33 Pages

# Lecture #9 Notes

Course: STATISTICS 5207, Fall 2012

School: Michigan State University

Word Count: 2727

Rating:

###### Document Preview

1 ST5207 Nonparametric Regression, Lecture 9 Lijian Yang Department of Statistics & Probability Michigan State University East Lansing, MI 48824 and Department of Statistics & Applied Probability National University of Singapore Singapore 117546 ST5207 Nonparametric Regression, 24th March 2005 2 Spline estimation mp (x) = mp (x) + p (x) with both terms as {b1p,p (x) , ..., bN,p (x)} {b1p,p (x) ,...

##### Unformatted Document Excerpt
Coursehero >> Michigan >> Michigan State University >> STATISTICS 5207

Course Hero has millions of student submitted documents similar to the one
below including study guides, practice problems, reference materials, practice exams, textbook help and tutor support.

Course Hero has millions of student submitted documents similar to the one below including study guides, practice problems, reference materials, practice exams, textbook help and tutor support.

Find millions of documents on Course Hero - Study Guides, Lecture Notes, Reference Materials, Practice Exams and more. Course Hero has millions of course specific materials providing students with the best way to expand their education.

Below is a small sample set of documents:

York University - MATH - 1090
in (¬ A) iff it occurs in A– p occurs in (A o B), o∈ {∧ , ∨ , →, ≡}, iff it occurs in A or Bor both.York University- MATH 109003-TruthTables7Tautology/ Satisfiable/ Contradiction• Definition. A formula A is a tautology iff
Cornell - AEP - 4900
110583905898674E-9i -4.020390836023968E-7+8.878417633805086E-8i1.4923608651122233E-4 2.0391788632774697E-4 -2.332212369880123E-4 5.968423716840449E-7-1.5557044315353457E-7i 4.299790313936439E-7-1.08141682664607E-7i 1.3
Michigan State University - STATISTICS - 5207
1ST5207 Nonparametric Regression, Lecture 8Lijian YangDepartment of Statistics &amp; ProbabilityMichigan State UniversityEast Lansing, MI 48824andDepartment of Statistics &amp; Applied ProbabilityNational University of SingaporeSingapore 117546ST5207 No
Michigan State University - STATISTICS - 5207
1ST5207 Nonparametric Regression, Lecture 7Lijian YangDepartment of Statistics &amp; ProbabilityMichigan State UniversityEast Lansing, MI 48824andDepartment of Statistics &amp; Applied ProbabilityNational University of SingaporeSingapore 117546ST5207 No
Michigan State University - STATISTICS - 5207
1ST5207 Nonparametric Regression, Lecture 6Lijian YangDepartment of Statistics &amp; ProbabilityMichigan State UniversityEast Lansing, MI 48824andDepartment of Statistics &amp; Applied ProbabilityNational University of SingaporeSingapore 117546ST5207 No
Michigan State University - STATISTICS - 5207
1ST5207 Nonparametric Regression, Lecture 5Lijian YangDepartment of Statistics &amp; ProbabilityMichigan State UniversityEast Lansing, MI 48824andDepartment of Statistics &amp; Applied ProbabilityNational University of SingaporeSingapore 117546ST5207 No
Michigan State University - STATISTICS - 5207
1ST5207 Nonparametric Regression, Lecture 4Lijian YangDepartment of Statistics &amp; ProbabilityMichigan State UniversityEast Lansing, MI 48824andDepartment of Statistics &amp; Applied ProbabilityNational University of SingaporeSingapore 117546ST5207 No
Michigan State University - STATISTICS - 5207
1ST5207 Nonparametric Regression, Lecture 3Lijian YangDepartment of Statistics &amp; ProbabilityMichigan State UniversityEast Lansing, MI 48824andDepartment of Statistics &amp; Applied ProbabilityNational University of SingaporeSingapore 117546ST5207 No
Michigan State University - STATISTICS - 5207
1ST5207 Nonparametric Regression, Lecture 2Lijian YangDepartment of Statistics &amp; ProbabilityMichigan State UniversityEast Lansing, MI 48824andDepartment of Statistics &amp; Applied ProbabilityNational University of SingaporeSingapore 117546ST5207 No
Michigan State University - STATISTICS - 5207
1ST5207 Nonparametric Regression, Lecture 1Lijian YangDepartment of Statistics &amp; ProbabilityMichigan State UniversityEast Lansing, MI 48824andDepartment of Statistics &amp; 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 &lt; 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 &lt;- .0001B &lt;- .00035c &lt;- 1.075px &lt;- 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 &lt; 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
South Forsyth High School - ENGLISH LA - American L
South Forsyth High School - ENGLISH LA - American L
South Forsyth High School - ENGLISH LA - American L
South Forsyth High School - ENGLISH LA - American L
South Forsyth High School - ENGLISH LA - American L