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ST522HW6
School: N.C. State
Course: Statistical Theory II
ST 522002: 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

ST5222014Mid
School: N.C. State
Course: Statistical Theory II
Midterm Exam, Spring 2014, Statistics 522002 Feb. 26, Wednesday 9:00am  11:15am Your Name (Print): Note: 1. The exam is closedbook and closednotes. 2. There are totally FOUR problems. 3. Show all your work in the space provided. If you need additional

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 nrestrietecl ('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

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

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

HW10_sol
School: N.C. State
Course: STATISTICAL THEORY II
ST 522002: Statistical Theory II Solution to Homework Assignment  10 Prepared by ChenYen 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

HW11_sol
School: N.C. State
Course: STATISTICAL THEORY II
ST 522002: Statistical Theory II Solution to Homework Assignment  11 Prepared by ChenYen Lin Spring, 2012 8.27 Let 2 > 1 , then consider the ratio g (t2 ) c(2 ) = exp cfw_t[w(2 ) w(1 )] g (t1 ) c(1 ) If w() is an increasing function, then w(2 ) w(1 )

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 loglikelihood 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/ +

HW08_sol
School: N.C. State
Course: STATISTICAL THEORY II
ST 522002: Statistical Theory II Solution to Homework Assignment  8 Prepared by ChenYen 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

HW07_sol
School: N.C. State
Course: STATISTICAL THEORY II
ST 522002: Statistical Theory II Solution to Homework Assignment  7 Prepared by ChenYen Lin Spring, 2012 7.38 (a) The loglikelihood function can be written as L(x) = n log + ( 1) loglikelihood function w.r.t , we have log L(x) n =+ i log xi . Dier

ST522HW5
School: N.C. State
Course: Statistical Theory II
ST 522002: 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

ST522HW4
School: N.C. State
Course: Statistical Theory II
ST 522002: 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

ST522HW3
School: N.C. State
Course: Statistical Theory II
ST 522002: 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

Lab 4 Solution
School: N.C. State
Course: Statistical Theory II
Lt\b4 @y fu(.ft)rllc.\+ictl To c.heC1( H11A+ 1t i!. )1r/.hue11+ "' J.hrn, /e.i Slt4'tTLll?.n+ C.vn,ple.+e q Por all B +na+ I) t'\"1I I'\ bewuse. lJ iJfld J_ I a.e it 1.s (o iollq I [()1 .lJ c.i Ecfw_ '( J cfw_) it, liI I I =:; 8 h.inltiOl

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" 63  ?:.1120.,_ :. o" I 1L1 is Becaw~e Cc + 2i1 . = 0 "V

Chapter 5 Lecture Notes
School: N.C. State
Course: Statistical Theory II
1 ST522002 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

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

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

Hw4solution
School: N.C. State
Course: Statistical Theory II
H vv4 I If c I ".2:Ti t5 v lif +hm PtlcroiT&ct1tiot1 rf I h / (f<At l2.Y;<, z 7l;Yi) 1.s o. f '4f+rcre<1t 6 ')  71t ' fi CZ.?/;;i_ AI> 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

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 Therefw.e., thm ry,1,.i.3j JCpeltMi'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,lGJ J die.). <RLB

ST522PracticeMidterm
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

Hw3solution
School: N.C. State
Course: Statistical Theory II
STS liW3 22 x: I  , '/. . . ' Io ( ~t1 on is " we can vvli +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 n1 .z R. ( ~11, +b, e (JI,  ,'fJJ,.,+hJ= 7/q) (lo.1t;+b)  =

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 re1Yl f h q I f) =:. I e( ;f , = ,11; 7 l,tl) I  ii" e h h II e Sr~+1s+1c. fuc+OtftG\+io() f:.or T ( J() = m i h ( ~i) i's (A,6) . +he. orem.) 14 (" .I (Jii JAJ .I. ~ X ?li"'ll bj .

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 ,

HW05_sol
School: N.C. State
Course: STATISTICAL THEORY II
ST 522002: Statistical Theory II Solution to Homework Assignment  5 Prepared by ChenYen 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

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

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

Hw1_sol
School: N.C. State
Course: Statist Theory II
ST 522, Spring 2008 HW 1 Solutions min , min , ,., ;

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: twosided interval [L(X), U (X)] [L(X), ) (, U (X)] (call L(X) the lower confidence bound) (call U (X) the upper confidence bo

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

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

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

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

Syllabus_ST522
School: N.C. State
Course: Statist Theory II
ST 522, Section 002, Spring 2009 (and ST 522L, 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

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 )

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

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)

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

Hw4_sol
School: N.C. State
Course: Statist Theory II
ST 522, Spring 2008 HW 4 Solutions 6.20 a. is minimal sufficient by LehmannScheffe Theorem. The pdfs in parts be are full exponential families. By writing each pdf in the form of a full exponential family and verifying that the parameter space c

HW04_sol
School: N.C. State
Course: STATISTICAL THEORY II
ST 522002: Statistical Theory II Solution to Homework Assignment  4 Prepared by ChenYen 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)

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

HW02_sol
School: N.C. State
Course: STATISTICAL THEORY II
ST 522002: Statistical Theory II Solution to Homework Assignment  2 Prepared by ChenYen 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

HW01_sol
School: N.C. State
Course: STATISTICAL THEORY II
ST 522002: Statistical Theory II Solution to Homework Assignment  1 Prepared by ChenYen 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  (xt) = which does not depend on 2 . 6.3 The joint

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) = ex , where x > 0, > 0. Suppose the prior distribution of is Gamma(, ). (a) Find the posterior density function of given X1 , ., Xn . (b)

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 CramerRao Lower Bound for the asymptotic variance of we compute 1 1 ln 1 2 and ; 1 (Using and simplifying). Therefor

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

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

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

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