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1 E ffi c i e n t a n d f a s t s p l i n e- b a c k fi t t e d k e r n e l s m o o t h i n g o f a d d i t i v e r e g r e s s i o n m o d e l * L i j i a n Y a n g D e p a r t m e n t o f S t a t i s t i c s a n d P r o b a b i l i t y M i c h i g a n S t a t e U n i v e r s i t y E a s t L a n s i n g , M I 4 8 8 2 4 a n d D e p a r t m e n t o f S t a t i s t i c s a n d A p p l i e d P r o b a b i l i t y N a t i o n a l U n i v e r s i t y o f S i n g a p o r e S i n g a p o r e 1 1 7 5 4 6 * J o i n t w o r k w i t h P h . D . s t u d e n t J i n g W a n g , p a r t i a l l y s u p p o r t e d b y N S F g r a n t s D M S 4 5 3 3 , B C S 3 8 4 2 , S E S 1 2 7 7 2 2 . S p l i n e- b a c k fi t t e d k e r n e l s m o o t h i n g o f a d d i t i v e r e g r e s s i o n m o d e l , M a y 1 1 , 2 5 d = 1 , n = 2 , e f f i c i e n c y = .8 4 4 1 .5 X Y 2 O u t l i n e N o n- a n d s e m i p a r a m e t r i c r e g r e s s i o n : u s e f u l , b u t i n t i m i d a t i n g S p l i n e- b a c k fi t t e d k e r n e l s m o o t h i n g o f a d d i t i v e r e g r e s s i o n m o d e l , M a y 1 1 , 2 5 d = 1 , n = 2 , e f f i c i e n c y = .8 4 4 1 .5 X Y 3 O u t l i n e N o n- a n d s e m i p a r a m e t r i c r e g r e s s i o n : u s e f u l , b u t i n t i m i d a t i n g C r i t i c i s m # 1 : c u r s e o f d i m e n s i o n a l i t y S p l i n e- b a c k fi t t e d k e r n e l s m o o t h i n g o f a d d i t i v e r e g r e s s i o n m o d e l , M a y 1 1 , 2 5 d = 1 , n = 2 , e f f i c i e n c y = .8 4 4 1 .5 X Y 4 O u t l i n e N o n- a n d s e m i p a r a m e t r i c r e g r e s s i o n : u s e f u l , b u t i n t i m i d a t i n g C r i t i c i s m # 1 : c u r s e o f d i m e n s i o n a l i t y C r i t i c i s m # 2 : c o m p u t i n g b u r d e n u n b e a r a b l e S p l i n e- b a c k fi t t e d k e r n e l s m o o t h i n g o f a d d i t i v e r e g r e s s i o n m o d e l , M a y 1 1 , 2 5 d = 1 , n = 2 , e f f i c i e n c y = .8 4 4 1 .5 X Y 5 O u t l i n e N o n- a n d s e m i p a r a m e t r i c r e g r e s s i o n : u s e f u l , b u t i n t i m i d a t i n g C r i t i c i s m # 1 : c u r s e o f d i m e n s i o n a l i t y C r i t i c i s m # 2 : c o m p u t i n g b u r d e n u n b e a r a b l e C r i t i c i s m # 3 : c o n fi d e n c e b a n d u n a v a i l a b l e S p l i n e- b a c k fi t t e d k e r n e l s m o o t h i n g o f a d d i t i v e r e g r e s s i o n m o d e l , M a y 1 1 , 2 5 d = 1 , n = 2 , e f f i c i e n c y = .8 4 4 1 .5 X Y 6 O u t l i n e N o n- a n d s e m i p a r a m e t r i c r e g r e s s i o n : u s e f u l , b u t i n t i m i d a t i n g C r i t i c i s m # 1 : c u r s e o f d i m e n s i o n a l i t y C r i t i c i s m # 2 : c o m p u t i n g b u r d e n u n b e a r a b l e C r i t i c i s m # 3 : c o n fi d e n c e b a n d u n a v a i l a b l e C r i t i c i s m # 4 : i n t u i t i o n l a c k i n g S p

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N.C. State - BUS - 426

i) maintenance of national policy autonomy. b. If exchange rates are fluctuating randomly, that may discourage international trade and encourage market segmentation. This, in turn, may lead to suboptimal allocation of resources. c. Economic agents ca

University of Texas - BIO 373 - 373

BIO 373 Oct. 16th Outline Population spatial structuremeasuring population sizemeasuring dispersal ratesspatial structure at the largest scales Population dynamicsdiscrete population growth, e.g. geometric growthcontinuous population growth, e.g. e

Rochester - PHL - 110

. If we succeed, we will have a casein which the argument's premises, lines 1 and 2, a re true. And this samecase will be one in which the conclusion is false, because the negation ofthe conclusion, line 3, will be true. So if we can make lines 1,

Kentucky - HIS - 462

etts Bay Colonya. Hundreds were killed and captured and sold into slaveryb. The Pequots were eliminated from Southern New Englandc. War resulted from escalating incidents regarding the fur trade32. Glorious Revolution: Overthrow of King James I

Michigan State University - STATISTICS - 5207

1ST5207 Nonparametric Regression, Lecture10Lijian YangDepartment of Statistics & ProbabilityMichigan State UniversityEast Lansing, MI 48824andDepartment of Statistics & Applied ProbabilityNational University of SingaporeSingapore 117546ST5207 N

Michigan State University - STATISTICS - 5207

1ST5207 Nonparametric Regression, Lecture 9Lijian YangDepartment of Statistics & ProbabilityMichigan State UniversityEast Lansing, MI 48824andDepartment of Statistics & Applied ProbabilityNational University of SingaporeSingapore 117546ST5207 No

Michigan State University - STATISTICS - 5207

1ST5207 Nonparametric Regression, Lecture 8Lijian YangDepartment of Statistics & ProbabilityMichigan State UniversityEast Lansing, MI 48824andDepartment of Statistics & Applied ProbabilityNational University of SingaporeSingapore 117546ST5207 No

Michigan State University - STATISTICS - 5207

1ST5207 Nonparametric Regression, Lecture 7Lijian YangDepartment of Statistics & ProbabilityMichigan State UniversityEast Lansing, MI 48824andDepartment of Statistics & Applied ProbabilityNational University of SingaporeSingapore 117546ST5207 No

Michigan State University - STATISTICS - 5207

1ST5207 Nonparametric Regression, Lecture 6Lijian YangDepartment of Statistics & ProbabilityMichigan State UniversityEast Lansing, MI 48824andDepartment of Statistics & Applied ProbabilityNational University of SingaporeSingapore 117546ST5207 No

Michigan State University - STATISTICS - 5207

1ST5207 Nonparametric Regression, Lecture 5Lijian YangDepartment of Statistics & ProbabilityMichigan State UniversityEast Lansing, MI 48824andDepartment of Statistics & Applied ProbabilityNational University of SingaporeSingapore 117546ST5207 No

Michigan State University - STATISTICS - 5207

1ST5207 Nonparametric Regression, Lecture 4Lijian YangDepartment of Statistics & ProbabilityMichigan State UniversityEast Lansing, MI 48824andDepartment of Statistics & Applied ProbabilityNational University of SingaporeSingapore 117546ST5207 No

Michigan State University - STATISTICS - 5207

1ST5207 Nonparametric Regression, Lecture 3Lijian YangDepartment of Statistics & ProbabilityMichigan State UniversityEast Lansing, MI 48824andDepartment of Statistics & Applied ProbabilityNational University of SingaporeSingapore 117546ST5207 No

Michigan State University - STATISTICS - 5207

1ST5207 Nonparametric Regression, Lecture 2Lijian YangDepartment of Statistics & ProbabilityMichigan State UniversityEast Lansing, MI 48824andDepartment of Statistics & Applied ProbabilityNational University of SingaporeSingapore 117546ST5207 No

Michigan State University - STATISTICS - 5207

1ST5207 Nonparametric Regression, Lecture 1Lijian YangDepartment of Statistics & ProbabilityMichigan State UniversityEast Lansing, MI 48824andDepartment of Statistics & Applied ProbabilityNational University of SingaporeSingapore 117546ST5207 No

Michigan State University - STATISTICS - 455

Course Overview and IntroductionLecture: Week 1Lecture: Week 1 (STT 455)Course Overview and IntroductionFall 2013 - Valdez1/9About the coursecourse instructorCourse instructorEmil ValdezOce: C337 Wells HallTelephone: 517-353-6332e-mail: valdez

Michigan State University - STATISTICS - 455

Life Tables and SelectionLecture: Weeks 4-5Lecture: Weeks 4-5 (STT 455)Life Tables and SelectionFall 2013 - Valdez1 / 28Chapter summaryChapter summaryWhat is a life table?also called a mortality tabletabulation of basic mortality functionsderiv

Michigan State University - STATISTICS - 455

Insurance BenetsLecture: Weeks 6-8Lecture: Weeks 6-8 (STT 455)Insurance BenetsFall 2013 - Valdez1 / 36An introductionAn introductionCentral theme: to quantify the value today of a (random) amount tobe paid at a random time in the future.main app

Michigan State University - STATISTICS - 455

Survival ModelsLecture: Weeks 2-3Lecture: Weeks 2-3 (STT 455)Survival ModelsFall 2013 - Valdez1 / 27Chapter summaryChapter summarySurvival modelsAge-at-death random variableTime-until-death random variablesForce of mortality (or hazard rate fun

Michigan State University - STATISTICS - 455

Suggested solutions to DHW textbook exercisesExercise 3.9(a) Let the constant force between ages [x + k, x + k + 1] be denoted by +k so thatxpx+k = ex+k ,from which it follows that +k = log px+k . Therefore, we havexPr[Rx s, Kx = k ]Pr[k < Tx k +

Michigan State University - STATISTICS - 455

Suggested solutions to DHW textbook exercisesExercise 3.8(a) Starting with p =xx+1 / x ,we note thatxBecause25= 98363 =26 ,24Starting with px]+2 =[x+1px==22x+3 / [x]+2 ,x+3px]+2[[x]+2=x+2 .[21]+2[x]+2 / [x]+1 ,=[20]+1=19Fi

Michigan State University - STATISTICS - 455

Suggested solutions to DHW textbook exercisesExercise 3.5(a) We have7p[70]= p[70] p[70]+1 p[70]+2 p[70]+3 p[70]+4 p75 p76= (1 q[70] )(1 q[70]+1 )(1 q[70]+2 )(1 q[70]+3 )(1 q[70]+4 )(1 q75 )(1 q76 )= (1 0.010373)(1 0.014330)(1 0.019192)(1 0.025023)(1

Michigan State University - STATISTICS - 455

Suggested solutions to DHW textbook exercisesExercise 3.2Whenx sare given, it is better to use the direct linear interpolation formula for UDD= (1 t)x+tx+tx+1and the direct exponential interpolation formula for constant forcex+t=1txtx+1 .

Michigan State University - STATISTICS - 455

Suggested solutions to DHW textbook exercisesExercise 3.1The gures below are based on the US Life Table, 2004 prepared by the Center for DiseaseControl and Prevention (CDC). The table typies pattern of human population mortality.0.10.010.00110090

Michigan State University - STATISTICS - 455

Suggested solutions to DHW textbook exercisesExercise 2.15(a) We know thatspx dsx =e=00S0 (x + s)1ds =S0 (x)S0 (x)S0 (x + s)ds.0Using a change of variable of integration t = x + s, we nd thatx =e1S0 (x)S0 (x + t)dt =01S0 (x)S0 (t)

Michigan State University - STATISTICS - 455

Suggested solutions to DHW textbook exercisesExercise 2.14(a) Starting withx =e1tpx dt =00 1+tpx dt +px t1px+1 dttpx dt = 1 +11 1+tpx dt1t1px+1 dtspx+1 ds=1+01= 1 + x+1eThe inequalities hold because we know that tpx 1 for all x an

Michigan State University - STATISTICS - 455

Suggested solutions to DHW textbook exercisesExercise 2.13(a) We are given = 2x where refers to smokers and unstarred, non-smokers. It is easyxto verify thatttpxt +s dsx= exp = exp 202tx+s ds= exp 0x+s ds= ( tpx )2 .0Note that because

Michigan State University - STATISTICS - 455

Suggested solutions to DHW textbook exercisesExercise 2.12(a) For Makehams law, it can easily be veried thatpx = exp A +Bxc (c 1)log(c).The following R code produces a table of px for x = 0 to x = 130:A <- .0001B <- .00035c <- 1.075px <- funct

Michigan State University - STATISTICS - 455

Suggested solutions to DHW textbook exercisesExercise 2.11(a) It is not dicult to show that under Makehams law, we havex(A + Bcz )dzS0 (x) = exp= exp Ax +0B(cx 1)log(c).It follows therefore thattpx= Sx (t) =S0 (x + t)S0 (x)exp A(x + t) +

Michigan State University - STATISTICS - 455

Suggested solutions to DHW textbook exercisesExercise 2.5Clearly, F0 (t) is the cdf of an Exponential with mean 1/. So T0 has an Exponential distribution.(a) Since S0 (t) = et , we haveSx (t) =e(x+t)S0 (x + t)== et .S0 (x)exThus, we see that Tx

Michigan State University - STATISTICS - 455

Suggested solutions to DHW textbook exercisesExercise 2.4(a) To show S0 is a legitimate survival function, we show 3 conditions:(i) S0 (0) = 1: trivial(ii) lim S0 (x) = 0: Since all parameters A, B , C and D are all positive, then the termxAx + 1 Bx

Michigan State University - STATISTICS - 455

Suggested solutions to DHW textbook exercisesExercise 2.3We are given(100 x)1/21100 x =, for 0 x 100.1010The probability that a newborn will die between ages 19 and 36 is given byS0 (x) =19|17 q0= Pr[19 < T0 36] = S0 (19) S0 (36)=Prepared by

Michigan State University - STATISTICS - 455

Suggested solutions to DHW textbook exercisesExercise 2.2(a) The implied limiting age is the solution to G( ) = 0 which leads us to18000 110 2 = ( 90)( + 200) = 0.Thus, = 90 since the limiting age cannot be negative.(b) For G to be a legitimate survi

Michigan State University - STATISTICS - 455

Suggested solutions to DHW textbook exercisesExercise 2.1(a) The probability that a newborn life dies before age 60 is given byPr[T0 60] = F0 (60) = 1 (1 60/105)1/5 = 1 (45/105)1/5 = 1 (3/7)1/5 = 0.1558791.(b) The probability that (30) survives to at

Michigan State University - STATISTICS - 455

Michigan State University - STATISTICS - 455

Michigan State UniversitySTT 455 - Actuarial Models IClass Test 1Monday, 7 October 2013Total Marks: 100 pointsPlease write your name and student number at the spaces provided:Name:Section No.: There are ve (5) multiple choice (MC) and one (1) writ

Michigan State University - STATISTICS - 455

Michigan State University - STATISTICS - 455

Michigan State University - STATISTICS - 455

Michigan State University - STATISTICS - 455

South Forsyth High School - MATH - Math 1

South Forsyth High School - MATH - Math 1

South Forsyth High School - MATH - Math 1

South Forsyth High School - CHEM - Chemistry

South Forsyth High School - CHEM - Chemistry

South Forsyth High School - CHEM - Chemistry

South Forsyth High School - CHEM - Chemistry

South Forsyth High School - CHEM - Chemistry

South Forsyth High School - CHEM - Chemistry

South Forsyth High School - CHEM - Chemistry

South Forsyth High School - CHEM - Chemistry

South Forsyth High School - CHEM - Chemistry

South Forsyth High School - CHEM - Chemistry

South Forsyth High School - CHEM - Chemistry

South Forsyth High School - CHEM - Chemistry

South Forsyth High School - CHEM - Chemistry

South Forsyth High School - CHEM - Chemistry

South Forsyth High School - CHEM - Chemistry

South Forsyth High School - CHEM - Chemistry

South Forsyth High School - CHEM - Chemistry

South Forsyth High School - CHEM - Chemistry

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