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Berkeley - ST 522
  • 5 Pages Lab 1 Solution
    Lab 1 Solution

    School: N.C. State

    Course: Statistical Theory II

  • Pages hw2solution
    Hw2solution

    School: N.C. State

    Course: Statistical Theory II

  • 1 Page ST522-HW6
    ST522-HW6

    School: N.C. State

    Course: Statistical Theory II

    ST 522-002: Statistical Theory II Homework 6, Due March 17, 2014 Spring 2014 1. 10.4 2. 10.10 3. 8.14 4. 8.15 5. (based on the Basic Exam 01/2014) Suppose that X1 , , Xn are i.i.d. random variables following the U nif orm(1 , 2 ) distribution. (a) Find th

  • 5 Pages ST522-2014-Mid
    ST522-2014-Mid

    School: N.C. State

    Course: Statistical Theory II

    Midterm Exam, Spring 2014, Statistics 522-002 Feb. 26, Wednesday 9:00am - 11:15am Your Name (Print): Note: 1. The exam is closed-book and closed-notes. 2. There are totally FOUR problems. 3. Show all your work in the space provided. If you need additional

  • 3 Pages Homework 9 Solution
    Homework 9 Solution

    School: N.C. State

    Course: Statistical Theory II

    SOLUTIONS FOR HOMEWORK 9

  • 7 Pages Homework 8 Solution
    Homework 8 Solution

    School: N.C. State

    Course: Statistical Theory II

    SOLUTIONS FOR HOMEWORK 8 c' . ,jo a gaill. e'" = ;r; a uo - I . - 1'L(" 1'1 et - 11J:~' ,.=1 i n t he u nrest-rietecl ('a.s(). I n t h(' f e,st.rided rr " ,') ' 'i - -:rj C ilSC) -,.2 J,.1 , _ ( J - -=r; :I;"' s et (/, = 1 i n t he n '1: + _ ~ix. , _ fj

  • 3 Pages Homework 5 Solution
    Homework 5 Solution

    School: N.C. State

    Course: Statistical Theory II

    SOLUTIONS FOR HOMEWORK 5 c. Under normality, d. To obtain the result, use part (c) with

  • 3 Pages Homework 6 Solution
    Homework 6 Solution

    School: N.C. State

    Course: Statistical Theory II

    SOLUTIONS FOR HOMEWORK 6

  • 5 Pages Final Exam Review Solution
    Final Exam Review Solution

    School: N.C. State

    Course: Statistical Theory II

    ST 522 FINAL EXAM REVIEW SOLUTIONS and 2 Pr(0 < nT < na ) + Pr(nT >nb) = , where nT n

  • 3 Pages HW10_sol
    HW10_sol

    School: N.C. State

    Course: STATISTICAL THEORY II

    ST 522-002: Statistical Theory II Solution to Homework Assignment - 10 Prepared by Chen-Yen Lin Spring, 2012 8.11 (a) From Exercise 7.23, the posterior distribution of 2 |s2 is inverted gamma + n1 2, (n1)s2 2 + 1 1 2 2 (s 1) 1 Denote (s2 ) = (n2 s + and W

  • 6 Pages HW11_sol
    HW11_sol

    School: N.C. State

    Course: STATISTICAL THEORY II

    ST 522-002: Statistical Theory II Solution to Homework Assignment - 11 Prepared by Chen-Yen Lin Spring, 2012 8.27 Let 2 > 1 , then consider the ratio g (t|2 ) c(2 ) = exp cfw_t[w(2 ) w(1 )] g (t|1 ) c(1 ) If w() is an increasing function, then w(2 ) w(1 )

  • 3 Pages HW09_sol
    HW09_sol

    School: N.C. State

    Course: STATISTICAL THEORY II

    ST522 Solutions to Homework #9 Prepared by Peng Yang April 11, 2012 8.5 (a) The log-likelihood is l(, |x) = n log + n log ( + 1)log (xi ) , x(1) This is increasing on , so both restricted and unrestricted MLEs of is = x(1) . Then, set l(, x(1) |x) = n/ +

  • 3 Pages HW08_sol
    HW08_sol

    School: N.C. State

    Course: STATISTICAL THEORY II

    ST 522-002: Statistical Theory II Solution to Homework Assignment - 8 Prepared by Chen-Yen Lin Spring, 2012 10.1 The Method of Moment Estimator (MOME) is = 3X , whose variance is given by the variance vanishes and therefore the MOME is a consistent estima

  • 2 Pages HW07_sol
    HW07_sol

    School: N.C. State

    Course: STATISTICAL THEORY II

    ST 522-002: Statistical Theory II Solution to Homework Assignment - 7 Prepared by Chen-Yen Lin Spring, 2012 7.38 (a) The log-likelihood function can be written as L(|x) = n log + ( 1) log-likelihood function w.r.t , we have log L(|x) n =+ i log xi . Dier

  • 1 Page ST522-HW5
    ST522-HW5

    School: N.C. State

    Course: Statistical Theory II

    ST 522-002: Statistical Theory II Homework 5, Due March 03, 2014 Spring 2014 1. 7.44 2. 7.55 3. 7.59 4. 7.60 5. 7.62 6. 10.1 1

  • 1 Page ST522-HW4
    ST522-HW4

    School: N.C. State

    Course: Statistical Theory II

    ST 522-002: Statistical Theory I Homework 4, Due February 12, 2014 Spring 2014 1. 7.19 2. 7.23 3. Let X1 , . . . , Xn i.i.d. U (, 1). Find the MLE and the MOME . Then compute their respective mean squared error. 4. 7.41 5. Let X1 , . . . , Xn i.i.d. Berno

  • 1 Page ST522-HW3
    ST522-HW3

    School: N.C. State

    Course: Statistical Theory II

    ST 522-002: Statistical Theory I Homework 3, Due February 05, 2014 Spring 2014 1. 6.40 2. 7.6 3. 7.9 4. Let X1 , . . . , Xn be a random sample from N (, 2 ). Find the maximum likelihood estimate of when x = 0 is observed. 5. (Based on the 01/2014 Basic Ex

  • 4 Pages Lab 4 Solution
    Lab 4 Solution

    School: N.C. State

    Course: Statistical Theory II

    Lt\b4 @y fu(.ft)rl-l-c.\+ictl To c.heC-1( -H11A+ 1-t i!. )1r/.-hue11+ "' J.hrn, /-e.i Slt4'tTLll?.n+ C.vn,pl-e.-+-e q Por- all B +na+ I) t'\"1-I I'\ bewuse. lJ iJfld- J_ I a.e it 1.s (o i-ollq I- [()-1 .lJ c.i Ecfw_ '( J cfw_) it, liI I I =:; 8 h.inlt-iOl

  • Pages Lab 2 Solution
    Lab 2 Solution

    School: N.C. State

    Course: Statistical Theory II

  • 5 Pages Lab 3 Solution
    Lab 3 Solution

    School: N.C. State

    Course: Statistical Theory II

    ST522 '~ .~J E ( X.J. J =- V-"rl Xl -t (t:IX) 1 ~ X.l (1 \~ 6.1 e ~-tlflllt+or- 111n bi.<.\Sed Gin J -:. of- t5 ~ -712 e. ~ tl 6 a .i.1c~L. - a 6 . "fhi,i~ I 6 -: _, - -t rv 64. :.x:z. J) 1(4" 6-3 - ?:.-112-0.,_ :. o" I 1L1 is Becaw~e Cc + 2i1 . -=- 0 "V

  • 14 Pages Chapter 5 Lecture Notes
    Chapter 5 Lecture Notes

    School: N.C. State

    Course: Statistical Theory II

    1 ST522-002 Statistical Theory IIStatistical Inference Spring 2014 Huixia Judy Wang hwang3@ncsu.edu Oce: 4270 SAS Hall Acknowledgement: part of the slides are modications of the handouts of Drs. Howard Bondell and Donald Martin. 2 Course Outline: Chapter

  • 63 Pages Chapter 6 Lecture Notes
    Chapter 6 Lecture Notes

    School: N.C. State

    Course: Statistical Theory II

    5 Order Statistics ( ) 5 Order Statistics ( ) 6 Principles of Data Reduction 6.1 Statistical Inference & Data Reduction Suppose data X = (X1 , . . . , Xn ) are from a probability distribution P , which is either completely or partially unknown, e.g. Poiss

  • 37 Pages Chapter 7 Lecture Notes
    Chapter 7 Lecture Notes

    School: N.C. State

    Course: Statistical Theory II

    5 Order Statistics (X) 5 Order Statistics (X) 6 Principles of Data Reduction (X) 7 Point Estimation 7.1 Basics Main goal: nd a point estimator (a function of the sample) to estimate either or some function of . Knowledge of the parameter yields knowledge

  • Pages midterm-solu
    Midterm-solu

    School: N.C. State

    Course: Statistical Theory II

  • 4 Pages hw4solution
    Hw4solution

    School: N.C. State

    Course: Statistical Theory II

    H vv4 I If c- I ".2:Ti t5 -v- lif +hm Ptlc-roi-T-&ct1t-iot1 rf I h /- (-f<At l2.Y;<-, z 7l;Yi) 1.s o. f '4f+rcre<1t 6 -') - 71t ' fi CZ.?/;-;i_ A-I> o J < is it E; 0 on " := 2 J t; E Y.- : Z?1i.2. l' ;>O (q vvT+h \ / t\ d meun ff" a nd p) :. :i a,.i. 11G1

  • 5 Pages hw5solution
    Hw5solution

    School: N.C. State

    Course: Statistical Theory II

    Hw5 ST - :i. X - \/'l 1;1" of. "\ .fvtflt+iOI" e \-\r11G\tOr of- 1.-1 n blt\Sed i't!Ae There-fw.e., thm ry,1,.i.3j JCpel-t-Mi'Ofl . .! .l I +& - E y & ( '( j 7=- '(31 t -e J Iy - )J G - To c.o.\cu \ 'lte we ' the Ee [ - q.21 r;a E Ht,lG-J J die-.). <-RLB

  • 2 Pages ST522-Practice-Midterm
    ST522-Practice-Midterm

    School: N.C. State

    Course: Statistical Theory II

    ST522 Practice Midterm Exam Note: the actual length of midterm exam will be dierent from the practice exam. 1. (a) Suppose X is a Binomial(n, p) random variable, 0 < p < 1. Find the MLE for p(1 p). Show that the MLE is not unbiased for p(1 p). Construct a

  • Pages practice midterm sol
    Practice Midterm Sol

    School: N.C. State

    Course: Statistical Theory II

  • Pages hw3solution
    Hw3solution

    School: N.C. State

    Course: Statistical Theory II

    STS liW3 22- x: I - -, '/. . . ' Io ( ~-t-1 on is " we can vvl-i +e T; IX, . - ~II I - .:. >X, T,( 6l,tM ) "Ind 'l/ ., +h I I +b) :=. +l,e SalhpJe T t.i~J Cl f7ltnJ - Jl11J) -:=. tcfw_ e -1- n-1 .z R. ( ~11, +b, e (JI, - ,'fJJ,.,+hJ-=- 7/q) (lo.1t;+b) - =

  • 6 Pages hw1solution
    Hw1solution

    School: N.C. State

    Course: Statistical Theory II

    \1 1j e So .1. . - .,., I .e, ,~ X- yn ., +or 1: l l t he. 0 r-e1Yl f h q I f) -=:. I e( ;f , = ,11; 7 l-,tl) I - ii" e h h II e S-r~+1s+1c. fuc+Otf-t-G\+io() f:.or- T ( J() = m i h ( ~i) i's (A,-6) . +he. or-em.) 14 (" .I (Jii -JAJ .I. ~ X -?li"'ll bj .

  • 3 Pages HW06_sol
    HW06_sol

    School: N.C. State

    Course: STATISTICAL THEORY II

    ST522 Solutions to Homework #6 Prepared by Peng Yang March 9, 2012 7.44 x is a complete sucient statistic for , and we have E (2 ) = var() + [E ()]2 = 2 + 1/n x x x which implies that E (2 (1/n) = 2 . Therefore, x2 (1/n) is an unbiased x 2 estimator of ,

  • 4 Pages HW05_sol
    HW05_sol

    School: N.C. State

    Course: STATISTICAL THEORY II

    ST 522-002: Statistical Theory II Solution to Homework Assignment - 5 Prepared by Chen-Yen Lin Spring, 2012 7.2 (a) The likelihood function of is ( L( |x) = 1 () )n ( )1 xi e i xi i By taking the derivative of log L w.r.t and set to zero, we arrive at d

  • 3 Pages hw3_sol
    Hw3_sol

    School: N.C. State

    Course: Statist Theory II

    ST 522, Spring 2008 HW 3 Solutions 6.16 ; ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! dim unknown) 1 , since knowing one of the 's enables us to find the

  • 2 Pages hw2_sol
    Hw2_sol

    School: N.C. State

    Course: Statist Theory II

    ST 522, Spring 2008 HW 2 Solutions 6.16 ; dim unknown) ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! 1 , since knowing one of the 's enables us to find the other three

  • 3 Pages hw1_sol
    Hw1_sol

    School: N.C. State

    Course: Statist Theory II

    ST 522, Spring 2008 HW 1 Solutions min , min , ,., ;

  • 19 Pages 08Chap9
    08Chap9

    School: N.C. State

    Course: Statist Theory II

    Chapter 9: Interval Estimation 1 Why Interval Estimators Interval estimator: [L(X), U (X)] Three types of intervals: two-sided interval [L(X), U (X)] [L(X), ) (-, U (X)] (call L(X) the lower confidence bound) (call U (X) the upper confidence bo

  • 46 Pages 08Chap7
    08Chap7

    School: N.C. State

    Course: Statist Theory II

    Chapter 7: Point Estimation The main task in this chapter: Assume X1 , , Xn iid f (x|), where is an unknown. We want to construct good estimators for or its function (). Important Questions: ^ How to construct an estimator using the random

  • 37 Pages 08Chap6
    08Chap6

    School: N.C. State

    Course: Statist Theory II

    Chapter 6: Principles of Data Reduction 1 Statistical Inference Data X = (X1 , . . . , Xn ): from a probability distribution f (x|), with unknown. Our task is Examples: to estimate to estimate to estimate to estimate based on data. the suc

  • 11 Pages 08Chap10
    08Chap10

    School: N.C. State

    Course: Statist Theory II

    Chapter 10: Asymptotic Evaluation Samples X1 , . . . , Xn i.i.d. f (x|), n large. We will see what happens if n . This assumption n generally makes life easier. Because limit theorems become available, distributions can be found approximately. L

  • 1 Page hint6.16
    Hint6.16

    School: N.C. State

    Course: Statist Theory II

    Hint for 6.16 The random vector (X1 , X2 , X3 , X4 ) follow a multinomial distribution with the density function f (x1 , x2 , x3 , x4 |) = = x2 x3 m! 1 x1 1 1 + (1 - ) (1 - ) x1 !x2 !x3 !x4 ! 2 4 4 4 4 m m! 1 (2 + )x1 (1 - )x2 (1 - )x3 x4 . x1 !x2

  • 2 Pages Syllabus_ST522
    Syllabus_ST522

    School: N.C. State

    Course: Statist Theory II

    ST 522, Section 002, Spring 2009 (and ST 522-L, Section 204) Statistical Theory II Course Meetings: MWF 11:20 AM 12:10 PM, Harrelson 325, W 10:15 11:05 AM, Harrelson 320 Instructor: Dr. Howard Bondell Office: Patterson Hall, Room 220D Email: bonde

  • 1 Page hint6.2
    Hint6.2

    School: N.C. State

    Course: Statist Theory II

    Hint for 6.2 Firstly, derive the joint density of (X1 , , Xn ). Here they are independently but not identically. The density of Xi is fXi (xi ) = eixi I(xi > i) = eixi I(xi /i > ) where I is the indicator function. The joint pdf of (X1 , , Xn )

  • 7 Pages 08Chapter5_order
    08Chapter5_order

    School: N.C. State

    Course: Statist Theory II

    Chapter 5: Order Statistics Given a random sample, we are interested in the smallest, largest, or middle observations. Examples: the highest flood waters (useful when planning for future emergencies) the lowest winter temperature recorded in the la

  • 2 Pages prac08_Final
    Prac08_Final

    School: N.C. State

    Course: Statist Theory II

    STAT 522 Practice Final Exam - Spring 2008 1. Let X1 , X2 , X3 be three random samples from Unif(0, ), where > 0 is unknown. (a) Show that X(1) / is distributed as Beta(1, 3). (b) Compute E[X(1) ]. Construct an unbiased estimator for using X(1)

  • 31 Pages 08Chap8
    08Chap8

    School: N.C. State

    Course: Statist Theory II

    Chapter 8: Hypothesis Testing 1 Hypotheses A hypothesis is a statement about a population parameter. Often, there are two complementary statements/hypotheses about , respectively called the null hypothesis and alternative hypothesis. Let be the par

  • 2 Pages hw4_sol
    Hw4_sol

    School: N.C. State

    Course: Statist Theory II

    ST 522, Spring 2008 HW 4 Solutions 6.20 a. is minimal sufficient by Lehmann-Scheffe Theorem. The pdfs in parts b-e are full exponential families. By writing each pdf in the form of a full exponential family and verifying that the parameter space c

  • 2 Pages hw5_sol
    Hw5_sol

    School: N.C. State

    Course: Statist Theory II

    ST 522, Spring 2008 HW 5 Solutions 6.31

  • 4 Pages hw6_sol
    Hw6_sol

    School: N.C. State

    Course: Statist Theory II

    ST 522, Spring 2008 HW 6 Solutions

  • 2 Pages HW04_sol
    HW04_sol

    School: N.C. State

    Course: STATISTICAL THEORY II

    ST 522-002: Statistical Theory II Solution to Homework Assignment - 4 Prepared by Chen-Yen Lin Spring, 2012 7.7 Since the likelihood function can only take two distinct values, L(1; x) = 2n the MLE of can be expressed as = 1 0 if 2n if 2n i xi and L(0; x)

  • 2 Pages HW03_sol
    HW03_sol

    School: N.C. State

    Course: STATISTICAL THEORY II

    ST522 Solutions to Homework #3 Prepared by Peng Yang February 15, 2012 6.20 (a)The joint pdf of Xi s is f (X |) = 2n 2n n=1 Xi I(0,) (X(n) ), thus X(n) is i sucient by Factorization Thm. Let Y = X(n) , the density of Y is f (y ) = 2n 2n1 y , 2n 0<y< For a

  • 1 Page HW02_sol
    HW02_sol

    School: N.C. State

    Course: STATISTICAL THEORY II

    ST 522-002: Statistical Theory II Solution to Homework Assignment - 2 Prepared by Chen-Yen Lin Spring, 2012 iid 6.13 Let Yi = Xi , i = 1, 2 then Yi exp(1). Then the statistics S (Y1 , Y2 ) = log Y1 log Y1 log Y1 log X1 = = = log Y2 log Y2 log Y2 log X2

  • 1 Page HW01_sol
    HW01_sol

    School: N.C. State

    Course: STATISTICAL THEORY II

    ST 522-002: Statistical Theory II Solution to Homework Assignment - 1 Prepared by Chen-Yen Lin Spring, 2012 6.1 Yes, since the conditional distribution of X given |X | becomes x = t, t o.w. 0.5 0 fX |X | (x|t) = which does not depend on 2 . 6.3 The joint

  • 2 Pages syllabus08
    Syllabus08

    School: N.C. State

    Course: Statist Theory II

  • 1 Page prob5
    Prob5

    School: N.C. State

    Course: Statist Theory II

    Prob 5. Let X1 , X2 , ., Xn be a random sample from a distribution with an exponential distribution f (x|) = e-x , where x > 0, > 0. Suppose the prior distribution of is Gamma(, ). (a) Find the posterior density function of given X1 , ., Xn . (b)

  • 3 Pages hw13_sol
    Hw13_sol

    School: N.C. State

    Course: Statist Theory II

    ST 522, Spring 2008 HW 13 Solutions 10.10 a. The asymptotic variance of 1 is by the Delta Method. 1 , To calculate the Cramer-Rao Lower Bound for the asymptotic variance of we compute 1 1 ln 1 2 and ; 1 (Using and simplifying). Therefor

  • 3 Pages hw12_sol
    Hw12_sol

    School: N.C. State

    Course: Statist Theory II

    ST 522, Spring 2008 HW 12 Solutions

  • 3 Pages hw11_sol
    Hw11_sol

    School: N.C. State

    Course: Statist Theory II

    ST 522, Spring 2008 HW 11 Solutions

  • 5 Pages hw10_sol
    Hw10_sol

    School: N.C. State

    Course: Statist Theory II

    ST 522, Spring 2008 HW 10 Solutions 1.96 1.96

  • 3 Pages hw9_sol
    Hw9_sol

    School: N.C. State

    Course: Statist Theory II

    ST 522, Spring 2008 HW 9 Solutions 7.60 With alpha known, the Gamma pdf is from a full exponential family with d=k=1. Thus completesufficient statistic. ~ 1 Therefore is a , 1 2 ! 1 ! 1 1 1 / . Unbiased and

  • 5 Pages hw8_sol
    Hw8_sol

    School: N.C. State

    Course: Statist Theory II

    ST 522, Spring 2008 HW8 Solutions c. From (a) 601.2 ; from (b) 124.8 7.58 a. If If 0, 1, ; ; 1 | |. 2 1 0 0 1 2 and the maximizing value of is 1. and the maximizing value of is 0

  • 3 Pages hw7_sol
    Hw7_sol

    School: N.C. State

    Course: Statist Theory II

    ST 522, Spring 2008 HW7 Solutions 7.42 a. . . 1 1 0 We must minimize Using the method of Lagrange Multipliers, let g= Then 2 And 1 1 , setting equal to zero gives 1 / Plugging into th

  • 1 Page prac08_Mid
    Prac08_Mid

    School: N.C. State

    Course: Statist Theory II

    STAT 522 Practice Midterm Exam 02/18/08 1. Let X1 , X2 , ., Xn be n independent samples from N (, 2 ). Let l1 , l2 , ., ln and m1 , m2 , ., mn n n n n 2 2 be known constants satisfying i=1 li = i=1 mi = 0, i=1 li = i=1 mi = 1. Dene n n U = i=1 li

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