Review Questions, Chapters 8, 9. f(y) = 0, elsewhere. F (y) = f Y(1) = n ( e y/θ) n 1 1 θ e y/θ = n θ e yn

Size: px
Start display at page:

Download "Review Questions, Chapters 8, 9. f(y) = 0, elsewhere. F (y) = f Y(1) = n ( e y/θ) n 1 1 θ e y/θ = n θ e yn"

Transcription

1 Stat 366 Lab 2 Solutios (September 2, 2006) page TA: Yury Petracheko, CAB 484, yuryp@ualberta.ca, yuryp/ Review Questios, Chapters 8, Suppose that Y, Y 2,..., Y deote a radom sample of size from a populatio with a expoetial distributio whose desity is give by (/θ)e y/θ, y > 0 If Y () = mi(y, Y 2,..., Y ) deotes the smallest-order statistic, show that ˆθ = Y () is a ubiased estimator for θ ad fid MSE(ˆθ). Solutio. Let s fid the distributio fuctio of Y : e y/θ, y > 0 F (y) = Now we ca use the formula F Y() (y) = [ F (y) ] or fy() = ( F (y) ) f(y) to fid the the desity fuctio for Y () : for y > 0, f Y() = ( e y/θ) θ e y/θ = θ e y θ. We ca recogize this desity fuctio to be the desity of the expoetial distributio with parameter θ /, Y () Exp ( θ ). Kowig the distributio of Y () allows us to compute the expectatio of ˆθ = Y () : E[ˆθ] = E[Y () ] = θ = θ. So, E[ˆθ] = θ, ad ˆθ is a ubiased estimator of θ. To fid MSE(ˆθ), use the formula MSE(ˆθ) = V [ˆθ] + ( B(ˆθ) ) 2. Sice the estimator is ubiased, its bias B(ˆθ) equals zero. For the variace, remember that Y () is expoetial. We have MSE(ˆθ) = V [ˆθ] + 0 = 2 V [ Y () ] = 2 θ2 2 = θ2.

2 Stat 366 Lab 2 Solutios (September 2, 2006) page Suppose that Y, Y 2,..., Y deote a radom sample of size from a expoetial distributio with desity fuctio give by (/θ)e y/θ, y > 0 I Exercise 8.5 we determied that ˆθ = Y () is a ubiased estimator of θ with MSE(ˆθ)= θ 2. Cosider the estimator ˆθ 2 = Ȳ, ad fid the efficiecy of ˆθ relative to ˆθ 2. Solutio. First compute the variace of ˆθ 2 : [ ] V [ˆθ 2 ] = V [Ȳ ] = V Y + + Y = V [Y Y ] = ( V [Y ] + + V [Y 2 ] ) ) θ 2 = = ( θ θ 2 2 }{{} times 2 = θ2. To fid the relative efficiecy, we eed to fid the ratio of two variaces: eff(ˆθ, ˆθ 2 ) = V (ˆθ 2 ) V (ˆθ ) = θ2 θ 2 =. We coclude that ˆθ 2 is preferable to ˆθ. 9.6 Let Y, Y 2,..., Y deote a radom sample from the probability desity fuctio (θ + )y θ, 0 < y < ; θ > Fid a estimator for θ by the method of momets. Solutio. Let s fid the first momet of this distributio: µ = The method of momets implies 0 (θ + ) y θ+ dy = (θ + ) yθ+2 θ + 2 = θ + 0 θ + 2. Ȳ = ˆθ + ˆθ + 2 ˆθ = 2Ȳ Ȳ.

3 Stat 366 Lab 2 Solutios (September 2, 2006) page Suppose that Y, Y 2,..., Y deote a radom sample from the Poisso distributio with mea λ. (a) Fid the maximum-likelihood estimator ˆλ for λ. (d) What is the MLE for P (Y = 0) = e λ? Solutio. Let s defie the likelihood fuctio L(λ y, y 2,..., y ): p(y i ) = λ y i e λ y i! = λ y i e λ y. i! The problem ow is to fid the maximum value of this fuctio of λ. Let s make a simplifyig trasformatio: ( ) l y i l λ λ Differetiatio with respect to λ yields: l(y i!). d dx l λ y i = 0. Solvig this equatio: The latter is the MLE for λ. λ = y i, or ˆλ = Y i = Ȳ. To aswer (b), recall the ivariace priciple for MLEs: if t(ˆθ) is a oe-to-oe fuctio, the t(θ) = t(ˆθ). I our case t(λ) = e λ, so ê λ = e ˆλ = e Ȳ. 9.75a Suppose that Y, Y 2,..., Y costitute a radom sample from a uiform distributio with probability desity fuctio 2θ +, 0 y 2θ + 0, elsewhere. Obtai the maximum-likelihood estimator of θ.

4 Stat 366 Lab 2 Solutios (September 2, 2006) page 4 Solutio. This is a somewhat differet problem from the previous oe because the support of the desity fuctio depeds o θ. Recall the idicator fuctio I(A). It is equal to oe whe A is true, ad zero if A is false. We ca write the likelihood fuctio i the followig way: f(y i ) = 2θ + I(0 y i 2θ + ) = (2θ + ) I(0 y i 2θ + ). We ca simplify this eve further if we ote that the product of idicator is o-zero oly whe all of the uderlyig coditios fulfill. That is, all y i are less that 2θ + ad positive. Notice that this statemet is equivalet to the followig: 0 y () ad y () 2θ +. (We use order statistics y () = mi(y,..., y ) ad y () = max(y,..., y ).) We have (2θ + ) I(0 y ()) I(y () 2θ + ). Now look at the first part of the likelihood fuctio L, (2θ + ). Notice that this is a decreasig (ad cotiuous) fuctio of θ. If we wat to maximize L, we should choose the value of θ as small as possible. Notice that if 2θ + is smaller tha y (), the the value of L(θ) is zero. So, the miimum of 2θ + is y (). This gives the miimum value for θ ad maximizes the likelihood L(θ). We coclude (provided at least oe observatio i the sample is positive) ( Y () = 2ˆθ + ˆθ = Y() ) Let Y, Y 2,..., Y deote a radom sample from the probability desity fuctio (θ + )y θ, 0 < y < ; θ > Fid the maximum-likelihood estimator for θ. Compare your aswer to the method of momets estimator foud i Exercise 9.6. Solutio. Defie the likelihood fuctio: ( ) θ. (θ + )yi θ = (θ + ) y i Take the logarithms: l l(θ + ) + θ l y i.

5 Stat 366 Lab 2 Solutios (September 2, 2006) page 5 Fid critical poits: so ad fially d dθ l θ + + l y i = 0, θ = l y, i ˆθ = l Y. i This is quite differet from the method of momets estimator foud i Exercise 9.6.

Direction: This test is worth 250 points. You are required to complete this test within 50 minutes.

Direction: This test is worth 250 points. You are required to complete this test within 50 minutes. Term Test October 3, 003 Name Math 56 Studet Number Directio: This test is worth 50 poits. You are required to complete this test withi 50 miutes. I order to receive full credit, aswer each problem completely

More information

Stat410 Probability and Statistics II (F16)

Stat410 Probability and Statistics II (F16) Some Basic Cocepts of Statistical Iferece (Sec 5.) Suppose we have a rv X that has a pdf/pmf deoted by f(x; θ) or p(x; θ), where θ is called the parameter. I previous lectures, we focus o probability problems

More information

Questions and Answers on Maximum Likelihood

Questions and Answers on Maximum Likelihood Questios ad Aswers o Maximum Likelihood L. Magee Fall, 2008 1. Give: a observatio-specific log likelihood fuctio l i (θ) = l f(y i x i, θ) the log likelihood fuctio l(θ y, X) = l i(θ) a data set (x i,

More information

This exam contains 19 pages (including this cover page) and 10 questions. A Formulae sheet is provided with the exam.

This exam contains 19 pages (including this cover page) and 10 questions. A Formulae sheet is provided with the exam. Probability ad Statistics FS 07 Secod Sessio Exam 09.0.08 Time Limit: 80 Miutes Name: Studet ID: This exam cotais 9 pages (icludig this cover page) ad 0 questios. A Formulae sheet is provided with the

More information

Estimation for Complete Data

Estimation for Complete Data Estimatio for Complete Data complete data: there is o loss of iformatio durig study. complete idividual complete data= grouped data A complete idividual data is the oe i which the complete iformatio of

More information

Let us give one more example of MLE. Example 3. The uniform distribution U[0, θ] on the interval [0, θ] has p.d.f.

Let us give one more example of MLE. Example 3. The uniform distribution U[0, θ] on the interval [0, θ] has p.d.f. Lecture 5 Let us give oe more example of MLE. Example 3. The uiform distributio U[0, ] o the iterval [0, ] has p.d.f. { 1 f(x =, 0 x, 0, otherwise The likelihood fuctio ϕ( = f(x i = 1 I(X 1,..., X [0,

More information

Lecture 11 and 12: Basic estimation theory

Lecture 11 and 12: Basic estimation theory Lecture ad 2: Basic estimatio theory Sprig 202 - EE 94 Networked estimatio ad cotrol Prof. Kha March 2 202 I. MAXIMUM-LIKELIHOOD ESTIMATORS The maximum likelihood priciple is deceptively simple. Louis

More information

Topic 9: Sampling Distributions of Estimators

Topic 9: Sampling Distributions of Estimators Topic 9: Samplig Distributios of Estimators Course 003, 2018 Page 0 Samplig distributios of estimators Sice our estimators are statistics (particular fuctios of radom variables), their distributio ca be

More information

MATH 472 / SPRING 2013 ASSIGNMENT 2: DUE FEBRUARY 4 FINALIZED

MATH 472 / SPRING 2013 ASSIGNMENT 2: DUE FEBRUARY 4 FINALIZED MATH 47 / SPRING 013 ASSIGNMENT : DUE FEBRUARY 4 FINALIZED Please iclude a cover sheet that provides a complete setece aswer to each the followig three questios: (a) I your opiio, what were the mai ideas

More information

1.010 Uncertainty in Engineering Fall 2008

1.010 Uncertainty in Engineering Fall 2008 MIT OpeCourseWare http://ocw.mit.edu.00 Ucertaity i Egieerig Fall 2008 For iformatio about citig these materials or our Terms of Use, visit: http://ocw.mit.edu.terms. .00 - Brief Notes # 9 Poit ad Iterval

More information

MATH 320: Probability and Statistics 9. Estimation and Testing of Parameters. Readings: Pruim, Chapter 4

MATH 320: Probability and Statistics 9. Estimation and Testing of Parameters. Readings: Pruim, Chapter 4 MATH 30: Probability ad Statistics 9. Estimatio ad Testig of Parameters Estimatio ad Testig of Parameters We have bee dealig situatios i which we have full kowledge of the distributio of a radom variable.

More information

Topic 9: Sampling Distributions of Estimators

Topic 9: Sampling Distributions of Estimators Topic 9: Samplig Distributios of Estimators Course 003, 2016 Page 0 Samplig distributios of estimators Sice our estimators are statistics (particular fuctios of radom variables), their distributio ca be

More information

Problem Set 4 Due Oct, 12

Problem Set 4 Due Oct, 12 EE226: Radom Processes i Systems Lecturer: Jea C. Walrad Problem Set 4 Due Oct, 12 Fall 06 GSI: Assae Gueye This problem set essetially reviews detectio theory ad hypothesis testig ad some basic otios

More information

Topic 9: Sampling Distributions of Estimators

Topic 9: Sampling Distributions of Estimators Topic 9: Samplig Distributios of Estimators Course 003, 2018 Page 0 Samplig distributios of estimators Sice our estimators are statistics (particular fuctios of radom variables), their distributio ca be

More information

Stat 421-SP2012 Interval Estimation Section

Stat 421-SP2012 Interval Estimation Section Stat 41-SP01 Iterval Estimatio Sectio 11.1-11. We ow uderstad (Chapter 10) how to fid poit estimators of a ukow parameter. o However, a poit estimate does ot provide ay iformatio about the ucertaity (possible

More information

Chapter 6 Principles of Data Reduction

Chapter 6 Principles of Data Reduction Chapter 6 for BST 695: Special Topics i Statistical Theory. Kui Zhag, 0 Chapter 6 Priciples of Data Reductio Sectio 6. Itroductio Goal: To summarize or reduce the data X, X,, X to get iformatio about a

More information

Unbiased Estimation. February 7-12, 2008

Unbiased Estimation. February 7-12, 2008 Ubiased Estimatio February 7-2, 2008 We begi with a sample X = (X,..., X ) of radom variables chose accordig to oe of a family of probabilities P θ where θ is elemet from the parameter space Θ. For radom

More information

IIT JAM Mathematical Statistics (MS) 2006 SECTION A

IIT JAM Mathematical Statistics (MS) 2006 SECTION A IIT JAM Mathematical Statistics (MS) 6 SECTION A. If a > for ad lim a / L >, the which of the followig series is ot coverget? (a) (b) (c) (d) (d) = = a = a = a a + / a lim a a / + = lim a / a / + = lim

More information

Lecture Note 8 Point Estimators and Point Estimation Methods. MIT Spring 2006 Herman Bennett

Lecture Note 8 Point Estimators and Point Estimation Methods. MIT Spring 2006 Herman Bennett Lecture Note 8 Poit Estimators ad Poit Estimatio Methods MIT 14.30 Sprig 2006 Herma Beett Give a parameter with ukow value, the goal of poit estimatio is to use a sample to compute a umber that represets

More information

LECTURE NOTES 9. 1 Point Estimation. 1.1 The Method of Moments

LECTURE NOTES 9. 1 Point Estimation. 1.1 The Method of Moments LECTURE NOTES 9 Poit Estimatio Uder the hypothesis that the sample was geerated from some parametric statistical model, a atural way to uderstad the uderlyig populatio is by estimatig the parameters of

More information

Statistical Theory MT 2008 Problems 1: Solution sketches

Statistical Theory MT 2008 Problems 1: Solution sketches Statistical Theory MT 008 Problems : Solutio sketches. Which of the followig desities are withi a expoetial family? Explai your reasoig. a) Let 0 < θ < ad put fx, θ) = θ)θ x ; x = 0,,,... b) c) where α

More information

Solutions: Homework 3

Solutions: Homework 3 Solutios: Homework 3 Suppose that the radom variables Y,...,Y satisfy Y i = x i + " i : i =,..., IID where x,...,x R are fixed values ad ",...," Normal(0, )with R + kow. Fid ˆ = MLE( ). IND Solutio: Observe

More information

Simulation. Two Rule For Inverting A Distribution Function

Simulation. Two Rule For Inverting A Distribution Function Simulatio Two Rule For Ivertig A Distributio Fuctio Rule 1. If F(x) = u is costat o a iterval [x 1, x 2 ), the the uiform value u is mapped oto x 2 through the iversio process. Rule 2. If there is a jump

More information

Maximum Likelihood Estimation

Maximum Likelihood Estimation ECE 645: Estimatio Theory Sprig 2015 Istructor: Prof. Staley H. Cha Maximum Likelihood Estimatio (LaTeX prepared by Shaobo Fag) April 14, 2015 This lecture ote is based o ECE 645(Sprig 2015) by Prof. Staley

More information

4. Partial Sums and the Central Limit Theorem

4. Partial Sums and the Central Limit Theorem 1 of 10 7/16/2009 6:05 AM Virtual Laboratories > 6. Radom Samples > 1 2 3 4 5 6 7 4. Partial Sums ad the Cetral Limit Theorem The cetral limit theorem ad the law of large umbers are the two fudametal theorems

More information

Since X n /n P p, we know that X n (n. Xn (n X n ) Using the asymptotic result above to obtain an approximation for fixed n, we obtain

Since X n /n P p, we know that X n (n. Xn (n X n ) Using the asymptotic result above to obtain an approximation for fixed n, we obtain Assigmet 9 Exercise 5.5 Let X biomial, p, where p 0, 1 is ukow. Obtai cofidece itervals for p i two differet ways: a Sice X / p d N0, p1 p], the variace of the limitig distributio depeds oly o p. Use the

More information

Properties of Point Estimators and Methods of Estimation

Properties of Point Estimators and Methods of Estimation CHAPTER 9 Properties of Poit Estimators ad Methods of Estimatio 9.1 Itroductio 9. Relative Efficiecy 9.3 Cosistecy 9.4 Sufficiecy 9.5 The Rao Blackwell Theorem ad Miimum-Variace Ubiased Estimatio 9.6 The

More information

Statistical Theory MT 2009 Problems 1: Solution sketches

Statistical Theory MT 2009 Problems 1: Solution sketches Statistical Theory MT 009 Problems : Solutio sketches. Which of the followig desities are withi a expoetial family? Explai your reasoig. (a) Let 0 < θ < ad put f(x, θ) = ( θ)θ x ; x = 0,,,... (b) (c) where

More information

Lecture 7: Properties of Random Samples

Lecture 7: Properties of Random Samples Lecture 7: Properties of Radom Samples 1 Cotiued From Last Class Theorem 1.1. Let X 1, X,...X be a radom sample from a populatio with mea µ ad variace σ

More information

7.1 Convergence of sequences of random variables

7.1 Convergence of sequences of random variables Chapter 7 Limit Theorems Throughout this sectio we will assume a probability space (, F, P), i which is defied a ifiite sequece of radom variables (X ) ad a radom variable X. The fact that for every ifiite

More information

EECS564 Estimation, Filtering, and Detection Hwk 2 Solns. Winter p θ (z) = (2θz + 1 θ), 0 z 1

EECS564 Estimation, Filtering, and Detection Hwk 2 Solns. Winter p θ (z) = (2θz + 1 θ), 0 z 1 EECS564 Estimatio, Filterig, ad Detectio Hwk 2 Sols. Witer 25 4. Let Z be a sigle observatio havig desity fuctio where. p (z) = (2z + ), z (a) Assumig that is a oradom parameter, fid ad plot the maximum

More information

Lecture 12: September 27

Lecture 12: September 27 36-705: Itermediate Statistics Fall 207 Lecturer: Siva Balakrisha Lecture 2: September 27 Today we will discuss sufficiecy i more detail ad the begi to discuss some geeral strategies for costructig estimators.

More information

Direction: This test is worth 150 points. You are required to complete this test within 55 minutes.

Direction: This test is worth 150 points. You are required to complete this test within 55 minutes. Term Test 3 (Part A) November 1, 004 Name Math 6 Studet Number Directio: This test is worth 10 poits. You are required to complete this test withi miutes. I order to receive full credit, aswer each problem

More information

An Introduction to Signal Detection and Estimation - Second Edition Chapter IV: Selected Solutions

An Introduction to Signal Detection and Estimation - Second Edition Chapter IV: Selected Solutions A Itroductio to Sigal Detectio Estimatio - Secod Editio Chapter IV: Selected Solutios H V Poor Priceto Uiversity April 6 5 Exercise 1: a b ] ˆθ MAP (y) = arg max log θ θ y log θ =1 1 θ e ˆθ MMSE (y) =

More information

STAT Homework 1 - Solutions

STAT Homework 1 - Solutions STAT-36700 Homework 1 - Solutios Fall 018 September 11, 018 This cotais solutios for Homework 1. Please ote that we have icluded several additioal commets ad approaches to the problems to give you better

More information

Random Variables, Sampling and Estimation

Random Variables, Sampling and Estimation Chapter 1 Radom Variables, Samplig ad Estimatio 1.1 Itroductio This chapter will cover the most importat basic statistical theory you eed i order to uderstad the ecoometric material that will be comig

More information

January 25, 2017 INTRODUCTION TO MATHEMATICAL STATISTICS

January 25, 2017 INTRODUCTION TO MATHEMATICAL STATISTICS Jauary 25, 207 INTRODUCTION TO MATHEMATICAL STATISTICS Abstract. A basic itroductio to statistics assumig kowledge of probability theory.. Probability I a typical udergraduate problem i probability, we

More information

Statistical Inference (Chapter 10) Statistical inference = learn about a population based on the information provided by a sample.

Statistical Inference (Chapter 10) Statistical inference = learn about a population based on the information provided by a sample. Statistical Iferece (Chapter 10) Statistical iferece = lear about a populatio based o the iformatio provided by a sample. Populatio: The set of all values of a radom variable X of iterest. Characterized

More information

Kurskod: TAMS11 Provkod: TENB 21 March 2015, 14:00-18:00. English Version (no Swedish Version)

Kurskod: TAMS11 Provkod: TENB 21 March 2015, 14:00-18:00. English Version (no Swedish Version) Kurskod: TAMS Provkod: TENB 2 March 205, 4:00-8:00 Examier: Xiagfeg Yag (Tel: 070 2234765). Please aswer i ENGLISH if you ca. a. You are allowed to use: a calculator; formel -och tabellsamlig i matematisk

More information

7.1 Convergence of sequences of random variables

7.1 Convergence of sequences of random variables Chapter 7 Limit theorems Throughout this sectio we will assume a probability space (Ω, F, P), i which is defied a ifiite sequece of radom variables (X ) ad a radom variable X. The fact that for every ifiite

More information

x = Pr ( X (n) βx ) =

x = Pr ( X (n) βx ) = Exercise 93 / page 45 The desity of a variable X i i 1 is fx α α a For α kow let say equal to α α > fx α α x α Pr X i x < x < Usig a Pivotal Quatity: x α 1 < x < α > x α 1 ad We solve i a similar way as

More information

Lecture 6 Simple alternatives and the Neyman-Pearson lemma

Lecture 6 Simple alternatives and the Neyman-Pearson lemma STATS 00: Itroductio to Statistical Iferece Autum 06 Lecture 6 Simple alteratives ad the Neyma-Pearso lemma Last lecture, we discussed a umber of ways to costruct test statistics for testig a simple ull

More information

Last Lecture. Wald Test

Last Lecture. Wald Test Last Lecture Biostatistics 602 - Statistical Iferece Lecture 22 Hyu Mi Kag April 9th, 2013 Is the exact distributio of LRT statistic typically easy to obtai? How about its asymptotic distributio? For testig

More information

Lecture 18: Sampling distributions

Lecture 18: Sampling distributions Lecture 18: Samplig distributios I may applicatios, the populatio is oe or several ormal distributios (or approximately). We ow study properties of some importat statistics based o a radom sample from

More information

AAEC/ECON 5126 FINAL EXAM: SOLUTIONS

AAEC/ECON 5126 FINAL EXAM: SOLUTIONS AAEC/ECON 5126 FINAL EXAM: SOLUTIONS SPRING 2015 / INSTRUCTOR: KLAUS MOELTNER This exam is ope-book, ope-otes, but please work strictly o your ow. Please make sure your ame is o every sheet you re hadig

More information

Resampling Methods. X (1/2), i.e., Pr (X i m) = 1/2. We order the data: X (1) X (2) X (n). Define the sample median: ( n.

Resampling Methods. X (1/2), i.e., Pr (X i m) = 1/2. We order the data: X (1) X (2) X (n). Define the sample median: ( n. Jauary 1, 2019 Resamplig Methods Motivatio We have so may estimators with the property θ θ d N 0, σ 2 We ca also write θ a N θ, σ 2 /, where a meas approximately distributed as Oce we have a cosistet estimator

More information

Introduction to Extreme Value Theory Laurens de Haan, ISM Japan, Erasmus University Rotterdam, NL University of Lisbon, PT

Introduction to Extreme Value Theory Laurens de Haan, ISM Japan, Erasmus University Rotterdam, NL University of Lisbon, PT Itroductio to Extreme Value Theory Laures de Haa, ISM Japa, 202 Itroductio to Extreme Value Theory Laures de Haa Erasmus Uiversity Rotterdam, NL Uiversity of Lisbo, PT Itroductio to Extreme Value Theory

More information

Distribution of Random Samples & Limit theorems

Distribution of Random Samples & Limit theorems STAT/MATH 395 A - PROBABILITY II UW Witer Quarter 2017 Néhémy Lim Distributio of Radom Samples & Limit theorems 1 Distributio of i.i.d. Samples Motivatig example. Assume that the goal of a study is to

More information

Question 1: The magnetic case

Question 1: The magnetic case September 6, 018 Corell Uiversity, Departmet of Physics PHYS 337, Advace E&M, HW # 4, due: 9/19/018, 11:15 AM Questio 1: The magetic case I class, we skipped over some details, so here you are asked to

More information

6. Sufficient, Complete, and Ancillary Statistics

6. Sufficient, Complete, and Ancillary Statistics Sufficiet, Complete ad Acillary Statistics http://www.math.uah.edu/stat/poit/sufficiet.xhtml 1 of 7 7/16/2009 6:13 AM Virtual Laboratories > 7. Poit Estimatio > 1 2 3 4 5 6 6. Sufficiet, Complete, ad Acillary

More information

STATISTICAL INFERENCE

STATISTICAL INFERENCE STATISTICAL INFERENCE POPULATION AND SAMPLE Populatio = all elemets of iterest Characterized by a distributio F with some parameter θ Sample = the data X 1,..., X, selected subset of the populatio = sample

More information

5. Likelihood Ratio Tests

5. Likelihood Ratio Tests 1 of 5 7/29/2009 3:16 PM Virtual Laboratories > 9. Hy pothesis Testig > 1 2 3 4 5 6 7 5. Likelihood Ratio Tests Prelimiaries As usual, our startig poit is a radom experimet with a uderlyig sample space,

More information

November 2002 Course 4 solutions

November 2002 Course 4 solutions November Course 4 solutios Questio # Aswer: B φ ρ = = 5. φ φ ρ = φ + =. φ Solvig simultaeously gives: φ = 8. φ = 6. Questio # Aswer: C g = [(.45)] = [5.4] = 5; h= 5.4 5 =.4. ˆ π =.6 x +.4 x =.6(36) +.4(4)

More information

SOLUTION FOR HOMEWORK 7, STAT np(1 p) (α + β + n) + ( np + α

SOLUTION FOR HOMEWORK 7, STAT np(1 p) (α + β + n) + ( np + α SOLUTION FOR HOMEWORK 7, STAT 6331 1 Exerc733 Here we just recall that MSE(ˆp B ) = p(1 p) (α + β + ) + ( p + α 2 α + β + p) 2 The you plug i α = β = (/4) 1/2 After simplificatios MSE(ˆp B ) = 4( 1/2 +

More information

EXAMINATIONS OF THE ROYAL STATISTICAL SOCIETY

EXAMINATIONS OF THE ROYAL STATISTICAL SOCIETY EXAMINATIONS OF THE ROYAL STATISTICAL SOCIETY GRADUATE DIPLOMA, 016 MODULE : Statistical Iferece Time allowed: Three hours Cadidates should aswer FIVE questios. All questios carry equal marks. The umber

More information

of the matrix is =-85, so it is not positive definite. Thus, the first

of the matrix is =-85, so it is not positive definite. Thus, the first BOSTON COLLEGE Departmet of Ecoomics EC771: Ecoometrics Sprig 4 Prof. Baum, Ms. Uysal Solutio Key for Problem Set 1 1. Are the followig quadratic forms positive for all values of x? (a) y = x 1 8x 1 x

More information

First Year Quantitative Comp Exam Spring, Part I - 203A. f X (x) = 0 otherwise

First Year Quantitative Comp Exam Spring, Part I - 203A. f X (x) = 0 otherwise First Year Quatitative Comp Exam Sprig, 2012 Istructio: There are three parts. Aswer every questio i every part. Questio I-1 Part I - 203A A radom variable X is distributed with the margial desity: >

More information

Frequentist Inference

Frequentist Inference Frequetist Iferece The topics of the ext three sectios are useful applicatios of the Cetral Limit Theorem. Without kowig aythig about the uderlyig distributio of a sequece of radom variables {X i }, for

More information

Linear regression. Daniel Hsu (COMS 4771) (y i x T i β)2 2πσ. 2 2σ 2. 1 n. (x T i β y i ) 2. 1 ˆβ arg min. β R n d

Linear regression. Daniel Hsu (COMS 4771) (y i x T i β)2 2πσ. 2 2σ 2. 1 n. (x T i β y i ) 2. 1 ˆβ arg min. β R n d Liear regressio Daiel Hsu (COMS 477) Maximum likelihood estimatio Oe of the simplest liear regressio models is the followig: (X, Y ),..., (X, Y ), (X, Y ) are iid radom pairs takig values i R d R, ad Y

More information

Econ 325 Notes on Point Estimator and Confidence Interval 1 By Hiro Kasahara

Econ 325 Notes on Point Estimator and Confidence Interval 1 By Hiro Kasahara Poit Estimator Eco 325 Notes o Poit Estimator ad Cofidece Iterval 1 By Hiro Kasahara Parameter, Estimator, ad Estimate The ormal probability desity fuctio is fully characterized by two costats: populatio

More information

Mathematical Statistics - MS

Mathematical Statistics - MS Paper Specific Istructios. The examiatio is of hours duratio. There are a total of 60 questios carryig 00 marks. The etire paper is divided ito three sectios, A, B ad C. All sectios are compulsory. Questios

More information

( θ. sup θ Θ f X (x θ) = L. sup Pr (Λ (X) < c) = α. x : Λ (x) = sup θ H 0. sup θ Θ f X (x θ) = ) < c. NH : θ 1 = θ 2 against AH : θ 1 θ 2

( θ. sup θ Θ f X (x θ) = L. sup Pr (Λ (X) < c) = α. x : Λ (x) = sup θ H 0. sup θ Θ f X (x θ) = ) < c. NH : θ 1 = θ 2 against AH : θ 1 θ 2 82 CHAPTER 4. MAXIMUM IKEIHOOD ESTIMATION Defiitio: et X be a radom sample with joit p.m/d.f. f X x θ. The geeralised likelihood ratio test g.l.r.t. of the NH : θ H 0 agaist the alterative AH : θ H 1,

More information

SOME THEORY AND PRACTICE OF STATISTICS by Howard G. Tucker

SOME THEORY AND PRACTICE OF STATISTICS by Howard G. Tucker SOME THEORY AND PRACTICE OF STATISTICS by Howard G. Tucker CHAPTER 9. POINT ESTIMATION 9. Covergece i Probability. The bases of poit estimatio have already bee laid out i previous chapters. I chapter 5

More information

Please do NOT write in this box. Multiple Choice. Total

Please do NOT write in this box. Multiple Choice. Total Istructor: Math 0560, Worksheet Alteratig Series Jauary, 3000 For realistic exam practice solve these problems without lookig at your book ad without usig a calculator. Multiple choice questios should

More information

Bayesian Methods: Introduction to Multi-parameter Models

Bayesian Methods: Introduction to Multi-parameter Models Bayesia Methods: Itroductio to Multi-parameter Models Parameter: θ = ( θ, θ) Give Likelihood p(y θ) ad prior p(θ ), the posterior p proportioal to p(y θ) x p(θ ) Margial posterior ( θ, θ y) is Iterested

More information

Discrete Mathematics for CS Spring 2008 David Wagner Note 22

Discrete Mathematics for CS Spring 2008 David Wagner Note 22 CS 70 Discrete Mathematics for CS Sprig 2008 David Wager Note 22 I.I.D. Radom Variables Estimatig the bias of a coi Questio: We wat to estimate the proportio p of Democrats i the US populatio, by takig

More information

MASSACHUSETTS INSTITUTE OF TECHNOLOGY 6.265/15.070J Fall 2013 Lecture 3 9/11/2013. Large deviations Theory. Cramér s Theorem

MASSACHUSETTS INSTITUTE OF TECHNOLOGY 6.265/15.070J Fall 2013 Lecture 3 9/11/2013. Large deviations Theory. Cramér s Theorem MASSACHUSETTS INSTITUTE OF TECHNOLOGY 6.265/5.070J Fall 203 Lecture 3 9//203 Large deviatios Theory. Cramér s Theorem Cotet.. Cramér s Theorem. 2. Rate fuctio ad properties. 3. Chage of measure techique.

More information

17. Joint distributions of extreme order statistics Lehmann 5.1; Ferguson 15

17. Joint distributions of extreme order statistics Lehmann 5.1; Ferguson 15 17. Joit distributios of extreme order statistics Lehma 5.1; Ferguso 15 I Example 10., we derived the asymptotic distributio of the maximum from a radom sample from a uiform distributio. We did this usig

More information

Exponential Families and Bayesian Inference

Exponential Families and Bayesian Inference Computer Visio Expoetial Families ad Bayesia Iferece Lecture Expoetial Families A expoetial family of distributios is a d-parameter family f(x; havig the followig form: f(x; = h(xe g(t T (x B(, (. where

More information

Chapter 6 Infinite Series

Chapter 6 Infinite Series Chapter 6 Ifiite Series I the previous chapter we cosidered itegrals which were improper i the sese that the iterval of itegratio was ubouded. I this chapter we are goig to discuss a topic which is somewhat

More information

ECE 901 Lecture 14: Maximum Likelihood Estimation and Complexity Regularization

ECE 901 Lecture 14: Maximum Likelihood Estimation and Complexity Regularization ECE 90 Lecture 4: Maximum Likelihood Estimatio ad Complexity Regularizatio R Nowak 5/7/009 Review : Maximum Likelihood Estimatio We have iid observatios draw from a ukow distributio Y i iid p θ, i,, where

More information

Lecture 19: Convergence

Lecture 19: Convergence Lecture 19: Covergece Asymptotic approach I statistical aalysis or iferece, a key to the success of fidig a good procedure is beig able to fid some momets ad/or distributios of various statistics. I may

More information

Sequences and Series of Functions

Sequences and Series of Functions Chapter 6 Sequeces ad Series of Fuctios 6.1. Covergece of a Sequece of Fuctios Poitwise Covergece. Defiitio 6.1. Let, for each N, fuctio f : A R be defied. If, for each x A, the sequece (f (x)) coverges

More information

MASSACHUSETTS INSTITUTE OF TECHNOLOGY 6.265/15.070J Fall 2013 Lecture 2 9/9/2013. Large Deviations for i.i.d. Random Variables

MASSACHUSETTS INSTITUTE OF TECHNOLOGY 6.265/15.070J Fall 2013 Lecture 2 9/9/2013. Large Deviations for i.i.d. Random Variables MASSACHUSETTS INSTITUTE OF TECHNOLOGY 6.265/15.070J Fall 2013 Lecture 2 9/9/2013 Large Deviatios for i.i.d. Radom Variables Cotet. Cheroff boud usig expoetial momet geeratig fuctios. Properties of a momet

More information

Element sampling: Part 2

Element sampling: Part 2 Chapter 4 Elemet samplig: Part 2 4.1 Itroductio We ow cosider uequal probability samplig desigs which is very popular i practice. I the uequal probability samplig, we ca improve the efficiecy of the resultig

More information

Machine Learning Brett Bernstein

Machine Learning Brett Bernstein Machie Learig Brett Berstei Week Lecture: Cocept Check Exercises Starred problems are optioal. Statistical Learig Theory. Suppose A = Y = R ad X is some other set. Furthermore, assume P X Y is a discrete

More information

10-701/ Machine Learning Mid-term Exam Solution

10-701/ Machine Learning Mid-term Exam Solution 0-70/5-78 Machie Learig Mid-term Exam Solutio Your Name: Your Adrew ID: True or False (Give oe setece explaatio) (20%). (F) For a cotiuous radom variable x ad its probability distributio fuctio p(x), it

More information

Convergence of random variables. (telegram style notes) P.J.C. Spreij

Convergence of random variables. (telegram style notes) P.J.C. Spreij Covergece of radom variables (telegram style otes).j.c. Spreij this versio: September 6, 2005 Itroductio As we kow, radom variables are by defiitio measurable fuctios o some uderlyig measurable space

More information

Read carefully the instructions on the answer book and make sure that the particulars required are entered on each answer book.

Read carefully the instructions on the answer book and make sure that the particulars required are entered on each answer book. THE UNIVERSITY OF WARWICK FIRST YEAR EXAMINATION: Jauary 2009 Aalysis I Time Allowed:.5 hours Read carefully the istructios o the aswer book ad make sure that the particulars required are etered o each

More information

f(x) dx as we do. 2x dx x also diverges. Solution: We compute 2x dx lim

f(x) dx as we do. 2x dx x also diverges. Solution: We compute 2x dx lim Math 3, Sectio 2. (25 poits) Why we defie f(x) dx as we do. (a) Show that the improper itegral diverges. Hece the improper itegral x 2 + x 2 + b also diverges. Solutio: We compute x 2 + = lim b x 2 + =

More information

MATHEMATICAL SCIENCES PAPER-II

MATHEMATICAL SCIENCES PAPER-II MATHEMATICAL SCIENCES PAPER-II. Let {x } ad {y } be two sequeces of real umbers. Prove or disprove each of the statemets :. If {x y } coverges, ad if {y } is coverget, the {x } is coverget.. {x + y } coverges

More information

ECONOMETRIC THEORY. MODULE XIII Lecture - 34 Asymptotic Theory and Stochastic Regressors

ECONOMETRIC THEORY. MODULE XIII Lecture - 34 Asymptotic Theory and Stochastic Regressors ECONOMETRIC THEORY MODULE XIII Lecture - 34 Asymptotic Theory ad Stochastic Regressors Dr. Shalabh Departmet of Mathematics ad Statistics Idia Istitute of Techology Kapur Asymptotic theory The asymptotic

More information

Math 132, Fall 2009 Exam 2: Solutions

Math 132, Fall 2009 Exam 2: Solutions Math 3, Fall 009 Exam : Solutios () a) ( poits) Determie for which positive real umbers p, is the followig improper itegral coverget, ad for which it is diverget. Evaluate the itegral for each value of

More information

Fall 2013 MTH431/531 Real analysis Section Notes

Fall 2013 MTH431/531 Real analysis Section Notes Fall 013 MTH431/531 Real aalysis Sectio 8.1-8. Notes Yi Su 013.11.1 1. Defiitio of uiform covergece. We look at a sequece of fuctios f (x) ad study the coverget property. Notice we have two parameters

More information

Sample questions. 8. Let X denote a continuous random variable with probability density function f(x) = 4x 3 /15 for

Sample questions. 8. Let X denote a continuous random variable with probability density function f(x) = 4x 3 /15 for Sample questios Suppose that humas ca have oe of three bloodtypes: A, B, O Assume that 40% of the populatio has Type A, 50% has type B, ad 0% has Type O If a perso has type A, the probability that they

More information

Math 152. Rumbos Fall Solutions to Review Problems for Exam #2. Number of Heads Frequency

Math 152. Rumbos Fall Solutions to Review Problems for Exam #2. Number of Heads Frequency Math 152. Rumbos Fall 2009 1 Solutios to Review Problems for Exam #2 1. I the book Experimetatio ad Measuremet, by W. J. Youde ad published by the by the Natioal Sciece Teachers Associatio i 1962, the

More information

Expectation and Variance of a random variable

Expectation and Variance of a random variable Chapter 11 Expectatio ad Variace of a radom variable The aim of this lecture is to defie ad itroduce mathematical Expectatio ad variace of a fuctio of discrete & cotiuous radom variables ad the distributio

More information

n outcome is (+1,+1, 1,..., 1). Let the r.v. X denote our position (relative to our starting point 0) after n moves. Thus X = X 1 + X 2 + +X n,

n outcome is (+1,+1, 1,..., 1). Let the r.v. X denote our position (relative to our starting point 0) after n moves. Thus X = X 1 + X 2 + +X n, CS 70 Discrete Mathematics for CS Sprig 2008 David Wager Note 9 Variace Questio: At each time step, I flip a fair coi. If it comes up Heads, I walk oe step to the right; if it comes up Tails, I walk oe

More information

Properties and Hypothesis Testing

Properties and Hypothesis Testing Chapter 3 Properties ad Hypothesis Testig 3.1 Types of data The regressio techiques developed i previous chapters ca be applied to three differet kids of data. 1. Cross-sectioal data. 2. Time series data.

More information

Lecture 3. Properties of Summary Statistics: Sampling Distribution

Lecture 3. Properties of Summary Statistics: Sampling Distribution Lecture 3 Properties of Summary Statistics: Samplig Distributio Mai Theme How ca we use math to justify that our umerical summaries from the sample are good summaries of the populatio? Lecture Summary

More information

5. Fractional Hot deck Imputation

5. Fractional Hot deck Imputation 5. Fractioal Hot deck Imputatio Itroductio Suppose that we are iterested i estimatig θ EY or eve θ 2 P ry < c where y fy x where x is always observed ad y is subject to missigess. Assume MAR i the sese

More information

Stat 319 Theory of Statistics (2) Exercises

Stat 319 Theory of Statistics (2) Exercises Kig Saud Uiversity College of Sciece Statistics ad Operatios Research Departmet Stat 39 Theory of Statistics () Exercises Refereces:. Itroductio to Mathematical Statistics, Sixth Editio, by R. Hogg, J.

More information

1. Parameter estimation point estimation and interval estimation. 2. Hypothesis testing methods to help decision making.

1. Parameter estimation point estimation and interval estimation. 2. Hypothesis testing methods to help decision making. Chapter 7 Parameter Estimatio 7.1 Itroductio Statistical Iferece Statistical iferece helps us i estimatig the characteristics of the etire populatio based upo the data collected from (or the evidece 0produced

More information

STA 260: Statistics and Probability II

STA 260: Statistics and Probability II Al Nosedal. University of Toronto. Winter 2017 1 Properties of Point Estimators and Methods of Estimation 2 3 If you can t explain it simply, you don t understand it well enough Albert Einstein. Definition

More information

NANYANG TECHNOLOGICAL UNIVERSITY SYLLABUS FOR ENTRANCE EXAMINATION FOR INTERNATIONAL STUDENTS AO-LEVEL MATHEMATICS

NANYANG TECHNOLOGICAL UNIVERSITY SYLLABUS FOR ENTRANCE EXAMINATION FOR INTERNATIONAL STUDENTS AO-LEVEL MATHEMATICS NANYANG TECHNOLOGICAL UNIVERSITY SYLLABUS FOR ENTRANCE EXAMINATION FOR INTERNATIONAL STUDENTS AO-LEVEL MATHEMATICS STRUCTURE OF EXAMINATION PAPER. There will be oe 2-hour paper cosistig of 4 questios.

More information

Maximum Likelihood Estimation and Complexity Regularization

Maximum Likelihood Estimation and Complexity Regularization ECE90 Sprig 004 Statistical Regularizatio ad Learig Theory Lecture: 4 Maximum Likelihood Estimatio ad Complexity Regularizatio Lecturer: Rob Nowak Scribe: Pam Limpiti Review : Maximum Likelihood Estimatio

More information

1 of 7 7/16/2009 6:06 AM Virtual Laboratories > 6. Radom Samples > 1 2 3 4 5 6 7 6. Order Statistics Defiitios Suppose agai that we have a basic radom experimet, ad that X is a real-valued radom variable

More information

IE 230 Seat # Name < KEY > Please read these directions. Closed book and notes. 60 minutes.

IE 230 Seat # Name < KEY > Please read these directions. Closed book and notes. 60 minutes. IE 230 Seat # Name < KEY > Please read these directios. Closed book ad otes. 60 miutes. Covers through the ormal distributio, Sectio 4.7 of Motgomery ad Ruger, fourth editio. Cover page ad four pages of

More information

Output Analysis and Run-Length Control

Output Analysis and Run-Length Control IEOR E4703: Mote Carlo Simulatio Columbia Uiversity c 2017 by Marti Haugh Output Aalysis ad Ru-Legth Cotrol I these otes we describe how the Cetral Limit Theorem ca be used to costruct approximate (1 α%

More information