = F ( q) Therefore High quatile VaR ad expected = q as the probability of ξ = h + τ ( h){[( / k)(1 q)] 1}/ ξ (10) is exceeded, defie the expected loss size, expected shortfall (ES), ES = E( X X > VaR ) ES q ca be computed usig VaR q Therefore, 3. Data ad Empirical Results 3.1 Data ad Data Exploratio q q VaR E( X VaR X VaR ) = + > q q q ES q = VaRq/(1 ξ ) + ( τ ( h) ξ h) /(1 ξ ) (11) ad the estimated mea excess fuctio of The upper pael of Figure 1 shows the daily gold prices per troy ouce (i US dollars) over the period of Jauary 1, 1985 through March 31, The lower pael of Figure 1 is the cotiuous percetage returs of the gold prices i the upper pael. Table 1 shows that daily gold returs have a positive skew (0.76>0) ad large kurtosis (14.16>3). It shows the distributio of returs has a fat tail.
6 Figure 1 Daily Gold Prices Per Troy Ouce (i US Dollars) ad Daily Gold Percetage Returs Daily Gold prices Daily Gold Percetage Chage Table 1 Descriptive Statistics of Daily Gold Percetage Returs Miimum st Quarter Mea Media 0 3rd Quarter 0.43 Maximum 9.64 Total N 5371 Std Deviatio 0.90 Skewess 0.76 Kurtosis Figure 2 shows that there is o autocorrelatio i the daily gold percetage retur series but the squared daily gold percetage returs appear sigificat autocorrelatio. Figure 2 suggest that the daily gold percetage returs ca be a GARCH time series model. It is reasoable to apply GEV distributio to model the maxima egative returs ad GPD distributio to model the excess distributio for egative daily gold percetage returs.
7 A Extreme Value Theory Approach for Aalyzig The Extreme Risk of the Gold Prices 103 Figure2 ACF of Daily Gold Percetage Returs ad Squared Daily Gold Percetage Returs Series : gold.r ACF Lag Series : gold.r^2 ACF Parametric Maximum Likelihood Estimates Lag The block maxima of egative returs have bee fitted to GEV model with three differet block sizes (yearly, quarterly, ad mothly). The results, listed i Table 2, show the estimates of three parameters ξ, σ, ad µ with icreasig block umbers ad the estimates of stadard error of the parameters are listed i paretheses. All block umbers show that shape parameter is positive except yearly block size case is egative but ot sigificat. All the parameters have decreasig estimated stadard errors as the umber of blocks icreased. I geeral, the block maxima of the egative returs follows a Fréchet family of GEV for quarter ad moth block frames. The Fréchet type of GEV cofirms that the origial series has fat tail. Table 2 Parametric Maximum Likelihood Estimates with three block frames (block sizes: year, quarter, ad moth) ad chace of a ew record will occur durig the ext period Year Quarter Moth ξ (0.19) (0.10) (0.06) σ 0.97 (0.18) 0.75 (0.07) 0.59 (0.03) µ (0.24) (0.09) (0.04) New record 1.77% 0.86% 0.44%
8 Figure 3, from left to right is the crude residual plot ad quatileversusquatile plot with expoetial distributio as the referece distributio for quarter block maxima. The QQ plot looks liear ad shows that the GEV distributio model is a good choice. It shows that the Fréchet distributios are fitted well for the block maxima egative returs. Figure 3 Residual plots versus quarter frame block maxima Residuals Expoetial Quatiles Orderig Ordered Data With the above parameters ad correspodig Fréchet distributios, there is 1.77%, 0.86% ad 0.44% chace that a ew record will be occurred durig the ext period (year, quarter ad moth). The poit estimates ad 95% cofidece iterval of 10 ad 20 years retur levels are listed below. Give the same time iterval (10 or 20 years), the poit estimate is icreased with decreasig block sizes. With the icreasig time iterval, the poit estimate icreased ad cofidece iterval is wider. It reveals that time is a risk factor. Table 3 Retur levels for the three block frames with 95% cofidece bouds 10 year retur level 20 year retur level Year Quarter Moth (4.31, 6.91) (4.28, 7.17) (4.57, 7.29) 5.58 (4.75, 9.13) 5.96 (4.77, 9.08) 6.53 (5.18, 9.11)
9 A Extreme Value Theory Approach for Aalyzig The Extreme Risk of the Gold Prices Extreme Over Thresholds ad Risk Measures A suitable threshold should be specified to fid the GPD approximatio of the egative returs excess distributio. Try several thresholds to fit the excess GPD. Do QQplots with correspodig GPD distributios for the daily egative returs over the thresholds (show o Figure 4) ad compare some sigificat umbers (Table 4). Figure 4 Residual diagostic check for GPD ( with threshold 2) fit to Daily Gold egative returs Fu(xu) F(x) (o log scale) x (o log scale) x (o log scale) Residuals Expoetial Quatiles Orderig Ordered Data Figure 4 show that the threshold 2 model fit the GPD well. O the topleft pael of Figure 4 is the GPD estimate of the excess distributio, topright pael is the tail estimate of equatio (9), bottomleft pael is the scatter plot of residuals, ad bottomright pael is the QQplot of residuals. Base o the diagostic plots show data fit GPD model well. Table 4 shows that the shape parameter is stable icreasig as threshold icreasig from 1.5 to 2, ad the shape parameter swigs to egative value. The details are displayed as Figure 5. Theoretically, shape parameter of GEV ad GPD should be the same. The shape parameters of threshold 2 ad the moth block size GEV are similar (see table 2). Accordig to the referred iformatio, the estimate value 2 for shape parameter is the best specified threshold. The GPD approximatio has shape parameter 0.15 ad scale parameter 0.56.
10 Table 4 GPD model fittig results with thresholds from 1 to 6 threshold # of exceed Probability less tha threshold ξ τ ( h) (0.05) 0.62 (0.04) (0.06) 0.66 (0.05) (0.07) 0.68 (0.06) (0.09) 0.60 (0.07) (0.12) 0.56 (0.09) (0.14) 0.72 (0.14) (0.18) 0.67 (0.16) Figure 5 Estimates of shape parameter for egative returs as a fuctio of the threshold values. Threshold Shape (xi) (CI, p = 0.95) Exceedaces To measure the risk, ValueatRisk ad expected shortfall (ES) ca be calculated based o GPD approximatio with threshold 2. The results is listed i row 1 of Table 5. For compariso, the values of the GPD approximatio with threshold 2 ad ormal distributio approximatios are listed i the followig table. All the values of Normal Distributio approximatio is uder estimate. GPD model with threshold 2 is a better chose.
11 A Extreme Value Theory Approach for Aalyzig The Extreme Risk of the Gold Prices 107 Table 5 Risk measures (VaR ad ES) for GPD model with thresholds 2 compared with ormal distributio VaR. 95 ES. 95 VaR. 99 ES. 99 Threshold= Normal Distr Cocludig Remarks Fisher ad Tippet theorem has a assumptio of iid, however, the GEV is suitable for statioary time series icludig statioary GARCH cases. Explorig the depedece properties o extreme values ad the joittail properties of multivariate extreme value cases are appeared ot log ago. Further ivestigatio is defiite welcome. Refereces Bali, T. G., (2003), A extreme value approach to estimatig volatility ad value at risk, Joural of Busiess, 76: Brooks, C., Clare, A. D., Dalle Molle, J.W., ad Persad, G., (2005), A compariso of extreme value theory approaches for determiig value at risk, Joural of Empirical Fiace, vol. 12, Byström, Has N.E., (2005), Extreme value theory ad extremely large electricity price chages, Iteratioal Review of Ecoomics & Fiace, 14:1, Chou, HegChih ad Che, SheYua, (2004), Price limits ad the applicatio of extreme value theory i margi settig, Joural of Risk Maagemet, 6:2, Coles, S., (2001), A itroductio to statistical modelig of extreme values, Lodo: Spriger. Embrechts, P., Klüppelberg, C, ad Mikosch, T., (1997),Modelig extremal evets for isurace ad fiace, Berli ad Heidelberg: Spriger. Feradez, V.,(2005), Risk Maagemet uder extreme evets, Iteratioal Review of Fiacial Aalysis, 14:2, Fisher, R. ad Tippett, L., (1928), Limitig forms of the frequecy distributio of the largest or smallest member of a sample, Proceegigs of the Cambridge Philosophical Society, 24,
12 Geçay, R. ad Selçuk, F., (2004), Extreme value theory ad valueatrisk: relative performace i emergig markets, Iteratioal Joural of Forecastig, 20:2, Jekiso, A. F., (1955), The frequecy distributio of the aual maximum (or miimum) values of meteorological elemets, Quarterly Joural of the Royal Meteorological Society, 87, Maroh, F., (2005), Tail idex estimatio i models of geeralized order statistics, Commuicatios i Statistics: Theory & Methods, 34:5, Neftci, S. N., (2000), Value at risk calculatios, extreme evets, ad tail Estimatio, Joural of Derivatives, 7:3, Ramaza, G. ad Faruk, S., (2006), Overight borrowig, iterest rates ad extreme value theory, Europea Ecoomic Review, Vol.50, Smith, R. L., (1985), Maximum likelihood estimatio i a class of oregular cases, Biometrika, 72,
13 A Extreme Value Theory Approach for Aalyzig The Extreme Risk of the Gold Prices 109
MOST PEOPLE WOULD RATHER LIVE WITH A PROBLEM THEY CAN'T SOLVE, THAN ACCEPT A SOLUTION THEY CAN'T UNDERSTAND.
XI1 (1074) MOST PEOPLE WOULD RATHER LIVE WITH A PROBLEM THEY CAN'T SOLVE, THAN ACCEPT A SOLUTION THEY CAN'T UNDERSTAND. R. E. D. WOOLSEY AND H. S. SWANSON XI2 (1075) STATISTICAL DECISION MAKING Advaced
More informationElementary Statistics
Elemetary Statistics M. Ghamsary, Ph.D. Sprig 004 Chap 0 Descriptive Statistics Raw Data: Whe data are collected i origial form, they are called raw data. The followig are the scores o the first test of
More informationA goodnessoffit test based on the empirical characteristic function and a comparison of tests for normality
A goodessoffit test based o the empirical characteristic fuctio ad a compariso of tests for ormality J. Marti va Zyl Departmet of Mathematical Statistics ad Actuarial Sciece, Uiversity of the Free State,
More information62. Power series Definition 16. (Power series) Given a sequence {c n }, the series. c n x n = c 0 + c 1 x + c 2 x 2 + c 3 x 3 +
62. Power series Defiitio 16. (Power series) Give a sequece {c }, the series c x = c 0 + c 1 x + c 2 x 2 + c 3 x 3 + is called a power series i the variable x. The umbers c are called the coefficiets of
More informationIt should be unbiased, or approximately unbiased. Variance of the variance estimator should be small. That is, the variance estimator is stable.
Chapter 10 Variace Estimatio 10.1 Itroductio Variace estimatio is a importat practical problem i survey samplig. Variace estimates are used i two purposes. Oe is the aalytic purpose such as costructig
More informationSTATISTICAL INFERENCE
STATISTICAL INFERENCE POPULATION AND SAMPLE Populatio = all elemets of iterest Characterized by a distributio F with some parameter θ Sample = the data X 1,..., X, selected subset of the populatio = sample
More informationKLMED8004 Medical statistics. Part I, autumn Estimation. We have previously learned: Population and sample. New questions
We have previously leared: KLMED8004 Medical statistics Part I, autum 00 How kow probability distributios (e.g. biomial distributio, ormal distributio) with kow populatio parameters (mea, variace) ca give
More informationBinomial Distribution
0.0 0.5 1.0 1.5 2.0 2.5 3.0 0 1 2 3 4 5 6 7 0.0 0.5 1.0 1.5 2.0 2.5 3.0 Overview Example: coi tossed three times Defiitio Formula Recall that a r.v. is discrete if there are either a fiite umber of possible
More informationSome Properties of the Exact and Score Methods for Binomial Proportion and Sample Size Calculation
Some Properties of the Exact ad Score Methods for Biomial Proportio ad Sample Size Calculatio K. KRISHNAMOORTHY AND JIE PENG Departmet of Mathematics, Uiversity of Louisiaa at Lafayette Lafayette, LA 705041010,
More informationREGRESSION (Physics 1210 Notes, Partial Modified Appendix A)
REGRESSION (Physics 0 Notes, Partial Modified Appedix A) HOW TO PERFORM A LINEAR REGRESSION Cosider the followig data poits ad their graph (Table I ad Figure ): X Y 0 3 5 3 7 4 9 5 Table : Example Data
More informationAsymptotic Results for the Linear Regression Model
Asymptotic Results for the Liear Regressio Model C. Fli November 29, 2000 1. Asymptotic Results uder Classical Assumptios The followig results apply to the liear regressio model y = Xβ + ε, where X is
More informationAnalysis of Algorithms. Introduction. Contents
Itroductio The focus of this module is mathematical aspects of algorithms. Our mai focus is aalysis of algorithms, which meas evaluatig efficiecy of algorithms by aalytical ad mathematical methods. We
More informationChapter 6 Principles of Data Reduction
Chapter 6 for BST 695: Special Topics i Statistical Theory. Kui Zhag, 0 Chapter 6 Priciples of Data Reductio Sectio 6. Itroductio Goal: To summarize or reduce the data X, X,, X to get iformatio about a
More informationPostedPrice, SealedBid Auctions
PostedPrice, SealedBid Auctios Professors Greewald ad Oyakawa 2070208 We itroduce the postedprice, sealedbid auctio. This auctio format itroduces the idea of approximatios. We describe how well this
More informationChapter 3. Strong convergence. 3.1 Definition of almost sure convergence
Chapter 3 Strog covergece As poited out i the Chapter 2, there are multiple ways to defie the otio of covergece of a sequece of radom variables. That chapter defied covergece i probability, covergece i
More informationA General Family of Estimators for Estimating Population Variance Using Known Value of Some Population Parameter(s)
Rajesh Sigh, Pakaj Chauha, Nirmala Sawa School of Statistics, DAVV, Idore (M.P.), Idia Floreti Smaradache Uiversity of New Meico, USA A Geeral Family of Estimators for Estimatig Populatio Variace Usig
More informationA sequence of numbers is a function whose domain is the positive integers. We can see that the sequence
Sequeces A sequece of umbers is a fuctio whose domai is the positive itegers. We ca see that the sequece,, 2, 2, 3, 3,... is a fuctio from the positive itegers whe we write the first sequece elemet as
More informationQuestion 1: Exercise 8.2
Questio 1: Exercise 8. (a) Accordig to the regressio results i colum (1), the house price is expected to icrease by 1% ( 100% 0.0004 500 ) with a additioal 500 square feet ad other factors held costat.
More informationIntegrable Functions. { f n } is called a determining sequence for f. If f is integrable with respect to, then f d does exist as a finite real number
MATH 532 Itegrable Fuctios Dr. Neal, WKU We ow shall defie what it meas for a measurable fuctio to be itegrable, show that all itegral properties of simple fuctios still hold, ad the give some coditios
More informationLecture 1 Probability and Statistics
Wikipedia: Lecture 1 Probability ad Statistics Bejami Disraeli, British statesma ad literary figure (1804 1881): There are three kids of lies: lies, damed lies, ad statistics. popularized i US by Mark
More informationON POINTWISE BINOMIAL APPROXIMATION
Iteratioal Joural of Pure ad Applied Mathematics Volume 71 No. 1 2011, 5766 ON POINTWISE BINOMIAL APPROXIMATION BY wfunctions K. Teerapabolar 1, P. Wogkasem 2 Departmet of Mathematics Faculty of Sciece
More informationOBJECTIVES. Chapter 1 INTRODUCTION TO INSTRUMENTATION FUNCTION AND ADVANTAGES INTRODUCTION. At the end of this chapter, students should be able to:
OBJECTIVES Chapter 1 INTRODUCTION TO INSTRUMENTATION At the ed of this chapter, studets should be able to: 1. Explai the static ad dyamic characteristics of a istrumet. 2. Calculate ad aalyze the measuremet
More informationDISCRIMINATING BETWEEN THE LOGISTIC AND THE NORMAL DISTRIBUTIONS BASED ON LIKELIHOOD RATIO
DISCRIMIATIG BETWEE THE OGISTIC AD THE ORMA DISTRIBUTIOS BASED O IKEIHOOD RATIO F. Aucoi 1 ad F. Ashkar ABSTRACT The problem of discrimiatig betwee the ormal ad the logistic distributios with ukow parameters
More informationIntroducing Sample Proportions
Itroducig Sample Proportios Probability ad statistics Aswers & Notes TINspire Ivestigatio Studet 60 mi 7 8 9 0 Itroductio A 00 survey of attitudes to climate chage, coducted i Australia by the CSIRO,
More informationMonte Carlo Integration
Mote Carlo Itegratio I these otes we first review basic umerical itegratio methods (usig Riema approximatio ad the trapezoidal rule) ad their limitatios for evaluatig multidimesioal itegrals. Next we itroduce
More informationStatistics. Chapter 10 TwoSample Tests. Copyright 2013 Pearson Education, Inc. publishing as Prentice Hall. Chap 101
Statistics Chapter 0 TwoSample Tests Copyright 03 Pearso Educatio, Ic. publishig as Pretice Hall Chap 0 Learig Objectives I this chapter, you lear How to use hypothesis testig for comparig the differece
More informationReference : Chapter 15 of Ebeling (The simple case of r = n) Statistical Inference (SI) When the Underlying Distribution is Weibull
Referece : Chapter 5 of Ebelig (The simple case of r ) Statistical Iferece (SI) Whe the Uderlyig Distributio is Weibull Maghsoodloo Recall that by SI we mea estimatio ad test of hypothesis. We will cover
More informationTwoStage Controlled Fractional Factorial Screening for Simulation Experiments
TwoStage Cotrolled Fractioal Factorial Screeig for Simulatio Experimets Hog Wa Assistat Professor School of Idustrial Egieerig Purdue Uiversity 35 N. Grat Street West Lafayette, IN 4797223 hwa@purdue.edu
More informationSingular Continuous Measures by Michael Pejic 5/14/10
Sigular Cotiuous Measures by Michael Peic 5/4/0 Prelimiaries Give a set X, a σalgebra o X is a collectio of subsets of X that cotais X ad ad is closed uder complemetatio ad coutable uios hece, coutable
More informationThe Ratio Test. THEOREM 9.17 Ratio Test Let a n be a series with nonzero terms. 1. a. n converges absolutely if lim. n 1
460_0906.qxd //04 :8 PM Page 69 SECTION 9.6 The Ratio ad Root Tests 69 Sectio 9.6 EXPLORATION Writig a Series Oe of the followig coditios guaratees that a series will diverge, two coditios guaratee that
More informationSupplementary Appendix for. Mandatory Portfolio Disclosure, Stock Liquidity, and Mutual Fund Performance
Supplemetary Appedix for Madatory Portfolio Disclosure, Stock Liquidity, ad Mutual Fud Performace This Supplemetary Appedix cosists of two sectios Sectio I provides the propositios ad their proofs about
More informationCH19 Confidence Intervals for Proportions. Confidence intervals Construct confidence intervals for population proportions
CH19 Cofidece Itervals for Proportios Cofidece itervals Costruct cofidece itervals for populatio proportios Motivatio Motivatio We are iterested i the populatio proportio who support Mr. Obama. This sample
More informationProbability and Statistics
ICME Refresher Course: robability ad Statistics Staford Uiversity robability ad Statistics Luyag Che September 20, 2016 1 Basic robability Theory 11 robability Spaces A probability space is a triple (Ω,
More informationNotes on iteration and Newton s method. Iteration
Notes o iteratio ad Newto s method Iteratio Iteratio meas doig somethig over ad over. I our cotet, a iteratio is a sequece of umbers, vectors, fuctios, etc. geerated by a iteratio rule of the type 1 f
More informationSupplementary Appendix for. Mandatory Portfolio Disclosure, Stock Liquidity, and Mutual Fund Performance
Supplemetary Appedix for Madatory Portfolio Disclosure, Stock Liquidity, ad Mutual Fud Performace This Supplemetary Appedix cosists of two sectios Sectio I provides the propositios ad their proofs about
More informationROLL CUTTING PROBLEMS UNDER STOCHASTIC DEMAND
PacificAsia Joural of Mathematics, Volume 5, No., JauaryJue 20 ROLL CUTTING PROBLEMS UNDER STOCHASTIC DEMAND SHAKEEL JAVAID, Z. H. BAKHSHI & M. M. KHALID ABSTRACT: I this paper, the roll cuttig problem
More informationChapter 4  Summarizing Numerical Data
Chapter 4  Summarizig Numerical Data 15.075 Cythia Rudi Here are some ways we ca summarize data umerically. Sample Mea: i=1 x i x :=. Note: i this class we will work with both the populatio mea µ ad the
More informationTesting Statistical Hypotheses for Compare. Means with Vague Data
Iteratioal Mathematical Forum 5 o. 3 656 Testig Statistical Hypotheses for Compare Meas with Vague Data E. Baloui Jamkhaeh ad A. adi Ghara Departmet of Statistics Islamic Azad iversity Ghaemshahr Brach
More informationSeunghee Ye Ma 8: Week 5 Oct 28
Week 5 Summary I Sectio, we go over the Mea Value Theorem ad its applicatios. I Sectio 2, we will recap what we have covered so far this term. Topics Page Mea Value Theorem. Applicatios of the Mea Value
More informationA Risk Comparison of Ordinary Least Squares vs Ridge Regression
Joural of Machie Learig Research 14 (2013) 15051511 Submitted 5/12; Revised 3/13; Published 6/13 A Risk Compariso of Ordiary Least Squares vs Ridge Regressio Paramveer S. Dhillo Departmet of Computer
More informationSearching for the Holy Grail of Index Number Theory
Searchig for the Holy Grail of Idex Number Theory Bert M. Balk Rotterdam School of Maagemet Erasmus Uiversity Rotterdam Email bbalk@rsm.l ad Statistics Netherlads Voorburg April 15, 28 Abstract The idex
More informationIntroduction to Machine Learning DIS10
CS 189 Fall 017 Itroductio to Machie Learig DIS10 1 Fu with Lagrage Multipliers (a) Miimize the fuctio such that f (x,y) = x + y x + y = 3. Solutio: The Lagragia is: L(x,y,λ) = x + y + λ(x + y 3) Takig
More informationMAXIMA OF POISSONLIKE VARIABLES AND RELATED TRIANGULAR ARRAYS 1
The Aals of Applied Probability 1997, Vol. 7, No. 4, 953 971 MAXIMA OF POISSONLIKE VARIABLES AND RELATED TRIANGULAR ARRAYS 1 By Clive W. Aderso, Stuart G. Coles ad Jürg Hüsler Uiversity of Sheffield,
More information3/3/2014. CDS M Phil Econometrics. Types of Relationships. Types of Relationships. Types of Relationships. Vijayamohanan Pillai N.
3/3/04 CDS M Phil Old Least Squares (OLS) Vijayamohaa Pillai N CDS M Phil Vijayamoha CDS M Phil Vijayamoha Types of Relatioships Oly oe idepedet variable, Relatioship betwee ad is Liear relatioships Curviliear
More informationDiscrete Orthogonal Moment Features Using Chebyshev Polynomials
Discrete Orthogoal Momet Features Usig Chebyshev Polyomials R. Mukuda, 1 S.H.Og ad P.A. Lee 3 1 Faculty of Iformatio Sciece ad Techology, Multimedia Uiversity 75450 Malacca, Malaysia. Istitute of Mathematical
More informationRecall the study where we estimated the difference between mean systolic blood pressure levels of users of oral contraceptives and nonusers, x  y.
Testig Statistical Hypotheses Recall the study where we estimated the differece betwee mea systolic blood pressure levels of users of oral cotraceptives ad ousers, x  y. Such studies are sometimes viewed
More informationVaranasi , India. Corresponding author
A Geeral Family of Estimators for Estimatig Populatio Mea i Systematic Samplig Usig Auxiliary Iformatio i the Presece of Missig Observatios Maoj K. Chaudhary, Sachi Malik, Jayat Sigh ad Rajesh Sigh Departmet
More informationTesting Statistical Hypotheses with Fuzzy Data
Iteratioal Joural of Statistics ad Systems ISS 973675 Volume 6, umber 4 (), pp. 44449 Research Idia Publicatios http://www.ripublicatio.com/ijss.htm Testig Statistical Hypotheses with Fuzzy Data E. Baloui
More informationarxiv: v1 [math.pr] 13 Oct 2011
A tail iequality for quadratic forms of subgaussia radom vectors Daiel Hsu, Sham M. Kakade,, ad Tog Zhag 3 arxiv:0.84v math.pr] 3 Oct 0 Microsoft Research New Eglad Departmet of Statistics, Wharto School,
More informationAbout the use of a result of Professor Alexandru Lupaş to obtain some properties in the theory of the number e 1
Geeral Mathematics Vol. 5, No. 2007), 75 80 About the use of a result of Professor Alexadru Lupaş to obtai some properties i the theory of the umber e Adrei Verescu Dedicated to Professor Alexadru Lupaş
More informationN < < # < # < # 1 < # 6 < ; < N1
Exercise: Get the mothly S&P 500 idex (^GSPC) ad plot the periodogram of the returs. Mae sure that the umber of returs is divisible by. Y
More informationTopic 6 Sampling, hypothesis testing, and the central limit theorem
CSE 103: Probability ad statistics Fall 2010 Topic 6 Samplig, hypothesis testig, ad the cetral limit theorem 61 The biomial distributio Let X be the umberofheadswhe acoiofbiaspistossedtimes The distributio
More informationSequences I. Chapter Introduction
Chapter 2 Sequeces I 2. Itroductio A sequece is a list of umbers i a defiite order so that we kow which umber is i the first place, which umber is i the secod place ad, for ay atural umber, we kow which
More informationρ = Chapter 2 Review of Statistics Summarize from Heij Chapter Descriptive Statistics
Chapter Review of Statistics Summarize from Heij Chapter. Descriptive Statistics Presetatio of Statistics: there are may methods to preset iformatio i data set such as table, graph, ad picture. The importat
More informationFrequency Response of FIR Filters
EEL335: DiscreteTime Sigals ad Systems. Itroductio I this set of otes, we itroduce the idea of the frequecy respose of LTI systems, ad focus specifically o the frequecy respose of FIR filters.. Steadystate
More informationArea under a CurveUsing a Limit
Area uder a CurveUsig a it Sice lettig be a very large umber will result i a huge amout of work, the process ca be simplified by usig sigma otatio ad summatio formulas to create a Riema Sum The ext example
More informationLecture 11 October 27
STATS 300A: Theory of Statistics Fall 205 Lecture October 27 Lecturer: Lester Mackey Scribe: Viswajith Veugopal, Vivek Bagaria, Steve Yadlowsky Warig: These otes may cotai factual ad/or typographic errors..
More informationVariance of Discrete Random Variables Class 5, Jeremy Orloff and Jonathan Bloom
Variace of Discrete Radom Variables Class 5, 18.05 Jeremy Orloff ad Joatha Bloom 1 Learig Goals 1. Be able to compute the variace ad stadard deviatio of a radom variable.. Uderstad that stadard deviatio
More information5 Sequences and Series
Bria E. Veitch 5 Sequeces ad Series 5. Sequeces A sequece is a list of umbers i a defiite order. a is the first term a 2 is the secod term a is the th term The sequece {a, a 2, a 3,..., a,..., } is a
More informationInference approaches for instrumental variable quantile regression
Ecoomics Letters 95 (2007) 272 277 www.elsevier.com/locate/ecobase Iferece approaches for istrumetal variable quatile regressio Victor Cherozhukov a, Christia Hase b,, Michael Jasso c a Massachusetts Istitute
More informationWhich Extreme Values are Really Extremes? Jesús Gozalo Λ Statistics ad Ecoometrics José Olmo Statistics ad Ecoometrics U. Carlos III de Madrid U. Carlos III de Madrid jgozalo@esteco.uc3m.es olmob@esteco.uc3m.es
More informationLecture 6 Ecient estimators. RaoCramer bound.
Lecture 6 Eciet estimators. RaoCramer boud. 1 MSE ad Suciecy Let X (X 1,..., X) be a radom sample from distributio f θ. Let θ ˆ δ(x) be a estimator of θ. Let T (X) be a suciet statistic for θ. As we have
More informationExample 3.3: Rainfall reported at a group of five stations (see Fig. 3.7) is as follows. Kundla. Sabli
3.4.4 Spatial Cosistecy Check Raifall data exhibit some spatial cosistecy ad this forms the basis of ivestigatig the observed raifall values. A estimate of the iterpolated raifall value at a statio is
More informationInteger Linear Programming
Iteger Liear Programmig Itroductio Iteger L P problem (P) Mi = s. t. a = b i =,, m = i i 0, iteger =,, c Eemple Mi z = 5 s. t. + 0 0, 0, iteger F(P) = feasible domai of P Itroductio Iteger L P problem
More informationAn investigation of load extrapolation according to IEC Ed. 3
A ivestigatio of load extrapolatio accordig to IEC 64 Ed. Rolf Gez, Siemes Wid Power A/S, rge@siemes.com Kristia Bedix Nielse, DNV Global Wid Eergy, Kristia.Bedix.Nielse@dv.com Peter Hauge Madse, Siemes
More informationNumber of fatalities X Sunday 4 Monday 6 Tuesday 2 Wednesday 0 Thursday 3 Friday 5 Saturday 8 Total 28. Day
LECTURE # 8 Mea Deviatio, Stadard Deviatio ad Variace & Coefficiet of variatio Mea Deviatio Stadard Deviatio ad Variace Coefficiet of variatio First, we will discuss it for the case of raw data, ad the
More informationBINOMIAL COEFFICIENT AND THE GAUSSIAN
BINOMIAL COEFFICIENT AND THE GAUSSIAN The biomial coefficiet is defied as! k!(! ad ca be writte out i the form of a Pascal Triagle startig at the zeroth row with elemet 0,0) ad followed by the two umbers,
More informationSEMIPARAMETRIC SINGLEINDEX MODELS. Joel L. Horowitz Department of Economics Northwestern University
SEMIPARAMETRIC SINGLEINDEX MODELS by Joel L. Horowitz Departmet of Ecoomics Northwester Uiversity INTRODUCTION Much of applied ecoometrics ad statistics ivolves estimatig a coditioal mea fuctio: E ( Y
More informationProbability, Expectation Value and Uncertainty
Chapter 1 Probability, Expectatio Value ad Ucertaity We have see that the physically observable properties of a quatum system are represeted by Hermitea operators (also referred to as observables ) such
More informationSome Thoughts on the Importance of Weighing the Tails
SEMINAR UIMA Probability ad Statistics Group Departmet of Mathematics Uiversity Aveiro 25 Jue 2008 Some Thoughts o the Importace of Weighig the Tails Isabel Fraga Alves CEAUL & DEIO Uiversity Lisbo Cláudia
More informationTopic 15: Maximum Likelihood Estimation
Topic 5: Maximum Likelihood Estimatio November ad 3, 20 Itroductio The priciple of maximum likelihood is relatively straightforward. As before, we begi with a sample X (X,..., X of radom variables chose
More informationJoint Probability Distributions and Random Samples. Jointly Distributed Random Variables. Chapter { }
UCLA STAT A Applied Probability & Statistics for Egieers Istructor: Ivo Diov, Asst. Prof. I Statistics ad Neurology Teachig Assistat: Neda Farziia, UCLA Statistics Uiversity of Califoria, Los Ageles, Sprig
More informationOn Orlicz Nframes. 1 Introduction. Renu Chugh 1,, Shashank Goel 2
Joural of Advaced Research i Pure Mathematics Olie ISSN: 19432380 Vol. 3, Issue. 1, 2010, pp. 104110 doi: 10.5373/jarpm.473.061810 O Orlicz Nframes Reu Chugh 1,, Shashak Goel 2 1 Departmet of Mathematics,
More informationA new iterative algorithm for reconstructing a signal from its dyadic wavelet transform modulus maxima
ol 46 No 6 SCIENCE IN CHINA (Series F) December 3 A ew iterative algorithm for recostructig a sigal from its dyadic wavelet trasform modulus maxima ZHANG Zhuosheg ( u ), LIU Guizhog ( q) & LIU Feg ( )
More informationSensitivity Analysis of Daubechies 4 Wavelet Coefficients for Reduction of Reconstructed Image Error
Proceedigs of the 6th WSEAS Iteratioal Coferece o SIGNAL PROCESSING, Dallas, Texas, USA, March 4, 7 67 Sesitivity Aalysis of Daubechies 4 Wavelet Coefficiets for Reductio of Recostructed Image Error DEVINDER
More informationHOMEWORK 2 SOLUTIONS
HOMEWORK SOLUTIONS CSE 55 RANDOMIZED AND APPROXIMATION ALGORITHMS 1. Questio 1. a) The larger the value of k is, the smaller the expected umber of days util we get all the coupos we eed. I fact if = k
More informationReview Questions, Chapters 8, 9. f(y) = 0, elsewhere. F (y) = f Y(1) = n ( e y/θ) n 1 1 θ e y/θ = n θ e yn
Stat 366 Lab 2 Solutios (September 2, 2006) page TA: Yury Petracheko, CAB 484, yuryp@ualberta.ca, http://www.ualberta.ca/ yuryp/ Review Questios, Chapters 8, 9 8.5 Suppose that Y, Y 2,..., Y deote a radom
More informationMath 2784 (or 2794W) University of Connecticut
ORDERS OF GROWTH PAT SMITH Math 2784 (or 2794W) Uiversity of Coecticut Date: Mar. 2, 22. ORDERS OF GROWTH. Itroductio Gaiig a ituitive feel for the relative growth of fuctios is importat if you really
More informationSequences of Definite Integrals, Factorials and Double Factorials
47 6 Joural of Iteger Sequeces, Vol. 8 (5), Article 5.4.6 Sequeces of Defiite Itegrals, Factorials ad Double Factorials Thierry DaaPicard Departmet of Applied Mathematics Jerusalem College of Techology
More informationResearch on Dependable level in Network Computing System Yongxia Li 1, a, Guangxia Xu 2,b and Shuangyan Liu 3,c
Applied Mechaics ad Materials Olie: 04006 ISSN: 66748, Vols. 5357, pp 0508 doi:0.408/www.scietific.et/amm.5357.05 04 Tras Tech Publicatios, Switzerlad Research o Depedable level i Network Computig
More informationUNIT ROOT MODEL SELECTION PETER C. B. PHILLIPS COWLES FOUNDATION PAPER NO. 1231
UNIT ROOT MODEL SELECTION BY PETER C. B. PHILLIPS COWLES FOUNDATION PAPER NO. 1231 COWLES FOUNDATION FOR RESEARCH IN ECONOMICS YALE UNIVERSITY Box 28281 New Have, Coecticut 6528281 28 http://cowles.eco.yale.edu/
More informationA Note on Sums of Independent Random Variables
Cotemorary Mathematics Volume 00 XXXX A Note o Sums of Ideedet Radom Variables Pawe l Hitczeko ad Stehe MotgomerySmith Abstract I this ote a two sided boud o the tail robability of sums of ideedet ad
More informationCentral Limit Theorem the Meaning and the Usage
Cetral Limit Theorem the Meaig ad the Usage Covetio about otatio. N, We are usig otatio X is variable with mea ad stadard deviatio. i lieu of sayig that X is a ormal radom Assume a sample of measuremets
More information1 Hypothesis test of a mean vector
THE UNIVERSITY OF CHICAGO Booth School of Busiess Busiess 41912, Sprig Quarter 2010, Mr Ruey S Tsay Lecture: Iferece about sample mea Key cocepts: 1 Hotellig s T 2 test 2 Likelihood ratio test 3 Various
More informationHOMEWORK #10 SOLUTIONS
Math 33  Aalysis I Sprig 29 HOMEWORK # SOLUTIONS () Prove that the fuctio f(x) = x 3 is (Riema) itegrable o [, ] ad show that x 3 dx = 4. (Without usig formulae for itegratio that you leart i previous
More informationExample 2. Find the upper bound for the remainder for the approximation from Example 1.
Lesso 8 Error Approimatios 0 Alteratig Series Remaider: For a coverget alteratig series whe approimatig the sum of a series by usig oly the first terms, the error will be less tha or equal to the absolute
More informationSTATISTICAL method is one branch of mathematical
40 INTERNATIONAL JOURNAL OF COMPUTING SCIENCE AND APPLIED MATHEMATICS, VOL 3, NO, AUGUST 07 Optimizig Forest Samplig by usig Lagrage Multipliers Suhud Wahyudi, Farida Agustii Widjajati ad Dea Oktaviati
More informationSECTION 1.5 : SUMMATION NOTATION + WORK WITH SEQUENCES
SECTION 1.5 : SUMMATION NOTATION + WORK WITH SEQUENCES Read Sectio 1.5 (pages 5 9) Overview I Sectio 1.5 we lear to work with summatio otatio ad formulas. We will also itroduce a brief overview of sequeces,
More informationSTAT 203 Chapter 18 Sampling Distribution Models
STAT 203 Chapter 18 Samplig Distributio Models Populatio vs. sample, parameter vs. statistic Recall that a populatio cotais the etire collectio of idividuals that oe wats to study, ad a sample is a subset
More informationSTRAIGHT LINES & PLANES
STRAIGHT LINES & PLANES PARAMETRIC EQUATIONS OF LINES The lie "L" is parallel to the directio vector "v". A fixed poit: "( a, b, c) " o the lie is give. Positio vectors are draw from the origi to the fixed
More informationNumerical Methods in Fourier Series Applications
Numerical Methods i Fourier Series Applicatios Recall that the basic relatios i usig the Trigoometric Fourier Series represetatio were give by f ( x) a o ( a x cos b x si ) () where the Fourier coefficiets
More informationMASSACHUSETTS INSTITUTE OF TECHNOLOGY 6.436J/15.085J Fall 2008 Lecture 6 9/24/2008 DISCRETE RANDOM VARIABLES AND THEIR EXPECTATIONS
MASSACHUSETTS INSTITUTE OF TECHNOLOGY 6.436J/15.085J Fall 2008 Lecture 6 9/24/2008 DISCRETE RANDOM VARIABLES AND THEIR EXPECTATIONS Cotets 1. A few useful discrete radom variables 2. Joit, margial, ad
More informationEfficient Estimation of Distributional Tail Shape and the Extremal Index with Applications to Risk Management
Journal of Mathematical Finance, 2016, 6, 626659 http://www.scirp.org/journal/jmf ISSN Online: 21622442 ISSN Print: 21622434 Efficient Estimation of Distributional Tail Shape and the Extremal Index
More information(b) What is the probability that a particle reaches the upper boundary n before the lower boundary m?
MATH 529 The Boudary Problem The drukard s walk (or boudary problem) is oe of the most famous problems i the theory of radom walks. Oe versio of the problem is described as follows: Suppose a particle
More informationSEPARABLE LEAST SQUARES, VARIABLE PROJECTION, AND THE GAUSSNEWTON ALGORITHM
SEPARABLE LEAS SQUARES, VARIABLE PROJECION, AND HE GAUSSNEWON ALGORIHM M.R.OSBORNE Key words. oliear least squares, scorig, Newto s method, expected Hessia, Kaufma s modificatio, rate of covergece, radom
More informationAsymptotic distribution of the firststage Fstatistic under weak IVs
November 6 Eco 59A WEAK INSTRUMENTS III Testig for Weak Istrumets From the results discussed i Weak Istrumets II we kow that at least i the case of a sigle edogeous regressor there are weakidetificatiorobust
More informationProbabilistic and Average Linear Widths in L Norm with Respect to rfold Wiener Measure
joural of approximatio theory 84, 3140 (1996) Article No. 0003 Probabilistic ad Average Liear Widths i L Norm with Respect to rfold Wieer Measure V. E. Maiorov Departmet of Mathematics, Techio, Haifa,
More informationEstimation of the Population Mean in Presence of NonResponse
Commuicatios of the Korea Statistical Society 0, Vol. 8, No. 4, 537 548 DOI: 0.535/CKSS.0.8.4.537 Estimatio of the Populatio Mea i Presece of NoRespose Suil Kumar,a, Sadeep Bhougal b a Departmet of Statistics,
More informationLecture #18
181 Variatioal Method (See CTDL 11481155, [Variatioal Method] 252263, 295307[Desity Matrices]) Last time: QuasiDegeeracy Diagoalize a part of ifiite H * submatrix : H (0) + H (1) * correctios for
More informationRefinements. New notes follow on next page. 1 of 8. Equilibrium path. Belief needed
efiemets Itro from ECO 744 Notes efiemets i Strategic Form: Elimiate Weakly Domiated Strategies  throwig out strictly domiated strategies (eve iteratively ever elimiates a Nash equilibrium (because Nash
More information