On the Maximum and Minimum of Multivariate Pareto Random Variables
|
|
- Margaret Copeland
- 6 years ago
- Views:
Transcription
1 ISSN X (rint) ISSN X (online) INFORMATION TECHNOLOGY AND CONTROL 213 Vol.42 No.3 On the Maximum Minimum of Multivariate Pareto Rom Variables Jungsoo Woo Deartment of Statistics Yeungnam University Gyongsan South Korea Saralees Nadarajah School of Mathematics University of Manchester Manchester M13 9PL UK htt://dx.doi.org/1.5755/j1.itc Abstract. Aksomaitis Burauskaite-Harju [Information Technology Control ] studied the distribution of max(x 1 X 2... X ) when (X 1 X 2... X ) follows the multivariate normal distribution. Here we study the moments of min(x 1 X 2... X ) max(x 1 X 2... X ) when (X 1 X 2... X ) follows the most commonly known multivariate Pareto distribution. Multivariate Pareto distributions are most relevant for modeling extreme values. Keywords: maximum; minimum; multivariate Pareto distribution. 1. Introduction Let (X 1 X 2... X ) be a continuous rom vector with means μ 1 μ 2... μ variances σ 1 2 σ σ 2 correlation coefficient ρ. The extreme values of ( X 1 X 2... X ) are M = mmm(x 1 X 2... X ) m = mmm(x 1 X 2... X ). It is often of interest to know how E(M) E(m) vary with resect to the means variances the correlation coefficient. For examle Smith Sardeshmukh (2) find that "The change of variance is associated both with altered skewness a change in high low extremes". Griffiths et al. (25) use the change in mean temerature as a redictor of extreme temerature change in the Asia-Pacific region. Karl et al. (28) observe that "A relatively small shift in the mean roduces a larger change in the number of extremes for both temerature reciitation". Nicholls (28) argues the need for careful statistical analysis "since the likelihood of individual extremes such as a late sring frost could change due to changes in variability as well as changes in the mean climate". Burton Allso (29) observe that "In recent years there has been an increasing tendency to use mean seeds to redict extremes". In the extreme value literature only four aers have studied the distributions of M m with resect to the means variances the correlation coefficient: Ker (21) Lien (25) Aksomaitis Burauskaite- Harju (29) Hakamiour et al. (211). Ker (21) suoses that (X 1 X 2 ) has a bivariate normal distribution. Lien (25) suoses that (X 1 X 2 ) has a bivariate lognormal distribution. Aksomaitis Burauskaite-Harju (29) suose that (X 1 X 2... X ) has a multivariate normal distribution. Hakamiour et al. (211) suose that ( X 1 X 2... X ) follows a multivariate Pareto distribution due to Muliere Scarsini (1987). As we can see the first two of these aers are limited to the case = 2. The distributions considered by the first three aers (bivariate normal bivariate lognormal multivariate normal) are not the most aroriate ones for modeling extreme values. Muliere Scarsini s (1987) multivariate Pareto distribution the distribution considered by the fourth aer suffers from discontinuity has limited alicability. Bivariate multivariate Pareto distributions have been the most oular distributions for modeling extremes. Among these the most commonly known distribution due to Arnold (1983 Chater 6) has the joint survivor function secified by x i F (x 1 x 2... x ) = 1 + (1.1) 242
2 On the Maximum Minimum of Multivariate Pareto Rom Variables for x i > i = > i = 12 >. This distribution has received widesread attention: Yeh (24) studies extreme order statistics of (1.1); Li (26) investigates tail deendence roerties of (1.1); Cai Tan (27) roose (1.1) as a model for deendent risks; Vernic (211) uses (1.1) to estimate tail conditional exectation a oular measure of risk. For the multivariate Pareto distribution given by (1.1) stard calculations show that μ i = > 1 (1.2) 1 σ i 2 = 2 ( 1) 2 ( 1) > 2 (1.3) ρ = 1 (1.4) The aim of this short note is to study how E(M) E(m) vary with resect to the means variances the correlation coefficient. The main results are Theorem 2.1 Theorem 2.2 Theorem 2.3. The level of mathematics required in Section 2 is not high but not less elementary than the mathematics used in Ker (21) Lien (25) Aksomaitis Burauskaite-Harju (29) Hakamiour et al. (211). We feel that the given results are imortant because of the rominence of multivariate Pareto distributions because of the rominence of (1.1) in modeling extreme values. Besides the results known on this toic have so far been limited for the bivariate case or distributions which are not most significant in modeling extreme values. 2. Main results The main results in this section need the following lemma. Lemma 2.1. Define I(k a ) = x k 1 (1 + aa). Then I(k a ) = a k B( k k) for a > < k < where B(a b) denotes the beta function defined by 1 B(a b) = t a 1 (1 t) b 1. Proof. Setting y = 1/(1 + ax) we can write I(k a ) = x k 1 (1 + aa) = 1 = a k y k 1 (1 y) k 1 = a k B( k k) where the final equality follows from the definition of the beta function. Theorem 2.1 derives exlicit exressions for the cumulative distribution functions of M = mmm(x 1 X 2... X ) m = mmm(x 1 X 2... X ). Theorem 2.2 derives exlicit exressions for the nth moment of M = mmm(x 1 X 2... X ) m = mmm(x 1 X 2... X ). Theorem 2.3 shows how E(M n ) vary with resect to the means variances the correlation coefficient in (1.2)-(1.4). Theorem 2.1. Let ( X 1 X 2... X ) be distributed mmm(x 1 X 2... X ) m = mmm(x 1 X 2... X ). Then Pr (M x) = x x 1 i<j 1 + x + x + x θ j θ k 1 i<j<k + + ( 1) 1 + x Pr (m x) = x for x >. Proof. We have Pr (M x) = Pr maxx 1 X x = Pr (X i x) = 1 Pr (X i > x) = 1 Pr (X i > x) + Pr X i > x X j > x 1 i<j Pr X i > x X j > x X k > x 1 i<j<k + + ( 1) Pr (X i > x) = x x 1 i<j 243
3 J. Woo S. Nadarajah 1 + x + x + x θ j θ k 1 i<j<k + + ( 1) 1 + x Pr (m x) = 1 Pr (m > x) = 1 Pr (min(x 1 X n ) > x) = 1 Pr (X i > x) = x The roof is comlete.. Theorem 2.2. Let ( X 1 X 2... X ) be distributed mmm(x 1 X 2... X ) m = mmm(x 1 X 2... X ). Let k A i 1 ik = 1 l=1 for 1 i 1 < < i k. Then l E(M n ) = mb( n n) A i A ij 1 i<j + A ijk 1 i<j<k = mb( n n)a 1 for n <. Proof. Using Theorem 2.1 we can write E(M n ) = n x n 1 PP(M > x) = n x n x n x n x + x θ j 1 i<j +n x n x + x θ k 1 i<j<k n( 1) ( 1) A 1 x n x = n I(n A i ) n I(n A i ) + n In A ijk n( 1) In A 1 1 i<j 1 i<j<k = n A i B( n n) A ij B( n n) +n A ijk B( n n) 1 i<j<k 1 i<j n( 1) A 1 1 B( n n) where the last ste follows from Lemma 2.1. Also using Theorem 2.1 we can write = n x n 1 Pr(m > x) = n x n x = min A 1 = na 1 B( n n) where the last ste follows from Lemma 2.1. The roof is comlete. Theorem 2.3. Let ( X 1 X 2... X ) be distributed mmm(x 1 X 2... X ) m = mmm(x 1 X 2... X ). Then (a) E(M n ) is an increasing function of ρ for ρ < 1/n; (b) is an increasing function of ρ for ρ < 1/n; (c) is an increasing function of µ i ; 244
4 On the Maximum Minimum of Multivariate Pareto Rom Variables (d) is an increasing function of σ i ; (e) V ar(m) is an increasing function of ρ for ρ < 1/2; (f ) V ar(m) is an increasing function of µ i ; (g) V ar(m) is an increasing function of σ i. Proof. Since E(M n ) = mb( n n) A i A ij Γ( n)γ(n) = n Γ() = 1 i<j A i A ij n! ( n) ( 1) A i 1 i<j A ij 1 i<j + A ijk 1 i<j<k + A ijk 1 i<j<k + A ijk 1 i<j<k ( 1) A 1 ( 1) A 1 ( 1) A 1 we see that E(M n ) is a decreasing function of for > n hence an increasing function of ρ for ρ < 1/n. Since = mb( n n)a 1 P Γ( n)γ(n) = NA 1 P Γ() = A 1 P n! ( n) ( 1) we see that is a decreasing function of for > n hence an increasing function of ρ for ρ < 1/n. Since = mb( n n)a 1 P = NN( n n) 1 = mb( n n)( 1) n 1 μ i where the last equality follows by (1.2) we see that is an increasing function of μ i. Since = mb( n n) 1 = mb( n n) ( 1)n ( 2) n/2 n/2 1 σ i where the last equality follows by (1.3) we see that is an increasing function of σ i. Since V aa(m) = E(m 2 ) E 2 (m) = A 1 [2B( 22) B 2 ( 11)] = A 2Γ( 2) 1 Γ() = A 1 ( 1) 2 ( 2) Γ2 ( 2) Γ 2 () we see that V aa(m) is a decreasing function of for > 2 hence an increasing function of ρ for ρ < 1/2. Since V aa(m) = A 1 = 1 μ i ( 1) 2 ( 2) 2 where the last equality follows by (1.2) we see that V aa(m) is an increasing function of μ i. Since V aa(m) = A 1 ( 1) 2 ( 2) = 1 σ i where the last equality follows by (1.3) we see that V aa(m) is an increasing function of σ i. Acknowledgments The authors would like to thank the Editor the referee for careful reading for their comments which greatly imroved the aer. References [1] A. Aksomaitis A. Burauskaite-Harju. The moments of the maximum of normally distributed deendent values. Information Technology Control 29 Vol [2] B. C. Arnold. Pareto Distributions. International Cooerative Publishing House: Maryl [3] M. D. Burton A. C. Allso. Predicting design wind seeds from Anemometer records: Some interesting findings. In: Proceedings of the 11th Americas Conference on Wind Engineering 29. [4] J. Cai K. S. Tan. Otimal retention for a sto-loss reinsurance under the VaR CTE risk measures. ASTIN Bulletin 27 Vol [5] G. M. Griffiths et al. Change in mean temerature as a redictor of extreme temerature change in the Asia- Pacific region. International Journal of Climatology 25 Vol
5 J. Woo S. Nadarajah [6] N. Hakamiour A. Mohammadour S. Nadarajah. Extremes of a bivariate Pareto distribution. Information Technology Control 211 Vol [7] T. R. Karl et al. Weather climate extremes in a changing climate: Regions of focus: North America Hawaii Caribbean U.S. Pacific Isls. Reort by the U.S. Climate Change Science Program 28. [8] A. P. Ker. On the maximum of bivariate normal rom variables. Extremes 21 Vol [9] H. Li. Tail deendence of multivariate Pareto distributions. Technical Reort 26-6 Deartment of Mathematics Washington State University Pullman WA USA 26. [1] D. Lien. On the minimum maximum of bivariate lognormal rom variables. Extremes 25 Vol [11] P. Muliere M. Scarsini. Characterization of a Marshall-Olkin tye class of distributions. Annals of the Institute of Statistical Mathematics 1987 Vol [12] N. Nicholls. Australian climate weather extremes: Past resent future. A Reort for the Deartment of Climate Change Australia 28. [13] C. A. Smith P. D. Sardeshmukh. The effect of ENSO on the intraseasonal variance of surface temeratures in winter. International Journal of Climatology 2 Vol [14] R. Vernic. Tail conditional exectation for the multivariate Pareto distribution of the second kind: Another aroach. Methodology Comuting in Alied Probability 211 Vol [15] H.-C. Yeh. Some roerties characterizations for generalized multivariate Pareto distributions. Journal of Multivariate Analysis 24 Vol Received August
Estimation of the large covariance matrix with two-step monotone missing data
Estimation of the large covariance matrix with two-ste monotone missing data Masashi Hyodo, Nobumichi Shutoh 2, Takashi Seo, and Tatjana Pavlenko 3 Deartment of Mathematical Information Science, Tokyo
More informationON GENERALIZED SARMANOV BIVARIATE DISTRIBUTIONS
TWMS J. A. Eng. Math. V., N., 2,. 86-97 ON GENERALIZED SARMANOV BIVARIATE DISTRIBUTIONS I. BAIRAMOV, B. (YAĞCI) ALTINSOY2, G. JAY KERNS 3 Abstract. A class of bivariate distributions which generalizes
More informationBivariate Weibull-power series class of distributions
Bivariate Weibull-power series class of distributions Saralees Nadarajah and Rasool Roozegar EM algorithm, Maximum likelihood estimation, Power series distri- Keywords: bution. Abstract We point out that
More informationApplicable Analysis and Discrete Mathematics available online at HENSEL CODES OF SQUARE ROOTS OF P-ADIC NUMBERS
Alicable Analysis and Discrete Mathematics available online at htt://efmath.etf.rs Al. Anal. Discrete Math. 4 (010), 3 44. doi:10.98/aadm1000009m HENSEL CODES OF SQUARE ROOTS OF P-ADIC NUMBERS Zerzaihi
More informationElements of Asymptotic Theory. James L. Powell Department of Economics University of California, Berkeley
Elements of Asymtotic Theory James L. Powell Deartment of Economics University of California, Berkeley Objectives of Asymtotic Theory While exact results are available for, say, the distribution of the
More informationHotelling s Two- Sample T 2
Chater 600 Hotelling s Two- Samle T Introduction This module calculates ower for the Hotelling s two-grou, T-squared (T) test statistic. Hotelling s T is an extension of the univariate two-samle t-test
More informationCHAPTER-II Control Charts for Fraction Nonconforming using m-of-m Runs Rules
CHAPTER-II Control Charts for Fraction Nonconforming using m-of-m Runs Rules. Introduction: The is widely used in industry to monitor the number of fraction nonconforming units. A nonconforming unit is
More informationOn Wrapping of Exponentiated Inverted Weibull Distribution
IJIRST International Journal for Innovative Research in Science & Technology Volume 3 Issue 11 Aril 217 ISSN (online): 2349-61 On Wraing of Exonentiated Inverted Weibull Distribution P.Srinivasa Subrahmanyam
More informationLinear diophantine equations for discrete tomography
Journal of X-Ray Science and Technology 10 001 59 66 59 IOS Press Linear diohantine euations for discrete tomograhy Yangbo Ye a,gewang b and Jiehua Zhu a a Deartment of Mathematics, The University of Iowa,
More informationSystem Reliability Estimation and Confidence Regions from Subsystem and Full System Tests
009 American Control Conference Hyatt Regency Riverfront, St. Louis, MO, USA June 0-, 009 FrB4. System Reliability Estimation and Confidence Regions from Subsystem and Full System Tests James C. Sall Abstract
More informationALTERNATIVE SOLUTION TO THE QUARTIC EQUATION by Farid A. Chouery 1, P.E. 2006, All rights reserved
ALTERNATIVE SOLUTION TO THE QUARTIC EQUATION b Farid A. Chouer, P.E. 006, All rights reserved Abstract A new method to obtain a closed form solution of the fourth order olnomial equation is roosed in this
More informationTowards understanding the Lorenz curve using the Uniform distribution. Chris J. Stephens. Newcastle City Council, Newcastle upon Tyne, UK
Towards understanding the Lorenz curve using the Uniform distribution Chris J. Stehens Newcastle City Council, Newcastle uon Tyne, UK (For the Gini-Lorenz Conference, University of Siena, Italy, May 2005)
More informationNumerical Methods for Particle Tracing in Vector Fields
On-Line Visualization Notes Numerical Methods for Particle Tracing in Vector Fields Kenneth I. Joy Visualization and Grahics Research Laboratory Deartment of Comuter Science University of California, Davis
More information#A64 INTEGERS 18 (2018) APPLYING MODULAR ARITHMETIC TO DIOPHANTINE EQUATIONS
#A64 INTEGERS 18 (2018) APPLYING MODULAR ARITHMETIC TO DIOPHANTINE EQUATIONS Ramy F. Taki ElDin Physics and Engineering Mathematics Deartment, Faculty of Engineering, Ain Shams University, Cairo, Egyt
More informationOn Introducing Asymmetry into Circular Distributions
Statistics in the Twenty-First Century: Secial Volume In Honour of Distinguished Professor Dr. Mir Masoom Ali On the Occasion of his 75th Birthday Anniversary PJSOR, Vol. 8, No. 3, ages 531-535, July 2012
More informationNew Information Measures for the Generalized Normal Distribution
Information,, 3-7; doi:.339/info3 OPEN ACCESS information ISSN 75-7 www.mdi.com/journal/information Article New Information Measures for the Generalized Normal Distribution Christos P. Kitsos * and Thomas
More informationAdditive results for the generalized Drazin inverse in a Banach algebra
Additive results for the generalized Drazin inverse in a Banach algebra Dragana S. Cvetković-Ilić Dragan S. Djordjević and Yimin Wei* Abstract In this aer we investigate additive roerties of the generalized
More informationEstimating function analysis for a class of Tweedie regression models
Title Estimating function analysis for a class of Tweedie regression models Author Wagner Hugo Bonat Deartamento de Estatística - DEST, Laboratório de Estatística e Geoinformação - LEG, Universidade Federal
More informationYixi Shi. Jose Blanchet. IEOR Department Columbia University New York, NY 10027, USA. IEOR Department Columbia University New York, NY 10027, USA
Proceedings of the 2011 Winter Simulation Conference S. Jain, R. R. Creasey, J. Himmelsach, K. P. White, and M. Fu, eds. EFFICIENT RARE EVENT SIMULATION FOR HEAVY-TAILED SYSTEMS VIA CROSS ENTROPY Jose
More informationNew weighing matrices and orthogonal designs constructed using two sequences with zero autocorrelation function - a review
University of Wollongong Research Online Faculty of Informatics - Paers (Archive) Faculty of Engineering and Information Sciences 1999 New weighing matrices and orthogonal designs constructed using two
More informationHölder Inequality with Variable Index
Theoretical Mathematics & Alications, vol.4, no.3, 204, 9-23 ISSN: 792-9687 (rint), 792-9709 (online) Scienress Ltd, 204 Hölder Ineuality with Variable Index Richeng Liu Abstract By using the Young ineuality,
More informationThe non-stochastic multi-armed bandit problem
Submitted for journal ublication. The non-stochastic multi-armed bandit roblem Peter Auer Institute for Theoretical Comuter Science Graz University of Technology A-8010 Graz (Austria) auer@igi.tu-graz.ac.at
More informationMATHEMATICAL MODELLING OF THE WIRELESS COMMUNICATION NETWORK
Comuter Modelling and ew Technologies, 5, Vol.9, o., 3-39 Transort and Telecommunication Institute, Lomonosov, LV-9, Riga, Latvia MATHEMATICAL MODELLIG OF THE WIRELESS COMMUICATIO ETWORK M. KOPEETSK Deartment
More informationThe Mathematics of Winning Streaks
The Mathematics of Winning Streaks Erik Leffler lefflererik@gmail.com under the direction of Prof. Henrik Eriksson Deartment of Comuter Science and Communications Royal Institute of Technology Research
More informationOutline. Markov Chains and Markov Models. Outline. Markov Chains. Markov Chains Definitions Huizhen Yu
and Markov Models Huizhen Yu janey.yu@cs.helsinki.fi Det. Comuter Science, Univ. of Helsinki Some Proerties of Probabilistic Models, Sring, 200 Huizhen Yu (U.H.) and Markov Models Jan. 2 / 32 Huizhen Yu
More information#A47 INTEGERS 15 (2015) QUADRATIC DIOPHANTINE EQUATIONS WITH INFINITELY MANY SOLUTIONS IN POSITIVE INTEGERS
#A47 INTEGERS 15 (015) QUADRATIC DIOPHANTINE EQUATIONS WITH INFINITELY MANY SOLUTIONS IN POSITIVE INTEGERS Mihai Ciu Simion Stoilow Institute of Mathematics of the Romanian Academy, Research Unit No. 5,
More informationElements of Asymptotic Theory. James L. Powell Department of Economics University of California, Berkeley
Elements of Asymtotic Theory James L. Powell Deartment of Economics University of California, Berkeley Objectives of Asymtotic Theory While exact results are available for, say, the distribution of the
More informationA Comparison between Biased and Unbiased Estimators in Ordinary Least Squares Regression
Journal of Modern Alied Statistical Methods Volume Issue Article 7 --03 A Comarison between Biased and Unbiased Estimators in Ordinary Least Squares Regression Ghadban Khalaf King Khalid University, Saudi
More informationLOGISTIC REGRESSION. VINAYANAND KANDALA M.Sc. (Agricultural Statistics), Roll No I.A.S.R.I, Library Avenue, New Delhi
LOGISTIC REGRESSION VINAANAND KANDALA M.Sc. (Agricultural Statistics), Roll No. 444 I.A.S.R.I, Library Avenue, New Delhi- Chairerson: Dr. Ranjana Agarwal Abstract: Logistic regression is widely used when
More informationA MIXED CONTROL CHART ADAPTED TO THE TRUNCATED LIFE TEST BASED ON THE WEIBULL DISTRIBUTION
O P E R A T I O N S R E S E A R C H A N D D E C I S I O N S No. 27 DOI:.5277/ord73 Nasrullah KHAN Muhammad ASLAM 2 Kyung-Jun KIM 3 Chi-Hyuck JUN 4 A MIXED CONTROL CHART ADAPTED TO THE TRUNCATED LIFE TEST
More informationJosé Alberto Mauricio. Instituto Complutense de Análisis Económico. Universidad Complutense de Madrid. Campus de Somosaguas Madrid - SPAIN
A CORRECTED ALGORITHM FOR COMPUTING THE THEORETICAL AUTOCOVARIANCE MATRICES OF A VECTOR ARMA MODEL José Alberto Mauricio Instituto Comlutense de Análisis Económico Universidad Comlutense de Madrid Camus
More informationCombining Logistic Regression with Kriging for Mapping the Risk of Occurrence of Unexploded Ordnance (UXO)
Combining Logistic Regression with Kriging for Maing the Risk of Occurrence of Unexloded Ordnance (UXO) H. Saito (), P. Goovaerts (), S. A. McKenna (2) Environmental and Water Resources Engineering, Deartment
More informationComputing the covariance of two Brownian area integrals
Statistica Neerlandica () Vol. 56, nr.,. ±9 Comuting the covariance of two Brownian area integrals J. A. Wellner* University of Washington, Statistics, Box 3543, Seattle, Washington 9895-43, U.S.A. R.
More informationState Estimation with ARMarkov Models
Deartment of Mechanical and Aerosace Engineering Technical Reort No. 3046, October 1998. Princeton University, Princeton, NJ. State Estimation with ARMarkov Models Ryoung K. Lim 1 Columbia University,
More informationFinite Mixture EFA in Mplus
Finite Mixture EFA in Mlus November 16, 2007 In this document we describe the Mixture EFA model estimated in Mlus. Four tyes of deendent variables are ossible in this model: normally distributed, ordered
More informationMULTIVARIATE STATISTICAL PROCESS OF HOTELLING S T CONTROL CHARTS PROCEDURES WITH INDUSTRIAL APPLICATION
Journal of Statistics: Advances in heory and Alications Volume 8, Number, 07, Pages -44 Available at htt://scientificadvances.co.in DOI: htt://dx.doi.org/0.864/jsata_700868 MULIVARIAE SAISICAL PROCESS
More informationPublished: 14 October 2013
Electronic Journal of Alied Statistical Analysis EJASA, Electron. J. A. Stat. Anal. htt://siba-ese.unisalento.it/index.h/ejasa/index e-issn: 27-5948 DOI: 1.1285/i275948v6n213 Estimation of Parameters of
More informationSAS for Bayesian Mediation Analysis
Paer 1569-2014 SAS for Bayesian Mediation Analysis Miočević Milica, Arizona State University; David P. MacKinnon, Arizona State University ABSTRACT Recent statistical mediation analysis research focuses
More informationDr. Shalabh Department of Mathematics and Statistics Indian Institute of Technology Kanpur
Analysis of Variance and Design of Exeriment-I MODULE II LECTURE -4 GENERAL LINEAR HPOTHESIS AND ANALSIS OF VARIANCE Dr. Shalabh Deartment of Mathematics and Statistics Indian Institute of Technology Kanur
More informationChapter 7: Special Distributions
This chater first resents some imortant distributions, and then develos the largesamle distribution theory which is crucial in estimation and statistical inference Discrete distributions The Bernoulli
More informationAdaptive estimation with change detection for streaming data
Adative estimation with change detection for streaming data A thesis resented for the degree of Doctor of Philosohy of the University of London and the Diloma of Imerial College by Dean Adam Bodenham Deartment
More informationElliptic Curves Spring 2015 Problem Set #1 Due: 02/13/2015
18.783 Ellitic Curves Sring 2015 Problem Set #1 Due: 02/13/2015 Descrition These roblems are related to the material covered in Lectures 1-2. Some of them require the use of Sage, and you will need to
More informationGenetic Algorithms, Selection Schemes, and the Varying Eects of Noise. IlliGAL Report No November Department of General Engineering
Genetic Algorithms, Selection Schemes, and the Varying Eects of Noise Brad L. Miller Det. of Comuter Science University of Illinois at Urbana-Chamaign David E. Goldberg Det. of General Engineering University
More information#A37 INTEGERS 15 (2015) NOTE ON A RESULT OF CHUNG ON WEIL TYPE SUMS
#A37 INTEGERS 15 (2015) NOTE ON A RESULT OF CHUNG ON WEIL TYPE SUMS Norbert Hegyvári ELTE TTK, Eötvös University, Institute of Mathematics, Budaest, Hungary hegyvari@elte.hu François Hennecart Université
More informationProbability Estimates for Multi-class Classification by Pairwise Coupling
Probability Estimates for Multi-class Classification by Pairwise Couling Ting-Fan Wu Chih-Jen Lin Deartment of Comuter Science National Taiwan University Taiei 06, Taiwan Ruby C. Weng Deartment of Statistics
More information2 K. ENTACHER 2 Generalized Haar function systems In the following we x an arbitrary integer base b 2. For the notations and denitions of generalized
BIT 38 :2 (998), 283{292. QUASI-MONTE CARLO METHODS FOR NUMERICAL INTEGRATION OF MULTIVARIATE HAAR SERIES II KARL ENTACHER y Deartment of Mathematics, University of Salzburg, Hellbrunnerstr. 34 A-52 Salzburg,
More informationDepartment of Mathematics
Deartment of Mathematics Ma 3/03 KC Border Introduction to Probability and Statistics Winter 209 Sulement : Series fun, or some sums Comuting the mean and variance of discrete distributions often involves
More informationSlash Distributions and Applications
CHAPTER 2 Slash Distributions and Alications 2.1 Introduction The concet of slash distributions was introduced by Kafadar (1988) as a heavy tailed alternative to the normal distribution. Further literature
More informationAsymptotically Optimal Simulation Allocation under Dependent Sampling
Asymtotically Otimal Simulation Allocation under Deendent Samling Xiaoing Xiong The Robert H. Smith School of Business, University of Maryland, College Park, MD 20742-1815, USA, xiaoingx@yahoo.com Sandee
More informationOn Wald-Type Optimal Stopping for Brownian Motion
J Al Probab Vol 34, No 1, 1997, (66-73) Prerint Ser No 1, 1994, Math Inst Aarhus On Wald-Tye Otimal Stoing for Brownian Motion S RAVRSN and PSKIR The solution is resented to all otimal stoing roblems of
More informationThree hours. To be supplied by the Examinations Office: Mathematical Formula Tables and Statistical Tables THE UNIVERSITY OF MANCHESTER.
Three hours To be supplied by the Examinations Office: Mathematical Formula Tables and Statistical Tables THE UNIVERSITY OF MANCHESTER EXTREME VALUES AND FINANCIAL RISK Examiner: Answer QUESTION 1, QUESTION
More informationChater Matrix Norms and Singular Value Decomosition Introduction In this lecture, we introduce the notion of a norm for matrices The singular value de
Lectures on Dynamic Systems and Control Mohammed Dahleh Munther A Dahleh George Verghese Deartment of Electrical Engineering and Comuter Science Massachuasetts Institute of Technology c Chater Matrix Norms
More informationUse of Transformations and the Repeated Statement in PROC GLM in SAS Ed Stanek
Use of Transformations and the Reeated Statement in PROC GLM in SAS Ed Stanek Introduction We describe how the Reeated Statement in PROC GLM in SAS transforms the data to rovide tests of hyotheses of interest.
More informationBivariate distributions characterized by one family of conditionals and conditional percentile or mode functions
Journal of Multivariate Analysis 99 2008) 1383 1392 www.elsevier.com/locate/jmva Bivariate istributions characterize by one family of conitionals an conitional ercentile or moe functions Barry C. Arnol
More informationLower bound solutions for bearing capacity of jointed rock
Comuters and Geotechnics 31 (2004) 23 36 www.elsevier.com/locate/comgeo Lower bound solutions for bearing caacity of jointed rock D.J. Sutcliffe a, H.S. Yu b, *, S.W. Sloan c a Deartment of Civil, Surveying
More informationResearch Article A New Method to Study Analytic Inequalities
Hindawi Publishing Cororation Journal of Inequalities and Alications Volume 200, Article ID 69802, 3 ages doi:0.55/200/69802 Research Article A New Method to Study Analytic Inequalities Xiao-Ming Zhang
More informationA New Class of Positively Quadrant Dependent Bivariate Distributions with Pareto
International Mathematical Forum, 2, 27, no. 26, 1259-1273 A New Class of Positively Quadrant Dependent Bivariate Distributions with Pareto A. S. Al-Ruzaiza and Awad El-Gohary 1 Department of Statistics
More informationMODELING THE RELIABILITY OF C4ISR SYSTEMS HARDWARE/SOFTWARE COMPONENTS USING AN IMPROVED MARKOV MODEL
Technical Sciences and Alied Mathematics MODELING THE RELIABILITY OF CISR SYSTEMS HARDWARE/SOFTWARE COMPONENTS USING AN IMPROVED MARKOV MODEL Cezar VASILESCU Regional Deartment of Defense Resources Management
More informationUsing the Divergence Information Criterion for the Determination of the Order of an Autoregressive Process
Using the Divergence Information Criterion for the Determination of the Order of an Autoregressive Process P. Mantalos a1, K. Mattheou b, A. Karagrigoriou b a.deartment of Statistics University of Lund
More informationRe-entry Protocols for Seismically Active Mines Using Statistical Analysis of Aftershock Sequences
Re-entry Protocols for Seismically Active Mines Using Statistical Analysis of Aftershock Sequences J.A. Vallejos & S.M. McKinnon Queen s University, Kingston, ON, Canada ABSTRACT: Re-entry rotocols are
More informationUniformly best wavenumber approximations by spatial central difference operators: An initial investigation
Uniformly best wavenumber aroximations by satial central difference oerators: An initial investigation Vitor Linders and Jan Nordström Abstract A characterisation theorem for best uniform wavenumber aroximations
More informationAn Improved Generalized Estimation Procedure of Current Population Mean in Two-Occasion Successive Sampling
Journal of Modern Alied Statistical Methods Volume 15 Issue Article 14 11-1-016 An Imroved Generalized Estimation Procedure of Current Poulation Mean in Two-Occasion Successive Samling G. N. Singh Indian
More informationMatching Transversal Edge Domination in Graphs
Available at htt://vamuedu/aam Al Al Math ISSN: 19-9466 Vol 11, Issue (December 016), 919-99 Alications and Alied Mathematics: An International Journal (AAM) Matching Transversal Edge Domination in Grahs
More informationNUMERICAL ANALYSIS OF THE IMPACT OF THE INLET AND OUTLET JETS FOR THE THERMAL STRATIFICATION INSIDE A STORAGE TANK
NUMERICAL ANALYSIS OF HE IMAC OF HE INLE AND OULE JES FOR HE HERMAL SRAIFICAION INSIDE A SORAGE ANK A. Zachár I. Farkas F. Szlivka Deartment of Comuter Science Szent IstvÆn University Æter K. u.. G d llı
More informationOn split sample and randomized confidence intervals for binomial proportions
On slit samle and randomized confidence intervals for binomial roortions Måns Thulin Deartment of Mathematics, Usala University arxiv:1402.6536v1 [stat.me] 26 Feb 2014 Abstract Slit samle methods have
More informationTopic: Lower Bounds on Randomized Algorithms Date: September 22, 2004 Scribe: Srinath Sridhar
15-859(M): Randomized Algorithms Lecturer: Anuam Guta Toic: Lower Bounds on Randomized Algorithms Date: Setember 22, 2004 Scribe: Srinath Sridhar 4.1 Introduction In this lecture, we will first consider
More informationSIMULATED ANNEALING AND JOINT MANUFACTURING BATCH-SIZING. Ruhul SARKER. Xin YAO
Yugoslav Journal of Oerations Research 13 (003), Number, 45-59 SIMULATED ANNEALING AND JOINT MANUFACTURING BATCH-SIZING Ruhul SARKER School of Comuter Science, The University of New South Wales, ADFA,
More informationREFINED STRAIN ENERGY OF THE SHELL
REFINED STRAIN ENERGY OF THE SHELL Ryszard A. Walentyński Deartment of Building Structures Theory, Silesian University of Technology, Gliwice, PL44-11, Poland ABSTRACT The aer rovides information on evaluation
More informationAN ALGORITHM FOR MATRIX EXTENSION AND WAVELET CONSTRUCTION W. LAWTON, S. L. LEE AND ZUOWEI SHEN Abstract. This paper gives a practical method of exten
AN ALGORITHM FOR MATRIX EXTENSION AND WAVELET ONSTRUTION W. LAWTON, S. L. LEE AND ZUOWEI SHEN Abstract. This aer gives a ractical method of extending an nr matrix P (z), r n, with Laurent olynomial entries
More informationIntroduction to Probability and Statistics
Introduction to Probability and Statistics Chater 8 Ammar M. Sarhan, asarhan@mathstat.dal.ca Deartment of Mathematics and Statistics, Dalhousie University Fall Semester 28 Chater 8 Tests of Hyotheses Based
More informationResearch Article Controllability of Linear Discrete-Time Systems with Both Delayed States and Delayed Inputs
Abstract and Alied Analysis Volume 203 Article ID 97546 5 ages htt://dxdoiorg/055/203/97546 Research Article Controllability of Linear Discrete-Time Systems with Both Delayed States and Delayed Inuts Hong
More informationComputation of the Smith Form for Multivariate Polynomial Matrices Using Maple
American Journal of Comutational Mathematics,,, -6 htt://dxdoiorg/436/ajcm3 Published Online March (htt://wwwscirporg/journal/ajcm) Comutation of the Smith Form for Multivariate Polynomial Matrices Using
More informationVarious Proofs for the Decrease Monotonicity of the Schatten s Power Norm, Various Families of R n Norms and Some Open Problems
Int. J. Oen Problems Comt. Math., Vol. 3, No. 2, June 2010 ISSN 1998-6262; Coyright c ICSRS Publication, 2010 www.i-csrs.org Various Proofs for the Decrease Monotonicity of the Schatten s Power Norm, Various
More informationComparing the kurtosis measures for symmetric-scale distribution functions considering a new kurtosis
Proceedings of the 8th WSAS International Conference on APPLID MATHMATICS, Tenerife, Sain, December 6-8, 00 (90-9 Comaring the urtosis measures for symmetric-scale distribution functions considering a
More informationAn Analysis of Reliable Classifiers through ROC Isometrics
An Analysis of Reliable Classifiers through ROC Isometrics Stijn Vanderlooy s.vanderlooy@cs.unimaas.nl Ida G. Srinkhuizen-Kuyer kuyer@cs.unimaas.nl Evgueni N. Smirnov smirnov@cs.unimaas.nl MICC-IKAT, Universiteit
More informationGRACEFUL NUMBERS. KIRAN R. BHUTANI and ALEXANDER B. LEVIN. Received 14 May 2001
IJMMS 29:8 2002 495 499 PII S06720200765 htt://immshindawicom Hindawi Publishing Cor GRACEFUL NUMBERS KIRAN R BHUTANI and ALEXANDER B LEVIN Received 4 May 200 We construct a labeled grah Dn that reflects
More informationDeveloping A Deterioration Probabilistic Model for Rail Wear
International Journal of Traffic and Transortation Engineering 2012, 1(2): 13-18 DOI: 10.5923/j.ijtte.20120102.02 Develoing A Deterioration Probabilistic Model for Rail Wear Jabbar-Ali Zakeri *, Shahrbanoo
More informationWeakly Short Memory Stochastic Processes: Signal Processing Perspectives
Weakly Short emory Stochastic Processes: Signal Processing Persectives by Garimella Ramamurthy Reort No: IIIT/TR/9/85 Centre for Security, Theory and Algorithms International Institute of Information Technology
More informationMultivariable Generalized Predictive Scheme for Gas Turbine Control in Combined Cycle Power Plant
Multivariable Generalized Predictive Scheme for Gas urbine Control in Combined Cycle Power Plant L.X.Niu and X.J.Liu Deartment of Automation North China Electric Power University Beiing, China, 006 e-mail
More informationChapter 3. GMM: Selected Topics
Chater 3. GMM: Selected oics Contents Otimal Instruments. he issue of interest..............................2 Otimal Instruments under the i:i:d: assumtion..............2. he basic result............................2.2
More informationCHAPTER 5 TANGENT VECTORS
CHAPTER 5 TANGENT VECTORS In R n tangent vectors can be viewed from two ersectives (1) they cature the infinitesimal movement along a ath, the direction, and () they oerate on functions by directional
More informationA New Asymmetric Interaction Ridge (AIR) Regression Method
A New Asymmetric Interaction Ridge (AIR) Regression Method by Kristofer Månsson, Ghazi Shukur, and Pär Sölander The Swedish Retail Institute, HUI Research, Stockholm, Sweden. Deartment of Economics and
More informationRatio Estimators in Simple Random Sampling Using Information on Auxiliary Attribute
ajesh Singh, ankaj Chauhan, Nirmala Sawan School of Statistics, DAVV, Indore (M.., India Florentin Smarandache Universit of New Mexico, USA atio Estimators in Simle andom Samling Using Information on Auxiliar
More informationEcon 3790: Business and Economics Statistics. Instructor: Yogesh Uppal
Econ 379: Business and Economics Statistics Instructor: Yogesh Ual Email: yual@ysu.edu Chater 9, Part A: Hyothesis Tests Develoing Null and Alternative Hyotheses Tye I and Tye II Errors Poulation Mean:
More informationInformation collection on a graph
Information collection on a grah Ilya O. Ryzhov Warren Powell October 25, 2009 Abstract We derive a knowledge gradient olicy for an otimal learning roblem on a grah, in which we use sequential measurements
More informationFactors Effect on the Saturation Parameter S and there Influences on the Gain Behavior of Ytterbium Doped Fiber Amplifier
Australian Journal of Basic and Alied Sciences, 5(12): 2010-2020, 2011 ISSN 1991-8178 Factors Effect on the Saturation Parameter S and there Influences on the Gain Behavior of Ytterbium Doed Fiber Amlifier
More informationUncorrelated Multilinear Principal Component Analysis for Unsupervised Multilinear Subspace Learning
TNN-2009-P-1186.R2 1 Uncorrelated Multilinear Princial Comonent Analysis for Unsuervised Multilinear Subsace Learning Haiing Lu, K. N. Plataniotis and A. N. Venetsanooulos The Edward S. Rogers Sr. Deartment
More informationResearch of PMU Optimal Placement in Power Systems
Proceedings of the 5th WSEAS/IASME Int. Conf. on SYSTEMS THEORY and SCIENTIFIC COMPUTATION, Malta, Setember 15-17, 2005 (38-43) Research of PMU Otimal Placement in Power Systems TIAN-TIAN CAI, QIAN AI
More informationPressure-sensitivity Effects on Toughness Measurements of Compact Tension Specimens for Strain-hardening Solids
American Journal of Alied Sciences (9): 19-195, 5 ISSN 1546-939 5 Science Publications Pressure-sensitivity Effects on Toughness Measurements of Comact Tension Secimens for Strain-hardening Solids Abdulhamid
More informationGENERALIZED NORMS INEQUALITIES FOR ABSOLUTE VALUE OPERATORS
International Journal of Analysis Alications ISSN 9-8639 Volume 5, Number (04), -9 htt://www.etamaths.com GENERALIZED NORMS INEQUALITIES FOR ABSOLUTE VALUE OPERATORS ILYAS ALI, HU YANG, ABDUL SHAKOOR Abstract.
More informationEcon 3790: Business and Economics Statistics. Instructor: Yogesh Uppal
Econ 379: Business and Economics Statistics Instructor: Yogesh Ual Email: yual@ysu.edu Chater 9, Part A: Hyothesis Tests Develoing Null and Alternative Hyotheses Tye I and Tye II Errors Poulation Mean:
More informationECON 4130 Supplementary Exercises 1-4
HG Set. 0 ECON 430 Sulementary Exercises - 4 Exercise Quantiles (ercentiles). Let X be a continuous random variable (rv.) with df f( x ) and cdf F( x ). For 0< < we define -th quantile (or 00-th ercentile),
More informationSupplement to the paper Accurate distributions of Mallows C p and its unbiased modifications with applications to shrinkage estimation
To aear in Economic Review (Otaru University of Commerce Vol. No..- 017. Sulement to the aer Accurate distributions of Mallows C its unbiased modifications with alications to shrinkage estimation Haruhiko
More information4. Score normalization technical details We now discuss the technical details of the score normalization method.
SMT SCORING SYSTEM This document describes the scoring system for the Stanford Math Tournament We begin by giving an overview of the changes to scoring and a non-technical descrition of the scoring rules
More informationA numerical implementation of a predictor-corrector algorithm for sufcient linear complementarity problem
A numerical imlementation of a redictor-corrector algorithm for sufcient linear comlementarity roblem BENTERKI DJAMEL University Ferhat Abbas of Setif-1 Faculty of science Laboratory of fundamental and
More informationUsing Factor Analysis to Study the Effecting Factor on Traffic Accidents
Using Factor Analysis to Study the Effecting Factor on Traffic Accidents Abstract Layla A. Ahmed Deartment of Mathematics, College of Education, University of Garmian, Kurdistan Region Iraq This aer is
More informationMATH 2710: NOTES FOR ANALYSIS
MATH 270: NOTES FOR ANALYSIS The main ideas we will learn from analysis center around the idea of a limit. Limits occurs in several settings. We will start with finite limits of sequences, then cover infinite
More informationarxiv: v1 [cond-mat.stat-mech] 1 Dec 2017
Suscetibility Proagation by Using Diagonal Consistency Muneki Yasuda Graduate School of Science Engineering, Yamagata University Kazuyuki Tanaka Graduate School of Information Sciences, Tohoku University
More informationOn Optimization of Power Coefficient of HAWT
Journal of Power and Energy Engineering, 14,, 198- Published Online Aril 14 in Scies htt://wwwscirorg/journal/jee htt://dxdoiorg/1436/jee1448 On Otimization of Power Coefficient of HAWT Marat Z Dosaev
More informationON THE LEAST SIGNIFICANT p ADIC DIGITS OF CERTAIN LUCAS NUMBERS
#A13 INTEGERS 14 (014) ON THE LEAST SIGNIFICANT ADIC DIGITS OF CERTAIN LUCAS NUMBERS Tamás Lengyel Deartment of Mathematics, Occidental College, Los Angeles, California lengyel@oxy.edu Received: 6/13/13,
More information