A NEW LOG-NORMAL DISTRIBUTION

Size: px
Start display at page:

Download "A NEW LOG-NORMAL DISTRIBUTION"

Transcription

1 Joural of Statstcs: Advaces Theory ad Applcatos Volume 6, Number, 06, Pages Avalable at DOI: A NEW LOG-NORMAL DISTRIBUTION Departmet of Statstcs Yale Uversty New Have, CT 065 USA e-mal: tahao.wu@yale.edu Abstract Log-ormal dstrbuto s ubqutous atural ad socal scece by ts smple form ad hgh relato to ormal dstrbuto. However, oftetmes log-ormal realzatos ca be depedet, so the applcato of -dmesoal log-ormal s mportat ad useful. I ths paper, we develop a ew form of log-ormal dstrbuto, whose margals are proved to be depedet of the power parameter. I addto, ths ew dstrbuto has specal propertes such as zero lower tal depedece ad equdstrbuto whch ca be fouded sghtful for future research.. Itroducto Oe-dmesoal log-ormal dstrbuto s ubqutous atural ad socal scece by ts smple form ad hgh relato to ormal dstrbuto, whose cumulatve probablty dstrbuto fucto s defed as l( X ) µ Φ( ). Itrgued by the depedece amog dfferet realzatos of log-ormal, bvarate as well as hgh-dmesoal log-ormal attracts researchers amog dfferet dscples. 00 Mathematcs Subject Classfcato: 6E0. Keywords ad phrases: log-ormal, -dmeso, copula, depedece. Receved August 5, 06; Revsed September 7, Scetfc Advaces Publshers

2 94 The most-wdely used multvarate log-ormal radom varate s defed as Y X, where ~ (, X LN µ ), so Y s dstrbuted as Y ~ LN( µ, ). That mpled that each margal has dfferet mea ad varace. Furthermore, multvarate log-ormal s fouded to perfectly descrbe may atural ad socal pheomeo. Yue [6] uses bvarate log-ormal dstrbuto to represet multple epsodc storm evets at the Motoyama meteorologcal observato stato Japa. Yerel ad Kouk [4] appled smlar dstrbuto to study Beylkova mageste ore depost mpurtes Esksehr (Turkey). Ad Yue [5] fts the flood-frequecy data also by a bvarate form of log-ormal dstrbuto. I the past studes, may work cocetrate o the theoretcal aspects of the exstg multvarate form metoed above, such lke the depedece measure (Nalbach-Leewska []). Nevertheless, attempts to exted log-ormal s rarely see, amog them Shaha ad Bhavsar [3] s a example to costruct a two pece multvarate dstrbuto. I ths paper, we develop a ew verso of -dmesoal log-ormal dstrbuto whch s a atural exteso of the oe-dmesoal oe. After provg t s a proper dstrbuto, some of ts propertes are gve. Oe of the promsg propertes s that ts margs are dstrbuted logormal wth same locato ad scale parameters, ad ts lower tal depedece attas zero whch dcates that a realzato of X has small probablty to be a vector of 0.. -Dmesoal Log-Normal Dstrbuto I ths secto, we wll preset the ew verso of log-ormal dstrbuto ad show t s deed a proper probablty measure. Suppose X { X, X,, } s a vector of radom varables ad we defe the X ew form of log-ormal dstrbuto whose jotly cumulatve probablty dstrbuto fucto s

3 A NEW LOG-NORMAL DISTRIBUTION 95 / ( ) l X µ Φ > ( ), f X 0, F X () 0, otherwse, where Φ deotes the cumulatve probablty fucto of a stadard ormal radom varable, < 0, µ > 0, ad > 0. Also, µ ad are both fte. I ths secto, we frst prove that t s deed a proper -dmesoal dstrbuto fucto f t satsfes the followg defto. Defto.. A -dmesoal dstrbuto fucto s a fucto F wth doma R such that () F ( t) 0 for all t R such that t k for at least oe k, ad F (,,, ) ; () F s -creasg. We the tur to prove that the dstrbuto defed Equato () follows the above two codtos. Ad Proof. () If t k for at least oe k, F ( t) 0 by our defto. / l( t ) lm t µ l 0 µ, () by the egatvty of. Thus, / l( t ) lm Φ t µ. (3) Hece codto () s satsfed. Ths defto comes from Page 45 o Nelse [].

4 96 () For two vectors a ad b such that b, F ( X) 0, we say that F s -creasg. Note that b a b a a f V ([ a, b] ) F / l( t ) lm t 0 µ, / l( t ) lm Φ t 0 µ 0. (4) So F ( X) s cotuous ( X, X,, X ). It s obvous that the followg equalty holds: F ( X) X X X > 0. (5) Therefore, the codto of -creasg s satsfed. By presetg the above proof, we have shows that the ew logormal we defed s deed a proper -dmesoal probablty dstrbuto. As a matter of fact, the dstrbuto ths paper has stadard logormal margs. Suppose F ( X ) s the cumulatve probablty dstrbuto fucto of X, the we ca deduce ts form by the followg steps: F ( X ) lm F ( X) X Φ l X µ l X µ Φ, (6) where X represets { X,.., X, X+,, X }. Ad that s exactly the oe-dmesoal log-ormal dstrbuto wth locato parameter µ

5 A NEW LOG-NORMAL DISTRIBUTION 97 ad scale parameter. Hece, we clam that the margal dstrbuto of X s dstrbuted log-ormal our case. 3. Propertes I ths secto, we show some propertes of the ew log-ormal dstrbuto whch ca be used for future research. 3.. Desty We ca see the jotly cumulatve dstrbuto fucto takes the followg form whe X > 0: t m F ( X ) exp( ) dm, (7) π / l( ) µ where t X, by shftg the tegral terval we ca get ( ) / l ( m ) F ( ) X µ X exp( ) dm. (8) π We take the partal dervatve w.r.t. X j, we get X ( l( X ) µ ) j Fj ( X ) exp( ), (9) X π / whch s the margal desty fucto of X j ad by lettg, t s easy to see the desty above s exactly that of uvarate log-ormal dstrbuto. Takg hgh-order partal dervatves gets us the jot desty, however, t s qute complcated. If we focus o the bvarate case, the followg stads for the cumulatve dstrbuto fucto:

6 98 Fgure. CDF of bvarate log-ormal. 3.. Order statstcs Defe the -order statstcs of X ( X, X,, ) as X ( ), the the dstrbuto of whch s X F ( X ( ) x ) F ( X x X x,, X x ), / l( x ) Φ µ l l x + µ Φ, (0) so the maxmum of X s dstrbuted log-ormal wth locato parameter µ l ad scale parameter. Recall the margs, so the dstrbuto of the maxmum always has larger mea but same varato. Moreover, the bgger the umber of varables the vector X, the bgger the mea.

7 3.3. Equdstrbuto A NEW LOG-NORMAL DISTRIBUTION 99 From the defto, we kow that ay subset of X s dstrbuted the same from of the ew log-ormal, so we coclude that F ( X) satsfes the equdstrbuto codto,.e., d F ( X,, X ) F ( X,, X ). () I fact, the dstrbuto fuctos have the same forms f the dmeso are same Copula The followg represets the copula of the -dmesoal log-ormal dstrbuto: l( exp( ( u ) )) Φ + µ µ C( u, u,, u ) Φ, u [ 0, ]. / () l u µ Defe a verse fucto ϕ ( u) as ϕ ( u ) Φ ( ), the ϕ( u ) exp( Φ ( u) + µ). The the -dmesoal copula ca be see as C( u, u,, u ) ϕ ( ϕ( u )). (3) We clam t s a Archmedea copula, whose defto s as follows. Defto If ϕ s a cotuous strctly decreasg fucto from I to [ 0, ] such that ϕ ( ) 0 ad ( 0). Archmedea -copula f ad oly f [ 0, ). ϕ The C s a ϕ s completely mootoc o Theorem. The -copula for multvarate log-ormal dstrbuto s a Archmedea copula.

8 00 Proof. () As Φ s the cumulatve probablty dstrbuto fucto for stad ormal radom varable so Φ s strctly creasg, as s Φ. The by the egatvty of ad postvty of, we kow that ϕ ( u) s strctly decreasg u. () The doma of ϕ s [ 0, ] by ts defto ad ( ) 0 ϕ u by the property of expoetal fucto. Hece ϕ s mappg from I to [ 0, ]. () Φ (), thus Φ () + µ addto, Φ ( 0), thus Φ ( 0) + µ ad ϕ ( 0 ). (v) The complete mootocty of ad ϕ ( ) 0. I ϕ s ped dow to show that of / k l u. It s easy to show that ( ) f ( k ) 0 for x > 0 ad k 0,,, / 3,, f ( u) l u. Ths completes the proof Tal depedece By the copula above, we are able to calculate the followg codtoal probablty: Tu Pr( F ( X ) u F ( X ) u ) j C( u, u) Φ( Φ ( u) + l ), u u (4) so the lower tal depedece s lm Tu u 0 Φ( Φ ( u) + l ) lm( ) 0. u u 0 (5) Proof. By L Hosptal s rule, we have Φ( Φ ( u) + l ) Φ ( u) t lm lm exp( ), u 0 u u 0 π u (6) where t Φ ( u) + l. By the property of verse fucto, moreover,

9 A NEW LOG-NORMAL DISTRIBUTION 0 lm u 0 π Φ ( u) t t exp( ) lm exp( ) u u 0 π Φ ( Φ ( u)) where t Φ ( u). Smplfyg t we get t t lm exp( ) exp( ), (7) u 0 t t lm exp( ) exp( ) 0. u 0 (8) That meas f equvaletly, X j s pretty small, the X has probablty 0 to be very small Codtoal probablty Recall the margal desty the Subsecto 3.. X caot be small, or / X ( l( X ) µ ) j F ( X ) exp( ), (9) X π whch fact has aother terpretato, Thus F ( X) Pr ( X x,, X j x j,, X x ). (0) x j Pr( X x,, X j x,, X x ) exp( x π ( l( x ) µ ) ), () whch s Pr( max X x arg max X j ). Itegratg the above probablty from x to,

10 0 x x π exp( ( l( x ) µ ) ) x x π ( l( x ) + l( ) µ ) exp( ), () whose tegratg factor s the desty of log-ormal dstrbuto wth locato parameter µ l( ) ad scale parameter. So the tegrato s l( x ) + l( ) µ Φ ( ), (3) whch s Pr( max X x arg max X j ). Furthermore, we have Pr( max X x ) Pr( max X x arg max X j ) j l( x ) + l( ) µ Φ ( ). (4) So the codto probablty s Pr( max X x arg max X j ) Pr( arg max X j max X x ) Pr( max X x ). (5) Also ote that Pr ( arg max X j ) whch s trval to prove. So the probablty that maxmzer of X s X j gve a threshold of the maxmum oly depeds o the umber of elemets X The role of We are terested the role of, we wll explore t of specal cases ths secto. Frst ote that by the uboudedess of l ( ) fucto,

11 A NEW LOG-NORMAL DISTRIBUTION 03 takg dervatve of F ( X) w.r.t. produces usged result. However, f / F ( x, x,, x) ( x ) we focus o, whose sg oly depeds o, ( x ) / x / l( ) < 0 sce, (6) F ( x, x,, x) so < 0, the larger, the smaller the probablty that all X fall dow to the same terval. Addtoally, / lm( ), for >, 0 X (7) so lm F ( X ), for >, (8) 0 thus the jot probablty coverges to whe goes to 0. / As a matter of fact, the fucto f ( x ) ( ) s called costat elastcty of substtuto (CES) fucto ad has a varety of applcatos Ecoomcs ad Egeerg. To see ths, we ca prove that x f ( tx) tf ( x), (9) so f X ~ F ( X; µ, ), ax ~ F ( X; µ + l( a), ). Ad s a measure of substtutablty ad the bgger the, the bgger the substtutablty of X. Accordgly, plays the role of measurg the across dffereces amog ( X, X,, X ) to some extet Dscusso The propertes proved above ca be potetally used future research. For example, ts equdstrbuto property ca draw atteto to fttg a sequece of evets whose every subset s dstrbuted equvaletly log-ormal. Ad by ts copula, oe ca easly compute the depedece betwee radom varables. Furthermore, as ts tal depedece s 0, the ew log-ormal dstrbuto s lkely to be appled to some real data whch ot all values ca be very small.

12 04 4. Cocluso I ths paper, we develop a ew form of -dmesoal log-ormal dstrbuto whch depeds o the shape parameter ad scale parameter. Ths ew log-ormal dstrbuto s proved to have stad log-ormal margs, dcatg that each X s dstrbuted uvarate log-ormal. Furthermore, some propertes such as ts copula, equdstrbuto, tal depedece, etc. are gve. Readers should see ths work as a framework ad some more research ca be doe regardg ths ew form of log-ormal, for example, the applcato of that to some real data ad see the ablty of ths dstrbuto of descrbg radom evets. Ackowledgemet The author thaks the aoymous referee for hs/her valuable commets. All errors are my ow. Refereces [] A. Nalbach-Leewska, Measures of depedece of the multvarate log-ormal dstrbuto, Seres Statstcs 0(3) (979), [] R. B. Nelse, A Itroducto to Copulas, Sprger Seres Statstcs, 006. [3] P. Shaha ad C. D. Bhavsar, Two-pece multvarate log-ormal dstrbuto, Iteratoal Joural of Sceces: Basc ad Appled Research 5() (04), [4] S. Yerel ad A. Kouk, Bvarate log-ormal dstrbuto model of cutoff grade mpurtes: A case study of mageste ore depost, Scetfc Research ad Essay 4() (009), [5] S. Yue, The bvarate log-ormal dstrbuto to model a multvarate flood epsode, Hydrologcal Processes 4 (000), [6] S. Yue, The bvarate log-ormal dstrbuto for descrbg jot statstcal propertes of a multvarate storm evet, Evrometrcs 3 (00), g

Functions of Random Variables

Functions of Random Variables Fuctos of Radom Varables Chapter Fve Fuctos of Radom Varables 5. Itroducto A geeral egeerg aalyss model s show Fg. 5.. The model output (respose) cotas the performaces of a system or product, such as weght,

More information

Lecture 3 Probability review (cont d)

Lecture 3 Probability review (cont d) STATS 00: Itroducto to Statstcal Iferece Autum 06 Lecture 3 Probablty revew (cot d) 3. Jot dstrbutos If radom varables X,..., X k are depedet, the ther dstrbuto may be specfed by specfyg the dvdual dstrbuto

More information

Chapter 5 Properties of a Random Sample

Chapter 5 Properties of a Random Sample Lecture 6 o BST 63: Statstcal Theory I Ku Zhag, /0/008 Revew for the prevous lecture Cocepts: t-dstrbuto, F-dstrbuto Theorems: Dstrbutos of sample mea ad sample varace, relatoshp betwee sample mea ad sample

More information

X ε ) = 0, or equivalently, lim

X ε ) = 0, or equivalently, lim Revew for the prevous lecture Cocepts: order statstcs Theorems: Dstrbutos of order statstcs Examples: How to get the dstrbuto of order statstcs Chapter 5 Propertes of a Radom Sample Secto 55 Covergece

More information

UNIVERSITY OF OSLO DEPARTMENT OF ECONOMICS

UNIVERSITY OF OSLO DEPARTMENT OF ECONOMICS UNIVERSITY OF OSLO DEPARTMENT OF ECONOMICS Postpoed exam: ECON430 Statstcs Date of exam: Jauary 0, 0 Tme for exam: 09:00 a.m. :00 oo The problem set covers 5 pages Resources allowed: All wrtte ad prted

More information

Multivariate Transformation of Variables and Maximum Likelihood Estimation

Multivariate Transformation of Variables and Maximum Likelihood Estimation Marquette Uversty Multvarate Trasformato of Varables ad Maxmum Lkelhood Estmato Dael B. Rowe, Ph.D. Assocate Professor Departmet of Mathematcs, Statstcs, ad Computer Scece Copyrght 03 by Marquette Uversty

More information

ρ < 1 be five real numbers. The

ρ < 1 be five real numbers. The Lecture o BST 63: Statstcal Theory I Ku Zhag, /0/006 Revew for the prevous lecture Deftos: covarace, correlato Examples: How to calculate covarace ad correlato Theorems: propertes of correlato ad covarace

More information

Chapter 4 Multiple Random Variables

Chapter 4 Multiple Random Variables Revew for the prevous lecture: Theorems ad Examples: How to obta the pmf (pdf) of U = g (, Y) ad V = g (, Y) Chapter 4 Multple Radom Varables Chapter 44 Herarchcal Models ad Mxture Dstrbutos Examples:

More information

UNIVERSITY OF OSLO DEPARTMENT OF ECONOMICS

UNIVERSITY OF OSLO DEPARTMENT OF ECONOMICS UNIVERSITY OF OSLO DEPARTMENT OF ECONOMICS Exam: ECON430 Statstcs Date of exam: Frday, December 8, 07 Grades are gve: Jauary 4, 08 Tme for exam: 0900 am 00 oo The problem set covers 5 pages Resources allowed:

More information

STK4011 and STK9011 Autumn 2016

STK4011 and STK9011 Autumn 2016 STK4 ad STK9 Autum 6 Pot estmato Covers (most of the followg materal from chapter 7: Secto 7.: pages 3-3 Secto 7..: pages 3-33 Secto 7..: pages 35-3 Secto 7..3: pages 34-35 Secto 7.3.: pages 33-33 Secto

More information

Lecture Notes Types of economic variables

Lecture Notes Types of economic variables Lecture Notes 3 1. Types of ecoomc varables () Cotuous varable takes o a cotuum the sample space, such as all pots o a le or all real umbers Example: GDP, Polluto cocetrato, etc. () Dscrete varables fte

More information

Analysis of Variance with Weibull Data

Analysis of Variance with Weibull Data Aalyss of Varace wth Webull Data Lahaa Watthaacheewaul Abstract I statstcal data aalyss by aalyss of varace, the usual basc assumptos are that the model s addtve ad the errors are radomly, depedetly, ad

More information

Extreme Value Theory: An Introduction

Extreme Value Theory: An Introduction (correcto d Extreme Value Theory: A Itroducto by Laures de Haa ad Aa Ferrera Wth ths webpage the authors ted to form the readers of errors or mstakes foud the book after publcato. We also gve extesos for

More information

X X X E[ ] E X E X. is the ()m n where the ( i,)th. j element is the mean of the ( i,)th., then

X X X E[ ] E X E X. is the ()m n where the ( i,)th. j element is the mean of the ( i,)th., then Secto 5 Vectors of Radom Varables Whe workg wth several radom varables,,..., to arrage them vector form x, t s ofte coveet We ca the make use of matrx algebra to help us orgaze ad mapulate large umbers

More information

The Mathematical Appendix

The Mathematical Appendix The Mathematcal Appedx Defto A: If ( Λ, Ω, where ( λ λ λ whch the probablty dstrbutos,,..., Defto A. uppose that ( Λ,,..., s a expermet type, the σ-algebra o λ λ λ are defed s deoted by ( (,,...,, σ Ω.

More information

Summary of the lecture in Biostatistics

Summary of the lecture in Biostatistics Summary of the lecture Bostatstcs Probablty Desty Fucto For a cotuos radom varable, a probablty desty fucto s a fucto such that: 0 dx a b) b a dx A probablty desty fucto provdes a smple descrpto of the

More information

CHAPTER VI Statistical Analysis of Experimental Data

CHAPTER VI Statistical Analysis of Experimental Data Chapter VI Statstcal Aalyss of Expermetal Data CHAPTER VI Statstcal Aalyss of Expermetal Data Measuremets do ot lead to a uque value. Ths s a result of the multtude of errors (maly radom errors) that ca

More information

Special Instructions / Useful Data

Special Instructions / Useful Data JAM 6 Set of all real umbers P A..d. B, p Posso Specal Istructos / Useful Data x,, :,,, x x Probablty of a evet A Idepedetly ad detcally dstrbuted Bomal dstrbuto wth parameters ad p Posso dstrbuto wth

More information

Extend the Borel-Cantelli Lemma to Sequences of. Non-Independent Random Variables

Extend the Borel-Cantelli Lemma to Sequences of. Non-Independent Random Variables ppled Mathematcal Sceces, Vol 4, 00, o 3, 637-64 xted the Borel-Catell Lemma to Sequeces of No-Idepedet Radom Varables olah Der Departmet of Statstc, Scece ad Research Campus zad Uversty of Tehra-Ira der53@gmalcom

More information

Bayes Estimator for Exponential Distribution with Extension of Jeffery Prior Information

Bayes Estimator for Exponential Distribution with Extension of Jeffery Prior Information Malaysa Joural of Mathematcal Sceces (): 97- (9) Bayes Estmator for Expoetal Dstrbuto wth Exteso of Jeffery Pror Iformato Hadeel Salm Al-Kutub ad Noor Akma Ibrahm Isttute for Mathematcal Research, Uverst

More information

Generating Multivariate Nonnormal Distribution Random Numbers Based on Copula Function

Generating Multivariate Nonnormal Distribution Random Numbers Based on Copula Function 7659, Eglad, UK Joural of Iformato ad Computg Scece Vol. 2, No. 3, 2007, pp. 9-96 Geeratg Multvarate Noormal Dstrbuto Radom Numbers Based o Copula Fucto Xaopg Hu +, Jam He ad Hogsheg Ly School of Ecoomcs

More information

Chapter 14 Logistic Regression Models

Chapter 14 Logistic Regression Models Chapter 4 Logstc Regresso Models I the lear regresso model X β + ε, there are two types of varables explaatory varables X, X,, X k ad study varable y These varables ca be measured o a cotuous scale as

More information

Complete Convergence and Some Maximal Inequalities for Weighted Sums of Random Variables

Complete Convergence and Some Maximal Inequalities for Weighted Sums of Random Variables Joural of Sceces, Islamc Republc of Ira 8(4): -6 (007) Uversty of Tehra, ISSN 06-04 http://sceces.ut.ac.r Complete Covergece ad Some Maxmal Iequaltes for Weghted Sums of Radom Varables M. Am,,* H.R. Nl

More information

Lecture 3. Sampling, sampling distributions, and parameter estimation

Lecture 3. Sampling, sampling distributions, and parameter estimation Lecture 3 Samplg, samplg dstrbutos, ad parameter estmato Samplg Defto Populato s defed as the collecto of all the possble observatos of terest. The collecto of observatos we take from the populato s called

More information

PROJECTION PROBLEM FOR REGULAR POLYGONS

PROJECTION PROBLEM FOR REGULAR POLYGONS Joural of Mathematcal Sceces: Advaces ad Applcatos Volume, Number, 008, Pages 95-50 PROJECTION PROBLEM FOR REGULAR POLYGONS College of Scece Bejg Forestry Uversty Bejg 0008 P. R. Cha e-mal: sl@bjfu.edu.c

More information

STATISTICAL PROPERTIES OF LEAST SQUARES ESTIMATORS. x, where. = y - ˆ " 1

STATISTICAL PROPERTIES OF LEAST SQUARES ESTIMATORS. x, where. = y - ˆ  1 STATISTICAL PROPERTIES OF LEAST SQUARES ESTIMATORS Recall Assumpto E(Y x) η 0 + η x (lear codtoal mea fucto) Data (x, y ), (x 2, y 2 ),, (x, y ) Least squares estmator ˆ E (Y x) ˆ " 0 + ˆ " x, where ˆ

More information

Ordinary Least Squares Regression. Simple Regression. Algebra and Assumptions.

Ordinary Least Squares Regression. Simple Regression. Algebra and Assumptions. Ordary Least Squares egresso. Smple egresso. Algebra ad Assumptos. I ths part of the course we are gog to study a techque for aalysg the lear relatoshp betwee two varables Y ad X. We have pars of observatos

More information

CHAPTER 3 POSTERIOR DISTRIBUTIONS

CHAPTER 3 POSTERIOR DISTRIBUTIONS CHAPTER 3 POSTERIOR DISTRIBUTIONS If scece caot measure the degree of probablt volved, so much the worse for scece. The practcal ma wll stck to hs apprecatve methods utl t does, or wll accept the results

More information

A New Family of Transformations for Lifetime Data

A New Family of Transformations for Lifetime Data Proceedgs of the World Cogress o Egeerg 4 Vol I, WCE 4, July - 4, 4, Lodo, U.K. A New Famly of Trasformatos for Lfetme Data Lakhaa Watthaacheewakul Abstract A famly of trasformatos s the oe of several

More information

Confidence Intervals for Double Exponential Distribution: A Simulation Approach

Confidence Intervals for Double Exponential Distribution: A Simulation Approach World Academy of Scece, Egeerg ad Techology Iteratoal Joural of Physcal ad Mathematcal Sceces Vol:6, No:, 0 Cofdece Itervals for Double Expoetal Dstrbuto: A Smulato Approach M. Alrasheed * Iteratoal Scece

More information

22 Nonparametric Methods.

22 Nonparametric Methods. 22 oparametrc Methods. I parametrc models oe assumes apror that the dstrbutos have a specfc form wth oe or more ukow parameters ad oe tres to fd the best or atleast reasoably effcet procedures that aswer

More information

Point Estimation: definition of estimators

Point Estimation: definition of estimators Pot Estmato: defto of estmators Pot estmator: ay fucto W (X,..., X ) of a data sample. The exercse of pot estmato s to use partcular fuctos of the data order to estmate certa ukow populato parameters.

More information

Qualifying Exam Statistical Theory Problem Solutions August 2005

Qualifying Exam Statistical Theory Problem Solutions August 2005 Qualfyg Exam Statstcal Theory Problem Solutos August 5. Let X, X,..., X be d uform U(,),

More information

Chapter 4 Multiple Random Variables

Chapter 4 Multiple Random Variables Revew o BST 63: Statstcal Theory I Ku Zhag, /0/008 Revew for Chapter 4-5 Notes: Although all deftos ad theorems troduced our lectures ad ths ote are mportat ad you should be famlar wth, but I put those

More information

( ) = ( ) ( ) Chapter 13 Asymptotic Theory and Stochastic Regressors. Stochastic regressors model

( ) = ( ) ( ) Chapter 13 Asymptotic Theory and Stochastic Regressors. Stochastic regressors model Chapter 3 Asmptotc Theor ad Stochastc Regressors The ature of eplaator varable s assumed to be o-stochastc or fed repeated samples a regresso aalss Such a assumpto s approprate for those epermets whch

More information

Discrete Mathematics and Probability Theory Fall 2016 Seshia and Walrand DIS 10b

Discrete Mathematics and Probability Theory Fall 2016 Seshia and Walrand DIS 10b CS 70 Dscrete Mathematcs ad Probablty Theory Fall 206 Sesha ad Walrad DIS 0b. Wll I Get My Package? Seaky delvery guy of some compay s out delverg packages to customers. Not oly does he had a radom package

More information

TESTS BASED ON MAXIMUM LIKELIHOOD

TESTS BASED ON MAXIMUM LIKELIHOOD ESE 5 Toy E. Smth. The Basc Example. TESTS BASED ON MAXIMUM LIKELIHOOD To llustrate the propertes of maxmum lkelhood estmates ad tests, we cosder the smplest possble case of estmatg the mea of the ormal

More information

Miin-Jye Wen National Cheng Kung University, City Tainan, Taiwan, R.O.C. Key words: general moment, multivariate survival function, set partition

Miin-Jye Wen National Cheng Kung University, City Tainan, Taiwan, R.O.C. Key words: general moment, multivariate survival function, set partition A Multvarate Webull Dstrbuto Cheg K. Lee chegl@uab.edu Charlotte, North Carola, USA M-Jye We Natoal Cheg Kug Uversty, Cty Taa, Tawa, R.O.C. Summary. A multvarate survval fucto of Webull Dstrbuto s developed

More information

ENGI 4421 Joint Probability Distributions Page Joint Probability Distributions [Navidi sections 2.5 and 2.6; Devore sections

ENGI 4421 Joint Probability Distributions Page Joint Probability Distributions [Navidi sections 2.5 and 2.6; Devore sections ENGI 441 Jot Probablty Dstrbutos Page 7-01 Jot Probablty Dstrbutos [Navd sectos.5 ad.6; Devore sectos 5.1-5.] The jot probablty mass fucto of two dscrete radom quattes, s, P ad p x y x y The margal probablty

More information

Module 7: Probability and Statistics

Module 7: Probability and Statistics Lecture 4: Goodess of ft tests. Itroducto Module 7: Probablty ad Statstcs I the prevous two lectures, the cocepts, steps ad applcatos of Hypotheses testg were dscussed. Hypotheses testg may be used to

More information

International Journal of Mathematical Archive-5(8), 2014, Available online through ISSN

International Journal of Mathematical Archive-5(8), 2014, Available online through   ISSN Iteratoal Joural of Mathematcal Archve-5(8) 204 25-29 Avalable ole through www.jma.fo ISSN 2229 5046 COMMON FIXED POINT OF GENERALIZED CONTRACTION MAPPING IN FUZZY METRIC SPACES Hamd Mottagh Golsha* ad

More information

Estimation of Stress- Strength Reliability model using finite mixture of exponential distributions

Estimation of Stress- Strength Reliability model using finite mixture of exponential distributions Iteratoal Joural of Computatoal Egeerg Research Vol, 0 Issue, Estmato of Stress- Stregth Relablty model usg fte mxture of expoetal dstrbutos K.Sadhya, T.S.Umamaheswar Departmet of Mathematcs, Lal Bhadur

More information

A Remark on the Uniform Convergence of Some Sequences of Functions

A Remark on the Uniform Convergence of Some Sequences of Functions Advaces Pure Mathematcs 05 5 57-533 Publshed Ole July 05 ScRes. http://www.scrp.org/joural/apm http://dx.do.org/0.436/apm.05.59048 A Remark o the Uform Covergece of Some Sequeces of Fuctos Guy Degla Isttut

More information

3. Basic Concepts: Consequences and Properties

3. Basic Concepts: Consequences and Properties : 3. Basc Cocepts: Cosequeces ad Propertes Markku Jutt Overvew More advaced cosequeces ad propertes of the basc cocepts troduced the prevous lecture are derved. Source The materal s maly based o Sectos.6.8

More information

Comparing Different Estimators of three Parameters for Transmuted Weibull Distribution

Comparing Different Estimators of three Parameters for Transmuted Weibull Distribution Global Joural of Pure ad Appled Mathematcs. ISSN 0973-768 Volume 3, Number 9 (207), pp. 55-528 Research Ida Publcatos http://www.rpublcato.com Comparg Dfferet Estmators of three Parameters for Trasmuted

More information

ON THE LOGARITHMIC INTEGRAL

ON THE LOGARITHMIC INTEGRAL Hacettepe Joural of Mathematcs ad Statstcs Volume 39(3) (21), 393 41 ON THE LOGARITHMIC INTEGRAL Bra Fsher ad Bljaa Jolevska-Tueska Receved 29:9 :29 : Accepted 2 :3 :21 Abstract The logarthmc tegral l(x)

More information

Comparison of Parameters of Lognormal Distribution Based On the Classical and Posterior Estimates

Comparison of Parameters of Lognormal Distribution Based On the Classical and Posterior Estimates Joural of Moder Appled Statstcal Methods Volume Issue Artcle 8 --03 Comparso of Parameters of Logormal Dstrbuto Based O the Classcal ad Posteror Estmates Raja Sulta Uversty of Kashmr, Sragar, Ida, hamzasulta8@yahoo.com

More information

Strong Convergence of Weighted Averaged Approximants of Asymptotically Nonexpansive Mappings in Banach Spaces without Uniform Convexity

Strong Convergence of Weighted Averaged Approximants of Asymptotically Nonexpansive Mappings in Banach Spaces without Uniform Convexity BULLETIN of the MALAYSIAN MATHEMATICAL SCIENCES SOCIETY Bull. Malays. Math. Sc. Soc. () 7 (004), 5 35 Strog Covergece of Weghted Averaged Appromats of Asymptotcally Noepasve Mappgs Baach Spaces wthout

More information

Generalized Convex Functions on Fractal Sets and Two Related Inequalities

Generalized Convex Functions on Fractal Sets and Two Related Inequalities Geeralzed Covex Fuctos o Fractal Sets ad Two Related Iequaltes Huxa Mo, X Su ad Dogya Yu 3,,3School of Scece, Bejg Uversty of Posts ad Telecommucatos, Bejg,00876, Cha, Correspodece should be addressed

More information

Comparison of Dual to Ratio-Cum-Product Estimators of Population Mean

Comparison of Dual to Ratio-Cum-Product Estimators of Population Mean Research Joural of Mathematcal ad Statstcal Sceces ISS 30 6047 Vol. 1(), 5-1, ovember (013) Res. J. Mathematcal ad Statstcal Sc. Comparso of Dual to Rato-Cum-Product Estmators of Populato Mea Abstract

More information

arxiv: v1 [math.st] 24 Oct 2016

arxiv: v1 [math.st] 24 Oct 2016 arxv:60.07554v [math.st] 24 Oct 206 Some Relatoshps ad Propertes of the Hypergeometrc Dstrbuto Peter H. Pesku, Departmet of Mathematcs ad Statstcs York Uversty, Toroto, Otaro M3J P3, Caada E-mal: pesku@pascal.math.yorku.ca

More information

Non-uniform Turán-type problems

Non-uniform Turán-type problems Joural of Combatoral Theory, Seres A 111 2005 106 110 wwwelsevercomlocatecta No-uform Turá-type problems DhruvMubay 1, Y Zhao 2 Departmet of Mathematcs, Statstcs, ad Computer Scece, Uversty of Illos at

More information

Random Variables. ECE 313 Probability with Engineering Applications Lecture 8 Professor Ravi K. Iyer University of Illinois

Random Variables. ECE 313 Probability with Engineering Applications Lecture 8 Professor Ravi K. Iyer University of Illinois Radom Varables ECE 313 Probablty wth Egeerg Alcatos Lecture 8 Professor Rav K. Iyer Uversty of Illos Iyer - Lecture 8 ECE 313 Fall 013 Today s Tocs Revew o Radom Varables Cumulatve Dstrbuto Fucto (CDF

More information

Q-analogue of a Linear Transformation Preserving Log-concavity

Q-analogue of a Linear Transformation Preserving Log-concavity Iteratoal Joural of Algebra, Vol. 1, 2007, o. 2, 87-94 Q-aalogue of a Lear Trasformato Preservg Log-cocavty Daozhog Luo Departmet of Mathematcs, Huaqao Uversty Quazhou, Fua 362021, P. R. Cha ldzblue@163.com

More information

Lecture Note to Rice Chapter 8

Lecture Note to Rice Chapter 8 ECON 430 HG revsed Nov 06 Lecture Note to Rce Chapter 8 Radom matrces Let Y, =,,, m, =,,, be radom varables (r.v. s). The matrx Y Y Y Y Y Y Y Y Y Y = m m m s called a radom matrx ( wth a ot m-dmesoal dstrbuto,

More information

Simple Linear Regression

Simple Linear Regression Statstcal Methods I (EST 75) Page 139 Smple Lear Regresso Smple regresso applcatos are used to ft a model descrbg a lear relatoshp betwee two varables. The aspects of least squares regresso ad correlato

More information

Research Article A New Iterative Method for Common Fixed Points of a Finite Family of Nonexpansive Mappings

Research Article A New Iterative Method for Common Fixed Points of a Finite Family of Nonexpansive Mappings Hdaw Publshg Corporato Iteratoal Joural of Mathematcs ad Mathematcal Sceces Volume 009, Artcle ID 391839, 9 pages do:10.1155/009/391839 Research Artcle A New Iteratve Method for Commo Fxed Pots of a Fte

More information

Solving Constrained Flow-Shop Scheduling. Problems with Three Machines

Solving Constrained Flow-Shop Scheduling. Problems with Three Machines It J Cotemp Math Sceces, Vol 5, 2010, o 19, 921-929 Solvg Costraed Flow-Shop Schedulg Problems wth Three Maches P Pada ad P Rajedra Departmet of Mathematcs, School of Advaced Sceces, VIT Uversty, Vellore-632

More information

Simulation Output Analysis

Simulation Output Analysis Smulato Output Aalyss Summary Examples Parameter Estmato Sample Mea ad Varace Pot ad Iterval Estmato ermatg ad o-ermatg Smulato Mea Square Errors Example: Sgle Server Queueg System x(t) S 4 S 4 S 3 S 5

More information

Chapter 13, Part A Analysis of Variance and Experimental Design. Introduction to Analysis of Variance. Introduction to Analysis of Variance

Chapter 13, Part A Analysis of Variance and Experimental Design. Introduction to Analysis of Variance. Introduction to Analysis of Variance Chapter, Part A Aalyss of Varace ad Epermetal Desg Itroducto to Aalyss of Varace Aalyss of Varace: Testg for the Equalty of Populato Meas Multple Comparso Procedures Itroducto to Aalyss of Varace Aalyss

More information

Midterm Exam 1, section 1 (Solution) Thursday, February hour, 15 minutes

Midterm Exam 1, section 1 (Solution) Thursday, February hour, 15 minutes coometrcs, CON Sa Fracsco State Uversty Mchael Bar Sprg 5 Mdterm am, secto Soluto Thursday, February 6 hour, 5 mutes Name: Istructos. Ths s closed book, closed otes eam.. No calculators of ay kd are allowed..

More information

A Study of the Reproducibility of Measurements with HUR Leg Extension/Curl Research Line

A Study of the Reproducibility of Measurements with HUR Leg Extension/Curl Research Line HUR Techcal Report 000--9 verso.05 / Frak Borg (borgbros@ett.f) A Study of the Reproducblty of Measuremets wth HUR Leg Eteso/Curl Research Le A mportat property of measuremets s that the results should

More information

. The set of these sums. be a partition of [ ab, ]. Consider the sum f( x) f( x 1)

. The set of these sums. be a partition of [ ab, ]. Consider the sum f( x) f( x 1) Chapter 7 Fuctos o Bouded Varato. Subject: Real Aalyss Level: M.Sc. Source: Syed Gul Shah (Charma, Departmet o Mathematcs, US Sargodha Collected & Composed by: Atq ur Rehma (atq@mathcty.org, http://www.mathcty.org

More information

Chapter 8: Statistical Analysis of Simulated Data

Chapter 8: Statistical Analysis of Simulated Data Marquette Uversty MSCS600 Chapter 8: Statstcal Aalyss of Smulated Data Dael B. Rowe, Ph.D. Departmet of Mathematcs, Statstcs, ad Computer Scece Copyrght 08 by Marquette Uversty MSCS600 Ageda 8. The Sample

More information

Part 4b Asymptotic Results for MRR2 using PRESS. Recall that the PRESS statistic is a special type of cross validation procedure (see Allen (1971))

Part 4b Asymptotic Results for MRR2 using PRESS. Recall that the PRESS statistic is a special type of cross validation procedure (see Allen (1971)) art 4b Asymptotc Results for MRR usg RESS Recall that the RESS statstc s a specal type of cross valdato procedure (see Alle (97)) partcular to the regresso problem ad volves fdg Y $,, the estmate at the

More information

Econometric Methods. Review of Estimation

Econometric Methods. Review of Estimation Ecoometrc Methods Revew of Estmato Estmatg the populato mea Radom samplg Pot ad terval estmators Lear estmators Ubased estmators Lear Ubased Estmators (LUEs) Effcecy (mmum varace) ad Best Lear Ubased Estmators

More information

MEASURES OF DISPERSION

MEASURES OF DISPERSION MEASURES OF DISPERSION Measure of Cetral Tedecy: Measures of Cetral Tedecy ad Dsperso ) Mathematcal Average: a) Arthmetc mea (A.M.) b) Geometrc mea (G.M.) c) Harmoc mea (H.M.) ) Averages of Posto: a) Meda

More information

5 Short Proofs of Simplified Stirling s Approximation

5 Short Proofs of Simplified Stirling s Approximation 5 Short Proofs of Smplfed Strlg s Approxmato Ofr Gorodetsky, drtymaths.wordpress.com Jue, 20 0 Itroducto Strlg s approxmato s the followg (somewhat surprsg) approxmato of the factoral,, usg elemetary fuctos:

More information

New Transformation of Dependent Input Variables Using Copula for RBDO

New Transformation of Dependent Input Variables Using Copula for RBDO 7 th World Cogresses of Structural ad Multdscplary Optmzato COE Seoul, May 5 May 7, Korea New Trasformato of Depedet Iput Varables Usg Copula for RBDO BUoojeog NohU, K.K. Cho*, ad Lu Du Departmet of Mechacal

More information

MATH 247/Winter Notes on the adjoint and on normal operators.

MATH 247/Winter Notes on the adjoint and on normal operators. MATH 47/Wter 00 Notes o the adjot ad o ormal operators I these otes, V s a fte dmesoal er product space over, wth gve er * product uv, T, S, T, are lear operators o V U, W are subspaces of V Whe we say

More information

02/15/04 INTERESTING FINITE AND INFINITE PRODUCTS FROM SIMPLE ALGEBRAIC IDENTITIES

02/15/04 INTERESTING FINITE AND INFINITE PRODUCTS FROM SIMPLE ALGEBRAIC IDENTITIES 0/5/04 ITERESTIG FIITE AD IFIITE PRODUCTS FROM SIMPLE ALGEBRAIC IDETITIES Thomas J Osler Mathematcs Departmet Rowa Uversty Glassboro J 0808 Osler@rowaedu Itroducto The dfferece of two squares, y = + y

More information

Lecture 7. Confidence Intervals and Hypothesis Tests in the Simple CLR Model

Lecture 7. Confidence Intervals and Hypothesis Tests in the Simple CLR Model Lecture 7. Cofdece Itervals ad Hypothess Tests the Smple CLR Model I lecture 6 we troduced the Classcal Lear Regresso (CLR) model that s the radom expermet of whch the data Y,,, K, are the outcomes. The

More information

4 Inner Product Spaces

4 Inner Product Spaces 11.MH1 LINEAR ALGEBRA Summary Notes 4 Ier Product Spaces Ier product s the abstracto to geeral vector spaces of the famlar dea of the scalar product of two vectors or 3. I what follows, keep these key

More information

Some Notes on the Probability Space of Statistical Surveys

Some Notes on the Probability Space of Statistical Surveys Metodološk zvezk, Vol. 7, No., 200, 7-2 ome Notes o the Probablty pace of tatstcal urveys George Petrakos Abstract Ths paper troduces a formal presetato of samplg process usg prcples ad cocepts from Probablty

More information

Study of Correlation using Bayes Approach under bivariate Distributions

Study of Correlation using Bayes Approach under bivariate Distributions Iteratoal Joural of Scece Egeerg ad Techolog Research IJSETR Volume Issue Februar 4 Stud of Correlato usg Baes Approach uder bvarate Dstrbutos N.S.Padharkar* ad. M.N.Deshpade** *Govt.Vdarbha Isttute of

More information

Introduction to local (nonparametric) density estimation. methods

Introduction to local (nonparametric) density estimation. methods Itroducto to local (oparametrc) desty estmato methods A slecture by Yu Lu for ECE 66 Sprg 014 1. Itroducto Ths slecture troduces two local desty estmato methods whch are Parze desty estmato ad k-earest

More information

Module 7. Lecture 7: Statistical parameter estimation

Module 7. Lecture 7: Statistical parameter estimation Lecture 7: Statstcal parameter estmato Parameter Estmato Methods of Parameter Estmato 1) Method of Matchg Pots ) Method of Momets 3) Mamum Lkelhood method Populato Parameter Sample Parameter Ubased estmato

More information

Arithmetic Mean and Geometric Mean

Arithmetic Mean and Geometric Mean Acta Mathematca Ntresa Vol, No, p 43 48 ISSN 453-6083 Arthmetc Mea ad Geometrc Mea Mare Varga a * Peter Mchalča b a Departmet of Mathematcs, Faculty of Natural Sceces, Costate the Phlosopher Uversty Ntra,

More information

On the Link Between the Concepts of Kurtosis and Bipolarization. Abstract

On the Link Between the Concepts of Kurtosis and Bipolarization. Abstract O the Lk etwee the Cocepts of Kurtoss ad polarzato Jacques SILE ar-ila Uversty Joseph Deutsch ar-ila Uversty Metal Haoka ar-ila Uversty h.d. studet) Abstract I a paper o the measuremet of the flatess of

More information

ENGI 4421 Propagation of Error Page 8-01

ENGI 4421 Propagation of Error Page 8-01 ENGI 441 Propagato of Error Page 8-01 Propagato of Error [Navd Chapter 3; ot Devore] Ay realstc measuremet procedure cotas error. Ay calculatos based o that measuremet wll therefore also cota a error.

More information

1 Solution to Problem 6.40

1 Solution to Problem 6.40 1 Soluto to Problem 6.40 (a We wll wrte T τ (X 1,...,X where the X s are..d. wth PDF f(x µ, σ 1 ( x µ σ g, σ where the locato parameter µ s ay real umber ad the scale parameter σ s > 0. Lettg Z X µ σ we

More information

Bayes (Naïve or not) Classifiers: Generative Approach

Bayes (Naïve or not) Classifiers: Generative Approach Logstc regresso Bayes (Naïve or ot) Classfers: Geeratve Approach What do we mea by Geeratve approach: Lear p(y), p(x y) ad the apply bayes rule to compute p(y x) for makg predctos Ths s essetally makg

More information

PTAS for Bin-Packing

PTAS for Bin-Packing CS 663: Patter Matchg Algorthms Scrbe: Che Jag /9/00. Itroducto PTAS for B-Packg The B-Packg problem s NP-hard. If we use approxmato algorthms, the B-Packg problem could be solved polyomal tme. For example,

More information

Chapter 9 Jordan Block Matrices

Chapter 9 Jordan Block Matrices Chapter 9 Jorda Block atrces I ths chapter we wll solve the followg problem. Gve a lear operator T fd a bass R of F such that the matrx R (T) s as smple as possble. f course smple s a matter of taste.

More information

Continuous Distributions

Continuous Distributions 7//3 Cotuous Dstrbutos Radom Varables of the Cotuous Type Desty Curve Percet Desty fucto, f (x) A smooth curve that ft the dstrbuto 3 4 5 6 7 8 9 Test scores Desty Curve Percet Probablty Desty Fucto, f

More information

Marcinkiewicz strong laws for linear statistics of ρ -mixing sequences of random variables

Marcinkiewicz strong laws for linear statistics of ρ -mixing sequences of random variables Aas da Academa Braslera de Cêcas 2006 784: 65-62 Aals of the Brazla Academy of Sceces ISSN 000-3765 www.scelo.br/aabc Marckewcz strog laws for lear statstcs of ρ -mxg sequeces of radom varables GUANG-HUI

More information

Maximum Likelihood Estimation

Maximum Likelihood Estimation Marquette Uverst Maxmum Lkelhood Estmato Dael B. Rowe, Ph.D. Professor Departmet of Mathematcs, Statstcs, ad Computer Scece Coprght 08 b Marquette Uverst Maxmum Lkelhood Estmato We have bee sag that ~

More information

Lebesgue Measure of Generalized Cantor Set

Lebesgue Measure of Generalized Cantor Set Aals of Pure ad Appled Mathematcs Vol., No.,, -8 ISSN: -8X P), -888ole) Publshed o 8 May www.researchmathsc.org Aals of Lebesgue Measure of Geeralzed ator Set Md. Jahurul Islam ad Md. Shahdul Islam Departmet

More information

1. The weight of six Golden Retrievers is 66, 61, 70, 67, 92 and 66 pounds. The weight of six Labrador Retrievers is 54, 60, 72, 78, 84 and 67.

1. The weight of six Golden Retrievers is 66, 61, 70, 67, 92 and 66 pounds. The weight of six Labrador Retrievers is 54, 60, 72, 78, 84 and 67. Ecoomcs 3 Itroducto to Ecoometrcs Sprg 004 Professor Dobk Name Studet ID Frst Mdterm Exam You must aswer all the questos. The exam s closed book ad closed otes. You may use your calculators but please

More information

Point Estimation: definition of estimators

Point Estimation: definition of estimators Pot Estmato: defto of estmators Pot estmator: ay fucto W (X,..., X ) of a data sample. The exercse of pot estmato s to use partcular fuctos of the data order to estmate certa ukow populato parameters.

More information

THE PROBABILISTIC STABILITY FOR THE GAMMA FUNCTIONAL EQUATION

THE PROBABILISTIC STABILITY FOR THE GAMMA FUNCTIONAL EQUATION Joural of Scece ad Arts Year 12, No. 3(2), pp. 297-32, 212 ORIGINAL AER THE ROBABILISTIC STABILITY FOR THE GAMMA FUNCTIONAL EQUATION DOREL MIHET 1, CLAUDIA ZAHARIA 1 Mauscrpt receved: 3.6.212; Accepted

More information

hp calculators HP 30S Statistics Averages and Standard Deviations Average and Standard Deviation Practice Finding Averages and Standard Deviations

hp calculators HP 30S Statistics Averages and Standard Deviations Average and Standard Deviation Practice Finding Averages and Standard Deviations HP 30S Statstcs Averages ad Stadard Devatos Average ad Stadard Devato Practce Fdg Averages ad Stadard Devatos HP 30S Statstcs Averages ad Stadard Devatos Average ad stadard devato The HP 30S provdes several

More information

Generalization of the Dissimilarity Measure of Fuzzy Sets

Generalization of the Dissimilarity Measure of Fuzzy Sets Iteratoal Mathematcal Forum 2 2007 o. 68 3395-3400 Geeralzato of the Dssmlarty Measure of Fuzzy Sets Faramarz Faghh Boformatcs Laboratory Naobotechology Research Ceter vesa Research Isttute CECR Tehra

More information

Investigation of Partially Conditional RP Model with Response Error. Ed Stanek

Investigation of Partially Conditional RP Model with Response Error. Ed Stanek Partally Codtoal Radom Permutato Model 7- vestgato of Partally Codtoal RP Model wth Respose Error TRODUCTO Ed Staek We explore the predctor that wll result a smple radom sample wth respose error whe a

More information

Outline. Point Pattern Analysis Part I. Revisit IRP/CSR

Outline. Point Pattern Analysis Part I. Revisit IRP/CSR Pot Patter Aalyss Part I Outle Revst IRP/CSR, frst- ad secod order effects What s pot patter aalyss (PPA)? Desty-based pot patter measures Dstace-based pot patter measures Revst IRP/CSR Equal probablty:

More information

A be a probability space. A random vector

A be a probability space. A random vector Statstcs 1: Probablty Theory II 8 1 JOINT AND MARGINAL DISTRIBUTIONS I Probablty Theory I we formulate the cocept of a (real) radom varable ad descrbe the probablstc behavor of ths radom varable by the

More information

Dimensionality Reduction and Learning

Dimensionality Reduction and Learning CMSC 35900 (Sprg 009) Large Scale Learg Lecture: 3 Dmesoalty Reducto ad Learg Istructors: Sham Kakade ad Greg Shakharovch L Supervsed Methods ad Dmesoalty Reducto The theme of these two lectures s that

More information

Chapter 4 (Part 1): Non-Parametric Classification (Sections ) Pattern Classification 4.3) Announcements

Chapter 4 (Part 1): Non-Parametric Classification (Sections ) Pattern Classification 4.3) Announcements Aoucemets No-Parametrc Desty Estmato Techques HW assged Most of ths lecture was o the blacboard. These sldes cover the same materal as preseted DHS Bometrcs CSE 90-a Lecture 7 CSE90a Fall 06 CSE90a Fall

More information

Lecture 02: Bounding tail distributions of a random variable

Lecture 02: Bounding tail distributions of a random variable CSCI-B609: A Theorst s Toolkt, Fall 206 Aug 25 Lecture 02: Boudg tal dstrbutos of a radom varable Lecturer: Yua Zhou Scrbe: Yua Xe & Yua Zhou Let us cosder the ubased co flps aga. I.e. let the outcome

More information

Reliability Based Design Optimization with Correlated Input Variables

Reliability Based Design Optimization with Correlated Input Variables 7--55 Relablty Based Desg Optmzato wth Correlated Iput Varables Copyrght 7 SAE Iteratoal Kyug K. Cho, Yoojeog Noh, ad Lu Du Departmet of Mechacal & Idustral Egeerg & Ceter for Computer Aded Desg, College

More information