9 U-STATISTICS. Eh =(m!) 1 Eh(X (1),..., X (m ) ) i.i.d
|
|
- Osborn Bridges
- 5 years ago
- Views:
Transcription
1 9 U-STATISTICS Suppose,,..., are P P..d. wth CDF F. Our goal s to estmate the expectato t (P)=Eh(,,..., m ). Note that ths expectato requres more tha oe cotrast to E, E, or Eh( ). Oe example s E or P((, ) S). For Eh( ), the emprcal estmator s asymptotcally optmal. Now, U-statstcs geeralze the dea. The advatage of usg U-statstcs s ubasedess. Notato. Let t (P) = R R h(x,..., x m )d F (x ) d F (x m ) where h s a kow fucto; h s called kerel. Assume that, the followg, wthout loss of geeralty, h s symmetrc,.e. h(x, x )=h(x, x ). If h s ot symmetrc, we ca replace t wth h (x,..., x m )= (m!) P h(x (),..., x (m ) ) where : N 7! N ad s set of all possble permutatos of {,..., m}. Note that h s also ubased, Eh =(m!) Eh( (),..., (m ) )..d =(m!) Eh(,..., m )=t(p). Defto 9.. Suppose h : R m 7! R s symmetrc ts argumets. The U-statstc for estmatg t (P)=Eh(,..., m ) s a symmetrc average u (,..., m )= m apple < < < m apple h(,..., m ). Example. Suppose m =, the u (, )= h(, )= h(, j ) () ( ) < <j = h(, j ) () ( ) 6=j Remark. E m apple < < < m apple Eh(,..., m )=Eh(,..., m )=t (P). Example 9.. (a) (b) Suppose t (P)=E = R x d F (x ). The, h(x )=x ad u (,..., m )= h( )= =. Suppose t (P)=(E ) = ÄR x d F (x ) ä. The u = from (a) s based sce 3 E( )= 6 4 E j + E( ) 7 5 6=(E ). = j 6= =(E ) = E 6=(E ) 35
2 Now, wrte t (P)= RR x x d F (x )d F (x ) ad h(x, x )=x x. The u (,..., )= j. ( ) (c) Suppose t (P)=P( apple t 0 )=F (t 0 )= R h(x )d F (x ). The h(x )= (,t 0 ](x ) ad whch s just the emprcal CDF. <j { apple t 0 } = ˆF (t 0 ) (d) Suppose t (P)=Var = RR x +x x x d F (x )d F (x ). The h(x, x )=(x +x x x )/ = (x x ) /. Note that Eh(x, x )={Var( )+[E( )] + Var( )+(E ) E( )}/..d. = Var( ). Hece, h(, j ) ( ) <j = + j j ( ) = 6= ˆ = 6=j ( ) = ( ). = Note that ˆ s a based estmator ad u s a ubased estmator. (e) Suppose t (P)=E = RR x x d F (x )d F (x ) (measure of dsperso). The j. ( ) u s called G s Mea Dfferece. (f ) Suppose t (P) =P( + apple 0) = RR {x + x apple 0}d F (x )d F (x ). The, h(x, x )= {x + x apple 0} ad {x + x j apple 0}. ( ) <j <j 36
3 Remark 9.3 (Prelmary Remark). Wrte () =( (),..., () ) as the order statstc. U-statstc ca be regarded as codtoal expectato gve (). For m =, h( )= h( ( ) )=EˆF [h( ) () ]. For m=, <j h(, j )= h( ( ), (j ) )=EˆF [h(, ) () ]. <j For arbtrary m, E ˆF [h(,,..., m ) () ]. Now: ay ubased estmator S = S(,..., ) ca be mproved by ts U-statstc verso (or S f S s ot symmetrc) E ˆF [S () ]. For example, s ubased for E. Now, E ˆF ( () )= P = ( ) = = P = = E ˆF ( ). Theorem 9.4. Let S = S(,..., ) be a ubased estmator of t (P) wth correspodg U- statstc u. The, u s ubased as well ad Varu apple VarS wth the equalty holdg f P( S)=. Proof. u s ubased: E(u )=E[EˆF (S () )] = t (P). E(S) Sce both u ad S are ubased we show Eu apple ES, Eu = E[E ˆF (S Jese equ. ())] apple E[E ˆF (S () )] = E(S ) wth "=" f the dstrbuto of E ˆF (S () ) s degeerate wth E ˆF (S () )=S almost surely. Note: Ths also follows from the Rao-Blackwell Theorem: Takg codtoal expectato of a ubased statstc codtoal o a suffcet statstc (eg. () here) wll gve us a estmator whch s as least as good the sese of lower rsk/varace. Remark 9.5 (The Varace for m apple (heurstcs)). m =. Var Var! h( ) = Varh( )=O(/). = 37
4 m =. Sce = h(, j )= h(, j ) ( ) <j <j [h(, )+h(, 3 )+ + h(, )], ot depedet the 0 B Var Var@ C h(, j ) A ( ) j > = Cov[h(, j ), h( k, l )]. ( ) j > k l >k =0 f,j,k,l dfferet The secod largest term (three sums): e.g. = k but k, j, l are dfferet (' u 3 ) Varu 3 3 ( ) = same order as m =. 4 Thus, we expect geeral, Var O(/). Notato. h (x,..., x )=E[h(,..., m ) = x,..., = x ] ad = Varh (,..., ) Lemma 9.6. (a) Eh (,..., )=t(p)(= Eh(,..., m )) for all apple apple m. (b) Cov[h(,...,, +,..., m ), h(,...,, 0 +,..., 0 )] = m Proof. We oly cosder m = here. (a) (b) =. Eh (, )=E{E[h(, ), ]} = E[h(, )] = t (P) Eh (, )=E{E[h(, ) ]} = E[h(, )] = t (P) Cov[h(, ), h(, )] = Var[h(, )] = Varh (, )=. The secod equalty s because h (, )=E[h(, ), ]=h(, ). =. Cov[h(, ), h(, 0 )] = E[h(, )h(, 0 )] E[h(, )]E[h(, 0 )]. ={E[h(, )]} =[t (P)] 38
5 Note that frst term o the rght had sde ca be computed by E[h(, )h(, 0 )] = E {E [h(, )h(, 0 ) ]} = E {E [h(, ) ]E [h(, 0 ) ]} = Eh ( ). Theorem 9.7 (Hoeffdg). (a) The varace of the U-statstc s Var m m m = m m. Note that oe ca compute from Lemma 9.6. (b) If > 0 ad < for =,,..., m, the Var( p u )! m. Proof. (a) For geeral proof, see Lee U-statstcs (990). We oly prove the case of m =. We wat to show that Var = [( ) ] wth = Cov[h(, ), h(, 0 )] ad = Cov[h(, ), h(, )] = Var[h(, )]. From Remark 9.5 we have Ç å Var Cov[h(, j ), h( k, l )] = j > k j > k l >k l >k Case, j, k, l are all dfferet. The Cov = 0. Case = k ad j = l : Cov[h(, j ), h( k, l )] =( ) + Cov[h(, j ), h( k, l )] = Cov[h(, j ), h(, j )] = Var[h(, j )] =.. Note that the umber of ways to choose, j out of {,,..., } s / =.. Ths gves us 39
6 Case 3 ( = k ad j 6= l ) or ( 6= k ad j = l ). Cov[h(, j ), h( k, l )] =. (b) Note that the umber of ways to choose, j, k, l s {z} ( ) {z} {z} = ( ). j 6= l or Ths gves ( ) = ( ) Cosder the varace formula from (a) m = k k!. ( m )( m ) ( m k + ) m k k! s large f k s large. Ths mples that s large f m s large,.e. =. Thus the m terms of the sum are domated by the = term,.e., by m m m m m (m )! m /m! = m. Hece, Var( p u )! m. m Theorem 9.8. p (u t (P)) D! N (0, m ). Proof. For m =, For geeral proof, see p. 78 of Serflg. m m u = c c c= apple < < c apple g c (,..., c )+o p ( / ). T = / (u t (P)) = / [h ( ) t (P)] + o p () := T. We wat to show that E(T T )! 0 () T T = o p()). E(T T ) = Var(T T ) = Var(T )+Var(T ) Cov(T,T ) = Var[ p (u t (P))] + Var p [h ( ) t (P)] p(u Cov t (P)), p [h ( ) t (P) = Var( p u ) + 4!m Var[h ( )] 4 4 Cov(u, h ( )) ( ) =!! 0. 40
7 For ( ), Cov(u, h ( )) = Cov[h( l, k ), h ( )] = ( ) k l 6=k ( ) =, f k = or l = The umber of o-zero terms s ( ).. For ( ), (9.6) = Cov[h(, ), h(, 0 )] = E[h(, )h(, 0 )] [t (P)] = E{E[h(, )h(, 0 ), ]} [t (P)] = E{h(, )E[h(, 0 ), ]} [t (P)] =E(h(, ) )=h ( ) = Cov[h(, ), h ( )]. Example 9.9. Suppose t (P)=P( + > 0), m =, h(, )=( + > 0). The ( + j > 0) ( ) <j ad, by Theorem 9.8, p (u P( + > 0)) D! N (0, 4 ). The ext thg s to calculate. (9.6) = Cov[h(, ), h(, 0 )] = E[h(, )h(, 0 )] [P( + > 0)] = E[( + > 0)( + 0 > 0)] [P( + > 0)] = E[( + > 0, + 0 > 0)] [P( + > 0)]. To obta a explct form we eed some assumptos. Suppose, for example, F s symmetrc aroud zero,.e. P( < a )=P( > a )=P( < a ) whch mples ad have the same dstrbuto. Moreover, suppose F s cotuous. The P( + > 0)=P( > 0)=P( + < 0) wth P( + = 0 = 0), ad therefore = P( + < 0)+P( + = 0)+P( + > 0)=P( + > 0) ) P( + > 0)=. O the other had, sce P( + > 0)=P( > )=P( > ), P( + > 0, + 0 > 0)=P( = max{,, 0 })=P( = max{,, 3 }, =,, 3)=/3. I sum, = /3 (/) = /. 4
8 Remark 9.0 (Geeralzato: Two-Sample Problems). Suppose,..., are..d. wth cdf F, Y,..., Y are..d. wth cdf G, ad F ad G are ukow, ad Y are depedet. Let h(,..., {z m, Y },..., Y my ) symmetrc symmetrc be a fucto of m +m Y argumets, wth m apple ad m Y apple Y. We wat to estmate t (P)= Eh(,..., m, Y,..., Y my ). For example, t (P)=P( < Y )=Eh(, Y ) where h(, Y )=( < Y ); or t (P)=E Y. Note also that h(,..., m, Y,..., Y my ) s trvally ubased for t (P)= Eh. Thus h(,..., m, Y j,..., Y j my ) wth apple apple... apple m apple ad apple j apple... apple j my apple Y s ubased as well. The umber of combatos s. Our ubased estamtor s m Y m Y m Y m Y h Ä ä,..., m, Y j,..., Y j my Example (m = m Y = ). Y h(, k, Y j, Y l ) <k j <l where h(,, Y, Y )=( < Y Y ). The Y ( k < Y j Y l ) <k j <l =the umber of (,j,k,l ) where y dstace s larger ad t (P) =P( Y Y > ). Note that ths U-statstc ca be used to test Y s more dspersed tha. Remark 9. (Propertes of the Two-Sample U-statstc). (a) Formula for Var u, see Lee or Serflg. (b) Asymptotc varace ad ormalty. Let = + Y ad /! c where c (0, ). Suppose Var[h(,,..., m, Y, Y,..., Y my )] > 0. The Var( p u )! = m c + m Y 0 c 0 ad p (u t (P)) D! N (0, ) where = Cov[h( 0,,..., m, Y, Y,..., Y my ), h(, 0,..., 0 m, Y 0, Y 0,..., Y 0 m Y ) = Cov[h( 0,,..., m, Y, Y,..., Y my ), h( 0, 0,..., 0 m, Y, Y 0,..., Y 0 m Y )] 4
9 Example 9.. Suppose t (P)=P( < Y )=E[( < Y )] (thus m = m Y = ). The ad ( < Y j ) Y = Cov[( 0 < Y ), ( < Y 0 )] = E[(( < Y )( < Y 0 )] {E[( < Y )]} = P( < Y, < Y 0 ) [P( < Y )] = Cov[( 0 < Y ), ( 0 < Y )] = E[(( < Y )( 0 < Y j )] {E[( < Y )]} = P( < Y, 0 < Y ) [P( < Y )]. Now suppose F = G s cotuous. The P( < Y )= P( Y )= P( > Y )= P(Y > ) ad hece P( < Y )= /. Furthermore, P( < Y, < Y 0)= P( = m{, Y, Y 0})=/3 ad P( < Y, 0 < Y )=P(Y = max{, 0, Y }) =/3. The = 0 = / ad the asymptotc varace s 0 j = m c + m Y 0 c = 0 c( c). I sum, p D (u P( < Y ))! N 0, c( c) Note: Y P P ( j < Y j ) s the Ma-Whtey test statstc wth H 0 : F = G (ad equvalet to a Wlcoxo rak sum statstc). Defto 9.3. Cosder a symmetrc fucto h : R m 7! R wth m apple. The V-statstc for estmatg t (P)=Eh(,..., m ) s V = V (,..., m )= m = h(,..., m ) m = Remark 9.4 (Comparg U- ad V-Statstcs). m =. P h( )=v m =. Frst ote that h(, j )= h(, j ). ( ) ( ) <j 6=j 43
10 O the other had, 3 v = h(, j )= h(, j )+ h(, ) 5 Moreover, Eh(, )=t (P). j j 6= = h(, j )+ h(, ) j 6= ( ) = u + h( {z }, ).! P!0 Ev = Eu + Eh(, ) = t (P) t (P)+ Eh(, ) 3 = t (P) Eh(, ) t (P) 5 =costat {z } =bas!0 Theorem 9.5. Let m =, = Varh (,..., ) (see 9.6), ad suppose 0 < <, <. The U- ad V-statstcs have the same asymptotc dstrbuto, Proof. From Remark 9.4, p p (V t (P)) = u + = p = =! p (V t (P)) D! N (0, 4 ).! h(, ) t (P) + (u t (P)) + h(, ) p (u t (P)) + p (u t (P)) + D!N (0,4 ) t (P) p h (h(, ) t (P)) D p (h(, ) t (P))! N (0, 4 ). {z } p!e[h(, ) t (P)]=costat Cocluso: U- ad V-statstcs are asymptotcally equvalet. The V-statstc s a more tutve estmator, the U-statstc s more coveet for proofs (ad ubased). 44
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 informationEconometric 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 informationTESTS 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 informationSTATISTICAL 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 informationChapter 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 informationLecture 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 informationUNIVERSITY 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{ }{ ( )} (, ) = ( ) ( ) ( ) Chapter 14 Exercises in Sampling Theory. Exercise 1 (Simple random sampling): Solution:
Chapter 4 Exercses Samplg Theory Exercse (Smple radom samplg: Let there be two correlated radom varables X ad A sample of sze s draw from a populato by smple radom samplg wthout replacemet The observed
More informationUNIVERSITY 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 informationOrdinary 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 informationLecture 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 information1 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 informationSTK4011 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 informationLecture 9: Tolerant Testing
Lecture 9: Tolerat Testg Dael Kae Scrbe: Sakeerth Rao Aprl 4, 07 Abstract I ths lecture we prove a quas lear lower boud o the umber of samples eeded to do tolerat testg for L dstace. Tolerat Testg We have
More informationX 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 informationQualifying Exam Statistical Theory Problem Solutions August 2005
Qualfyg Exam Statstcal Theory Problem Solutos August 5. Let X, X,..., X be d uform U(,),
More informationSpecial 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 informationChapter 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 informationρ < 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 informationPoint 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 informationLecture 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 informationPart 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 informationThird handout: On the Gini Index
Thrd hadout: O the dex Corrado, a tala statstca, proposed (, 9, 96) to measure absolute equalt va the mea dfferece whch s defed as ( / ) where refers to the total umber of dvduals socet. Assume that. The
More informationLecture 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 information3. 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( ) = ( ) ( ) 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 informationChapter 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 informationCHAPTER 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 informationParameter, Statistic and Random Samples
Parameter, Statstc ad Radom Samples A parameter s a umber that descrbes the populato. It s a fxed umber, but practce we do ot kow ts value. A statstc s a fucto of the sample data,.e., t s a quatty whose
More informationBounds on the expected entropy and KL-divergence of sampled multinomial distributions. Brandon C. Roy
Bouds o the expected etropy ad KL-dvergece of sampled multomal dstrbutos Brado C. Roy bcroy@meda.mt.edu Orgal: May 18, 2011 Revsed: Jue 6, 2011 Abstract Iformato theoretc quattes calculated from a sampled
More informationSampling Theory MODULE X LECTURE - 35 TWO STAGE SAMPLING (SUB SAMPLING)
Samplg Theory ODULE X LECTURE - 35 TWO STAGE SAPLIG (SUB SAPLIG) DR SHALABH DEPARTET OF ATHEATICS AD STATISTICS IDIA ISTITUTE OF TECHOLOG KAPUR Two stage samplg wth uequal frst stage uts: Cosder two stage
More informationA tighter lower bound on the circuit size of the hardest Boolean functions
Electroc Colloquum o Computatoal Complexty, Report No. 86 2011) A tghter lower boud o the crcut sze of the hardest Boolea fuctos Masak Yamamoto Abstract I [IPL2005], Fradse ad Mlterse mproved bouds o the
More informationChapter 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 information22 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 informationPROPERTIES OF GOOD ESTIMATORS
ESTIMATION INTRODUCTION Estmato s the statstcal process of fdg a appromate value for a populato parameter. A populato parameter s a characterstc of the dstrbuto of a populato such as the populato mea,
More informationHomework 1: Solutions Sid Banerjee Problem 1: (Practice with Asymptotic Notation) ORIE 4520: Stochastics at Scale Fall 2015
Fall 05 Homework : Solutos Problem : (Practce wth Asymptotc Notato) A essetal requremet for uderstadg scalg behavor s comfort wth asymptotc (or bg-o ) otato. I ths problem, you wll prove some basc facts
More informationChapter 3 Sampling For Proportions and Percentages
Chapter 3 Samplg For Proportos ad Percetages I may stuatos, the characterstc uder study o whch the observatos are collected are qualtatve ature For example, the resposes of customers may marketg surveys
More information18.413: Error Correcting Codes Lab March 2, Lecture 8
18.413: Error Correctg Codes Lab March 2, 2004 Lecturer: Dael A. Spelma Lecture 8 8.1 Vector Spaces A set C {0, 1} s a vector space f for x all C ad y C, x + y C, where we take addto to be compoet wse
More informationClass 13,14 June 17, 19, 2015
Class 3,4 Jue 7, 9, 05 Pla for Class3,4:. Samplg dstrbuto of sample mea. The Cetral Lmt Theorem (CLT). Cofdece terval for ukow mea.. Samplg Dstrbuto for Sample mea. Methods used are based o CLT ( Cetral
More informationCHAPTER 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 informationTHE ROYAL STATISTICAL SOCIETY 2016 EXAMINATIONS SOLUTIONS HIGHER CERTIFICATE MODULE 5
THE ROYAL STATISTICAL SOCIETY 06 EAMINATIONS SOLUTIONS HIGHER CERTIFICATE MODULE 5 The Socety s provdg these solutos to assst cadtes preparg for the examatos 07. The solutos are teded as learg ads ad should
More informationCS286.2 Lecture 4: Dinur s Proof of the PCP Theorem
CS86. Lecture 4: Dur s Proof of the PCP Theorem Scrbe: Thom Bohdaowcz Prevously, we have prove a weak verso of the PCP theorem: NP PCP 1,1/ (r = poly, q = O(1)). Wth ths result we have the desred costat
More informationSimulation 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 informationhp 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 informationRademacher Complexity. Examples
Algorthmc Foudatos of Learg Lecture 3 Rademacher Complexty. Examples Lecturer: Patrck Rebesch Verso: October 16th 018 3.1 Itroducto I the last lecture we troduced the oto of Rademacher complexty ad showed
More information5 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 informationChapter -2 Simple Random Sampling
Chapter - Smple Radom Samplg Smple radom samplg (SRS) s a method of selecto of a sample comprsg of umber of samplg uts out of the populato havg umber of samplg uts such that every samplg ut has a equal
More informationD KL (P Q) := p i ln p i q i
Cheroff-Bouds 1 The Geeral Boud Let P 1,, m ) ad Q q 1,, q m ) be two dstrbutos o m elemets, e,, q 0, for 1,, m, ad m 1 m 1 q 1 The Kullback-Lebler dvergece or relatve etroy of P ad Q s defed as m D KL
More informationArithmetic Mean Suppose there is only a finite number N of items in the system of interest. Then the population arithmetic mean is
Topc : Probablty Theory Module : Descrptve Statstcs Measures of Locato Descrptve statstcs are measures of locato ad shape that perta to probablty dstrbutos The prmary measures of locato are the arthmetc
More informationChapter 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 informationSTATISTICAL INFERENCE
(STATISTICS) STATISTICAL INFERENCE COMPLEMENTARY COURSE B.Sc. MATHEMATICS III SEMESTER ( Admsso) UNIVERSITY OF CALICUT SCHOOL OF DISTANCE EDUCATION CALICUT UNIVERSITY P.O., MALAPPURAM, KERALA, INDIA -
More informationChapter -2 Simple Random Sampling
Chapter - Smple Radom Samplg Smple radom samplg (SRS) s a method of selecto of a sample comprsg of umber of samplg uts out of the populato havg umber of samplg uts such that every samplg ut has a equal
More information18.657: Mathematics of Machine Learning
8.657: Mathematcs of Mache Learg Lecturer: Phlppe Rgollet Lecture 3 Scrbe: James Hrst Sep. 6, 205.5 Learg wth a fte dctoary Recall from the ed of last lecture our setup: We are workg wth a fte dctoary
More informationInvestigation 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 informationDimensionality 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 informationC-1: Aerodynamics of Airfoils 1 C-2: Aerodynamics of Airfoils 2 C-3: Panel Methods C-4: Thin Airfoil Theory
ROAD MAP... AE301 Aerodyamcs I UNIT C: 2-D Arfols C-1: Aerodyamcs of Arfols 1 C-2: Aerodyamcs of Arfols 2 C-3: Pael Methods C-4: Th Arfol Theory AE301 Aerodyamcs I Ut C-3: Lst of Subects Problem Solutos?
More informationM2S1 - EXERCISES 8: SOLUTIONS
MS - EXERCISES 8: SOLUTIONS. As X,..., X P ossoλ, a gve that T ˉX, the usg elemetary propertes of expectatos, we have E ft [T E fx [X λ λ, so that T s a ubase estmator of λ. T X X X Furthermore X X X From
More informationLecture 4 Sep 9, 2015
CS 388R: Radomzed Algorthms Fall 205 Prof. Erc Prce Lecture 4 Sep 9, 205 Scrbe: Xagru Huag & Chad Voegele Overvew I prevous lectures, we troduced some basc probablty, the Cheroff boud, the coupo collector
More informationECONOMETRIC THEORY. MODULE VIII Lecture - 26 Heteroskedasticity
ECONOMETRIC THEORY MODULE VIII Lecture - 6 Heteroskedastcty Dr. Shalabh Departmet of Mathematcs ad Statstcs Ida Isttute of Techology Kapur . Breusch Paga test Ths test ca be appled whe the replcated data
More informationå 1 13 Practice Final Examination Solutions - = CS109 Dec 5, 2018
Chrs Pech Fal Practce CS09 Dec 5, 08 Practce Fal Examato Solutos. Aswer: 4/5 8/7. There are multle ways to obta ths aswer; here are two: The frst commo method s to sum over all ossbltes for the rak of
More informationENGI 3423 Simple Linear Regression Page 12-01
ENGI 343 mple Lear Regresso Page - mple Lear Regresso ometmes a expermet s set up where the expermeter has cotrol over the values of oe or more varables X ad measures the resultg values of aother varable
More informationENGI 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 informationLikelihood Ratio, Wald, and Lagrange Multiplier (Score) Tests. Soccer Goals in European Premier Leagues
Lkelhood Rato, Wald, ad Lagrage Multpler (Score) Tests Soccer Goals Europea Premer Leagues - 4 Statstcal Testg Prcples Goal: Test a Hpothess cocerg parameter value(s) a larger populato (or ature), based
More informationEcon 388 R. Butler 2016 rev Lecture 5 Multivariate 2 I. Partitioned Regression and Partial Regression Table 1: Projections everywhere
Eco 388 R. Butler 06 rev Lecture 5 Multvarate I. Parttoed Regresso ad Partal Regresso Table : Projectos everywhere P = ( ) ad M = I ( ) ad s a vector of oes assocated wth the costat term Sample Model Regresso
More informationChapter 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 informationDr. Shalabh Department of Mathematics and Statistics Indian Institute of Technology Kanpur
Aalyss of Varace ad Desg of Exermets-I MODULE II LECTURE - GENERAL LINEAR HYPOTHESIS AND ANALYSIS OF VARIANCE Dr Shalabh Deartmet of Mathematcs ad Statstcs Ida Isttute of Techology Kaur Tukey s rocedure
More informationSummary 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 informationLINEAR REGRESSION ANALYSIS
LINEAR REGRESSION ANALYSIS MODULE V Lecture - Correctg Model Iadequaces Through Trasformato ad Weghtg Dr. Shalabh Departmet of Mathematcs ad Statstcs Ida Isttute of Techology Kapur Aalytcal methods for
More informationECON 482 / WH Hong The Simple Regression Model 1. Definition of the Simple Regression Model
ECON 48 / WH Hog The Smple Regresso Model. Defto of the Smple Regresso Model Smple Regresso Model Expla varable y terms of varable x y = β + β x+ u y : depedet varable, explaed varable, respose varable,
More informationIntroduction 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 informationLECTURE - 4 SIMPLE RANDOM SAMPLING DR. SHALABH DEPARTMENT OF MATHEMATICS AND STATISTICS INDIAN INSTITUTE OF TECHNOLOGY KANPUR
amplg Theory MODULE II LECTURE - 4 IMPLE RADOM AMPLIG DR. HALABH DEPARTMET OF MATHEMATIC AD TATITIC IDIA ITITUTE OF TECHOLOGY KAPUR Estmato of populato mea ad populato varace Oe of the ma objectves after
More informationLecture Notes to Rice Chapter 5
ECON 430 Revsed Sept. 06 Lecture Notes to Rce Chapter 5 By H. Goldste. Chapter 5 gves a troducto to probablstc approxmato methods, but s suffcet for the eeds of a adequate study of ecoometrcs. The commo
More informationarxiv:math/ v1 [math.gm] 8 Dec 2005
arxv:math/05272v [math.gm] 8 Dec 2005 A GENERALIZATION OF AN INEQUALITY FROM IMO 2005 NIKOLAI NIKOLOV The preset paper was spred by the thrd problem from the IMO 2005. A specal award was gve to Yure Boreko
More informationWu-Hausman Test: But if X and ε are independent, βˆ. ECON 324 Page 1
Wu-Hausma Test: Detectg Falure of E( ε X ) Caot drectly test ths assumpto because lack ubased estmator of ε ad the OLS resduals wll be orthogoal to X, by costructo as ca be see from the momet codto X'
More informationEconometrics. 3) Statistical properties of the OLS estimator
30C0000 Ecoometrcs 3) Statstcal propertes of the OLS estmator Tmo Kuosmae Professor, Ph.D. http://omepre.et/dex.php/tmokuosmae Today s topcs Whch assumptos are eeded for OLS to work? Statstcal propertes
More informationPTAS 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 informationIdea is to sample from a different distribution that picks points in important regions of the sample space. Want ( ) ( ) ( ) E f X = f x g x dx
Importace Samplg Used for a umber of purposes: Varace reducto Allows for dffcult dstrbutos to be sampled from. Sestvty aalyss Reusg samples to reduce computatoal burde. Idea s to sample from a dfferet
More informationL(θ X) s 0 (1 θ 0) m s. (s/m) s (1 s/m) m s
Hw 4 (due March ) 83 The LRT statstcs s λ(x) sup θ θ 0 L(θ X) The lkelhood s L(θ) θ P x ( sup Θ L(θ X) θ) m P x ad ad the log-lkelhood s (θ) x log θ +(m x ) log( θ) Let S X Note that the ucostraed MLE
More informationENGI 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 informationChapter 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 informationMultivariate 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 informationChapter 8. Inferences about More Than Two Population Central Values
Chapter 8. Ifereces about More Tha Two Populato Cetral Values Case tudy: Effect of Tmg of the Treatmet of Port-We tas wth Lasers ) To vestgate whether treatmet at a youg age would yeld better results tha
More informationLecture 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 informationChapter 10 Two Stage Sampling (Subsampling)
Chapter 0 To tage amplg (usamplg) I cluster samplg, all the elemets the selected clusters are surveyed oreover, the effcecy cluster samplg depeds o sze of the cluster As the sze creases, the effcecy decreases
More informationIntroduction to Matrices and Matrix Approach to Simple Linear Regression
Itroducto to Matrces ad Matrx Approach to Smple Lear Regresso Matrces Defto: A matrx s a rectagular array of umbers or symbolc elemets I may applcatos, the rows of a matrx wll represet dvduals cases (people,
More informationChapter 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 informationBayes (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 informationMA 524 Homework 6 Solutions
MA 524 Homework 6 Solutos. Sce S(, s the umber of ways to partto [] to k oempty blocks, ad c(, s the umber of ways to partto to k oempty blocks ad also the arrage each block to a cycle, we must have S(,
More informationCIS 800/002 The Algorithmic Foundations of Data Privacy October 13, Lecture 9. Database Update Algorithms: Multiplicative Weights
CIS 800/002 The Algorthmc Foudatos of Data Prvacy October 13, 2011 Lecturer: Aaro Roth Lecture 9 Scrbe: Aaro Roth Database Update Algorthms: Multplcatve Weghts We ll recall aga) some deftos from last tme:
More informationLecture 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 informationNaïve Bayes MIT Course Notes Cynthia Rudin
Thaks to Şeyda Ertek Credt: Ng, Mtchell Naïve Bayes MIT 5.097 Course Notes Cytha Rud The Naïve Bayes algorthm comes from a geeratve model. There s a mportat dstcto betwee geeratve ad dscrmatve models.
More informationArithmetic 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 informationChain Rules for Entropy
Cha Rules for Etroy The etroy of a collecto of radom varables s the sum of codtoal etroes. Theorem: Let be radom varables havg the mass robablty x x.x. The...... The roof s obtaed by reeatg the alcato
More informationPermutation Tests for More Than Two Samples
Permutato Tests for ore Tha Two Samples Ferry Butar Butar, Ph.D. Abstract A F statstc s a classcal test for the aalyss of varace where the uderlyg dstrbuto s a ormal. For uspecfed dstrbutos, the permutato
More informationCOV. Violation of constant variance of ε i s but they are still independent. The error term (ε) is said to be heteroscedastic.
c Pogsa Porchawseskul, Faculty of Ecoomcs, Chulalogkor Uversty olato of costat varace of s but they are stll depedet. C,, he error term s sad to be heteroscedastc. c Pogsa Porchawseskul, Faculty of Ecoomcs,
More informationi 2 σ ) i = 1,2,...,n , and = 3.01 = 4.01
ECO 745, Homework 6 Le Cabrera. Assume that the followg data come from the lear model: ε ε ~ N, σ,,..., -6. -.5 7. 6.9 -. -. -.9. -..6.4.. -.6 -.7.7 Fd the mamum lkelhood estmates of,, ad σ ε s.6. 4. ε
More informationTHE ROYAL STATISTICAL SOCIETY 2010 EXAMINATIONS SOLUTIONS GRADUATE DIPLOMA MODULE 2 STATISTICAL INFERENCE
THE ROYAL STATISTICAL SOCIETY 00 EXAMINATIONS SOLUTIONS GRADUATE DIPLOMA MODULE STATISTICAL INFERENCE The Socety provdes these solutos to assst caddates preparg for the examatos future years ad for the
More informationMean is only appropriate for interval or ratio scales, not ordinal or nominal.
Mea Same as ordary average Sum all the data values ad dvde by the sample sze. x = ( x + x +... + x Usg summato otato, we wrte ths as x = x = x = = ) x Mea s oly approprate for terval or rato scales, ot
More informationSolutions to Odd-Numbered End-of-Chapter Exercises: Chapter 17
Itroucto to Ecoometrcs (3 r Upate Eto) by James H. Stock a Mark W. Watso Solutos to O-Numbere E-of-Chapter Exercses: Chapter 7 (Ths erso August 7, 04) 05 Pearso Eucato, Ic. Stock/Watso - Itroucto to Ecoometrcs
More informationMidterm Exam 1, section 2 (Solution) Thursday, February hour, 15 minutes
coometrcs, CON Sa Fracsco State Uverst Mchael Bar Sprg 5 Mdterm xam, secto Soluto Thursda, Februar 6 hour, 5 mutes Name: Istructos. Ths s closed book, closed otes exam.. No calculators of a kd are allowed..
More information