Statistical Theory MT 2009 Problems 1: Solution sketches

Size: px
Start display at page:

Download "Statistical Theory MT 2009 Problems 1: Solution sketches"

Transcription

1 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 α > 0, λ > 0; f(x; α, λ) = λα Γ(α) e λx x α, x > 0, f(x, θ) = e (x θ), x θ. Solutio: (a) where 0 < θ <, ca be writte as f(x, θ) = ( θ)θ x ; x = 0,,,..., (b) (c) f(x, θ) = exp{x log θ + log( θ) + 0}; x = 0,,, so k =, c =, h (x) = x, φ (θ) = log θ, c(θ) = log( θ), d(x) = 0, ad X = {0,,,...} does ot deped o θ, so f is i a -parameter expoetial family. f(x; α, λ) = where α > 0, λ > 0, ca be writte as λα Γ(α) e λx x α, x > 0, f(x; α, λ) = exp { λx + (α ) log(x) + log(λ α /Γ(α)) + 0} so with θ = (α, λ), we ca choose k =, c = c =, h (x) = x, h (x) = log(x), φ (θ) = λ, φ (θ) = α, c(θ) = log(λ α /Γ(α)), d(x) = 0, ad X = (0, ) does ot deped o θ, so f is i a -parameter expoetial family. f(x, θ) = e (x θ), x θ, ca be writte i the expoetial family form with k =, c =, h (x) =, φ (θ) = θ, d(x) = x, but X = [θ, ) depeds o θ, so it is o-regular.

2 . Suppose X, X,..., X is a radom sample from the Pareto distributio f(x, λ) = λαλ x λ+ with x > α, λ > 0, ad α > 0 kow. Fid a miimal sufficiet statistic for λ. Solutio: The likelihood is so T = j= X j is sufficiet. For x, y we have L(λ, x) = (λαλ ) λ α λ = j= xλ+ ( j j= x j), λ+ L(λ, x) L(λ, y) = ( j= y j j= x j ) λ+ is costat i λ if ad oly if t(x) = t(y), so T is miimal sufficiet. 3. Suppose X, X,..., X is a radom sample from the log-ormal distributio with desity { f(x, µ, φ) = x πφ exp } (log x µ) φ with φ > 0 (so that log X j N (µ, φ)). Fid a miimal sufficiet statistic for the parameter θ = (µ, φ). Solutio: L(θ, x) = { ( exp x i )(πφ) / φ } (log x i µ) i= ad the expressio i the expoetial ca be writte as (log xi ) + µ log xi µ φ φ φ, so T = ( log X i, (log X i ) ) is sufficiet. By cosiderig the likelihood ratio we see that T is ideed miimal sufficiet. 4. Suppose X,..., X are idepedet ad expoetially distributed, each with desity fuctio Let X = i= X i ad put f(x; θ) = θ e x/θ, x 0. T = X. i= (X i X) Show that T is a acillary statistic. What does this say about t-tests o expoetial data?

3 Solutio: Note that Y i = θ X i has the expoetial distributio with parameter, so its distributio does ot deped o θ. Next ote that T = Y. i= (Y i Y ) Hece the distributio of T does ot deped o θ either; it is a acillary statistic. Thus a t-test based o expoetial data would ot reveal ay iformatio about the parameter θ. 5. Let X,..., X be i.i.d. uiform U [ θ, θ + ] radom variables. a) Show that ( ) X (), X () is miimal sufficiet for θ. b) Show that (S, A) = ( (X ) () + X () ), X () X () is miimal sufficiet for θ, ad that the distributio of A is idepedet of θ (so A is a acillary statistic). c) Show that ay value i the iterval [ x (), x () + ] is a maximum-likelihood-estimator for θ. Solutio: a) The likelihood is ( L(θ, x) = θ x,..., x θ + ) ( = θ x (), x () θ + ) which is a fuctio of (θ, x (), x () ), so ( X (), X () ) is sufficiet. For miimal sufficiecy, ote that L(θ, x) = L(θ, y) if ad oly if (x (), x () ) = (y (), y () ), provig the first assertio. b) We have that X () = S A X () = S + A so L(θ, x) is a fuctio of (θ, s, a), hece (S, A) is sufficiet, ad (S, A) is a fuctio of a miimal sufficiet statistic, so must be miimal sufficiet itself. To see that the distributio of A is idepedet of θ, write Y i = X i θ, the Y i U( /, /), ad Y (i) = X (i) θ. Hece A = X () X () = Y () Y (), the differece of order statistics from a distributio which does ot ivolve θ. So the distributio of A does ot deped o θ. c) Follows directly from the likelihood - the likelihood ( L(θ, x) = x () θ x () + ) is costat equal to o the iterval [ x (), x () + ], ad 0 otherwise. 6. The radom variables X,..., X are idepedet with geometric distributio P(X i = x) = p( p) x for x =,,.... Let θ = p. 3

4 (i) Fid the maximum likelihood estimator for p. Cosiderig =, is it ubiased? (ii) Show that ˆθ = X is the maximum likelihood estimator for θ. Is it ubiased? (iii) Compute the expected Fisher iformatio for θ. (iv) Does ˆθ attai the Cramer-Rao lower boud? Solutio: a) We have that L(p, x) = p ( p) xi i= ad so ( ) l(p) = log p + x i log( p); i= differetiate: l (p) = p i= x i ; p the oly solutio for l (p) = 0 is p = x. The secod derivative is l (θ) = p i= x i ( p) < 0, so that ˆp = x For =, is the mle. ( ) E X > p, = p( p)x x x= = p + p( p)x x x= so ˆp is ot ubiased. b) We have that L(θ, x) = θ ( θ ) xi i= ad so l(θ) = log θ + (x i ) log( θ ); i= 4

5 differetiate: l (θ) = (x θ); θ(θ ) the oly solutio for l (θ) = 0 is θ = x. The secod derivative is so that l (θ) = { θ(θ ) (x θ)(θ )}; (θ(θ )) l (ˆθ) = if ˆθ = x >, so if ˆθ >, the mle is ˆθ = x. (ˆθ(ˆθ )) ( ˆθ(ˆθ )) < 0 If x =, the L(θ, x) = θ, ad we kow that θ, so the likelihood is maximized for θ =. c) Calculate that E θ X = θ, V ar θ (X) = θ(θ ), so, from l (θ), I(θ) = = θ (θ ) V ar θ(x) θ(θ ), which equals the iverse of the variace of X. So it attais the Cramer-Rao lower boud. Note: As E θ X = θ, the estimator is ubiased, ad we have see that it attais the Cramer-Rao lower boud, hece it has miimum variace amog all ubiased estimators. 7. Let X,..., X be i.i.d. U[0, θ], havig desity f(x; θ) = θ, 0 x θ with θ > 0. (i) Estimate θ usig both the method of momets ad maximum likelihood. (ii) Calculate the meas ad variaces of the two estimators. (iii) Which oe should be preferred ad why? Solutio: a) We have E θ (X) = θ, so the m.o.m. is θ = X. The likelihood is L(θ, x) = θ (0 x,..., x θ) = θ ( x () θ ) 5

6 ad this is maximized for ˆθ = x (). b) We have by costructio E θ ( θ) = θ, ad V ar θ ( θ) = 4θ = θ 3. For ˆθ, calculate (see secod year probability, or the book by Rice) E θ (ˆθ) = + θ, V ar θ(ˆθ) = ( + ) ( + ) θ. c) The m.l.e. is ot ubiased but asymptotically ubiased, ad has much smaller variace tha the m.o.m., so I would prefer it. You may decide otherwise if you put more emphasis o ubiasedess. 8. Suppose X,..., X are a radom sample with mea µ ad fiite variace σ. Use the delta method to show that, i distributio, (X µ ) N (0, 4µ σ ). What would you suggest if µ = 0? Solutio: From the cetral limit theorem we kow that (X µ) N (0, σ ). We use the fuctio g(s) = s, so that g (s) = s; ow the delta method gives that Stadardizig gives the result. X N (µ, 4µ σ /). If µ = 0 the the result just gives covergece to poit mass at 0, which is ot iformative. Istead we could use that X σ N (0, ), so that X σ X σ χ. χ, ad, i distributio, 6

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

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

Homework for 2/3. 1. Determine the values of the following quantities: a. t 0.1,15 b. t 0.05,15 c. t 0.1,25 d. t 0.05,40 e. t 0.

Homework for 2/3. 1. Determine the values of the following quantities: a. t 0.1,15 b. t 0.05,15 c. t 0.1,25 d. t 0.05,40 e. t 0. Name: ID: Homework for /3. Determie the values of the followig quatities: a. t 0.5 b. t 0.055 c. t 0.5 d. t 0.0540 e. t 0.00540 f. χ 0.0 g. χ 0.0 h. χ 0.00 i. χ 0.0050 j. χ 0.990 a. t 0.5.34 b. t 0.055.753

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

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

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

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

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

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

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

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

APPM 5720 Solutions to Problem Set Five

APPM 5720 Solutions to Problem Set Five APPM 5720 Solutios to Problem Set Five 1. The pdf is f(x; α, β) 1 Γ(α) βα x α 1 e βx I (0, ) (x). The joit pdf is f( x; α, β) 1 [Γ(α)] β α ( i1 x i ) α 1 e β i1 x i i1 I (0, ) (x i ) 1 [Γ(α)] βα } {{ }

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

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

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

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

Lecture 6 Ecient estimators. Rao-Cramer bound.

Lecture 6 Ecient estimators. Rao-Cramer bound. Lecture 6 Eciet estimators. Rao-Cramer boud. 1 MSE ad Suciecy Let X (X 1,..., X) be a radom sample from distributio f θ. Let θ ˆ δ(x) be a estimator of θ. Let T (X) be a suciet statistic for θ. As we have

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

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

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

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

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

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

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

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 Stat 366 Lab 2 Solutios (September 2, 2006) page TA: Yury Petracheko, CAB 484, yuryp@ualberta.ca, http://www.ualberta.ca/ yuryp/ Review Questios, Chapters 8, 9 8.5 Suppose that Y, Y 2,..., Y deote a radom

More information

This section is optional.

This section is optional. 4 Momet Geeratig Fuctios* This sectio is optioal. The momet geeratig fuctio g : R R of a radom variable X is defied as g(t) = E[e tx ]. Propositio 1. We have g () (0) = E[X ] for = 1, 2,... Proof. Therefore

More information

Introductory statistics

Introductory statistics CM9S: Machie Learig for Bioiformatics Lecture - 03/3/06 Itroductory statistics Lecturer: Sriram Sakararama Scribe: Sriram Sakararama We will provide a overview of statistical iferece focussig o the key

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

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

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

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

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

MASSACHUSETTS INSTITUTE OF TECHNOLOGY 6.436J/15.085J Fall 2008 Lecture 19 11/17/2008 LAWS OF LARGE NUMBERS II THE STRONG LAW OF LARGE NUMBERS

MASSACHUSETTS INSTITUTE OF TECHNOLOGY 6.436J/15.085J Fall 2008 Lecture 19 11/17/2008 LAWS OF LARGE NUMBERS II THE STRONG LAW OF LARGE NUMBERS MASSACHUSTTS INSTITUT OF TCHNOLOGY 6.436J/5.085J Fall 2008 Lecture 9 /7/2008 LAWS OF LARG NUMBRS II Cotets. The strog law of large umbers 2. The Cheroff boud TH STRONG LAW OF LARG NUMBRS While the weak

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

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

Summary. Recap ... Last Lecture. Summary. Theorem

Summary. Recap ... Last Lecture. Summary. Theorem Last Lecture Biostatistics 602 - Statistical Iferece Lecture 23 Hyu Mi Kag April 11th, 2013 What is p-value? What is the advatage of p-value compared to hypothesis testig procedure with size α? How ca

More information

2. The volume of the solid of revolution generated by revolving the area bounded by the

2. The volume of the solid of revolution generated by revolving the area bounded by the IIT JAM Mathematical Statistics (MS) Solved Paper. A eigevector of the matrix M= ( ) is (a) ( ) (b) ( ) (c) ( ) (d) ( ) Solutio: (a) Eigevalue of M = ( ) is. x So, let x = ( y) be the eigevector. z (M

More information

Asymptotics. Hypothesis Testing UMP. Asymptotic Tests and p-values

Asymptotics. Hypothesis Testing UMP. Asymptotic Tests and p-values of the secod half Biostatistics 6 - Statistical Iferece Lecture 6 Fial Exam & Practice Problems for the Fial Hyu Mi Kag Apil 3rd, 3 Hyu Mi Kag Biostatistics 6 - Lecture 6 Apil 3rd, 3 / 3 Rao-Blackwell

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

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

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

( 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

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

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

Lecture 23: Minimal sufficiency

Lecture 23: Minimal sufficiency Lecture 23: Miimal sufficiecy Maximal reductio without loss of iformatio There are may sufficiet statistics for a give problem. I fact, X (the whole data set) is sufficiet. If T is a sufficiet statistic

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

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

Recap! Good statistics, cont.! Sufficiency! What are good statistics?! 2/20/14

Recap! Good statistics, cont.! Sufficiency! What are good statistics?! 2/20/14 Recap Cramér-Rao iequality Best ubiased estimators What are good statistics? Parameter: ukow umber that we are tryig to get a idea about usig a sample X 1,,X Statistic: A fuctio of the sample. It is a

More information

Limit Theorems. Convergence in Probability. Let X be the number of heads observed in n tosses. Then, E[X] = np and Var[X] = np(1-p).

Limit Theorems. Convergence in Probability. Let X be the number of heads observed in n tosses. Then, E[X] = np and Var[X] = np(1-p). Limit Theorems Covergece i Probability Let X be the umber of heads observed i tosses. The, E[X] = p ad Var[X] = p(-p). L O This P x p NM QP P x p should be close to uity for large if our ituitio is correct.

More information

Lecture Notes 15 Hypothesis Testing (Chapter 10)

Lecture Notes 15 Hypothesis Testing (Chapter 10) 1 Itroductio Lecture Notes 15 Hypothesis Testig Chapter 10) Let X 1,..., X p θ x). Suppose we we wat to kow if θ = θ 0 or ot, where θ 0 is a specific value of θ. For example, if we are flippig a coi, we

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

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

TAMS24: Notations and Formulas

TAMS24: Notations and Formulas TAMS4: Notatios ad Formulas Basic otatios ad defiitios X: radom variable stokastiska variabel Mea Vätevärde: µ = X = by Xiagfeg Yag kpx k, if X is discrete, xf Xxdx, if X is cotiuous Variace Varias: =

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

Last Lecture. Unbiased Test

Last Lecture. Unbiased Test Last Lecture Biostatistics 6 - Statistical Iferece Lecture Uiformly Most Powerful Test Hyu Mi Kag March 8th, 3 What are the typical steps for costructig a likelihood ratio test? Is LRT statistic based

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

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

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

Final Examination Statistics 200C. T. Ferguson June 10, 2010

Final Examination Statistics 200C. T. Ferguson June 10, 2010 Fial Examiatio Statistics 00C T. Ferguso Jue 0, 00. (a State the Borel-Catelli Lemma ad its coverse. (b Let X,X,... be i.i.d. from a distributio with desity, f(x =θx (θ+ o the iterval (,. For what value

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

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

Lecture 16: UMVUE: conditioning on sufficient and complete statistics

Lecture 16: UMVUE: conditioning on sufficient and complete statistics Lecture 16: UMVUE: coditioig o sufficiet ad complete statistics The 2d method of derivig a UMVUE whe a sufficiet ad complete statistic is available Fid a ubiased estimator of ϑ, say U(X. Coditioig o a

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

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

Point Estimation: properties of estimators 1 FINITE-SAMPLE PROPERTIES. finite-sample properties (CB 7.3) large-sample properties (CB 10.

Point Estimation: properties of estimators 1 FINITE-SAMPLE PROPERTIES. finite-sample properties (CB 7.3) large-sample properties (CB 10. Poit Estimatio: properties of estimators fiite-sample properties CB 7.3) large-sample properties CB 10.1) 1 FINITE-SAMPLE PROPERTIES How a estimator performs for fiite umber of observatios. Estimator:

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

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

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

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

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

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

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

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

Summary. Recap. Last Lecture. Let W n = W n (X 1,, X n ) = W n (X) be a sequence of estimators for

Summary. Recap. Last Lecture. Let W n = W n (X 1,, X n ) = W n (X) be a sequence of estimators for Last Lecture Biostatistics 602 - Statistical Iferece Lecture 17 Asymptotic Evaluatio of oit Estimators Hyu Mi Kag March 19th, 2013 What is a Bayes Risk? What is the Bayes rule Estimator miimizig square

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

Mathmatical Statisticals

Mathmatical Statisticals Mathmatical Statisticals Che, L.-A. Chapter 4. Distributio of Fuctio of Radom variables Sample space S : set of possible outcome i a experimet. Probability set fuctio P: ()P (A) 0, A S. ()P (S) =. (3)P

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

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

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

Probability and Statistics

Probability and Statistics ICME Refresher Course: robability ad Statistics Staford Uiversity robability ad Statistics Luyag Che September 20, 2016 1 Basic robability Theory 11 robability Spaces A probability space is a triple (Ω,

More information

Lecture 3: MLE and Regression

Lecture 3: MLE and Regression STAT/Q SCI 403: Itroductio to Resamplig Methods Sprig 207 Istructor: Ye-Chi Che Lecture 3: MLE ad Regressio 3. Parameters ad Distributios Some distributios are idexed by their uderlyig parameters. Thus,

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

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

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

LECTURE 8: ASYMPTOTICS I

LECTURE 8: ASYMPTOTICS I LECTURE 8: ASYMPTOTICS I We are iterested i the properties of estimators as. Cosider a sequece of radom variables {, X 1}. N. M. Kiefer, Corell Uiversity, Ecoomics 60 1 Defiitio: (Weak covergece) A sequece

More information

Probability and statistics: basic terms

Probability and statistics: basic terms Probability ad statistics: basic terms M. Veeraraghava August 203 A radom variable is a rule that assigs a umerical value to each possible outcome of a experimet. Outcomes of a experimet form the sample

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

Monte Carlo Integration

Monte Carlo Integration Mote Carlo Itegratio I these otes we first review basic umerical itegratio methods (usig Riema approximatio ad the trapezoidal rule) ad their limitatios for evaluatig multidimesioal itegrals. Next we itroduce

More information

1 Introduction to reducing variance in Monte Carlo simulations

1 Introduction to reducing variance in Monte Carlo simulations Copyright c 010 by Karl Sigma 1 Itroductio to reducig variace i Mote Carlo simulatios 11 Review of cofidece itervals for estimatig a mea I statistics, we estimate a ukow mea µ = E(X) of a distributio by

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

G. R. Pasha Department of Statistics Bahauddin Zakariya University Multan, Pakistan

G. R. Pasha Department of Statistics Bahauddin Zakariya University Multan, Pakistan Deviatio of the Variaces of Classical Estimators ad Negative Iteger Momet Estimator from Miimum Variace Boud with Referece to Maxwell Distributio G. R. Pasha Departmet of Statistics Bahauddi Zakariya Uiversity

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 01: the Central Limit Theorem. 1 Central Limit Theorem for i.i.d. random variables

Lecture 01: the Central Limit Theorem. 1 Central Limit Theorem for i.i.d. random variables CSCI-B609: A Theorist s Toolkit, Fall 06 Aug 3 Lecture 0: the Cetral Limit Theorem Lecturer: Yua Zhou Scribe: Yua Xie & Yua Zhou Cetral Limit Theorem for iid radom variables Let us say that we wat to aalyze

More information

Department of Mathematics

Department of Mathematics Departmet of Mathematics Ma 3/103 KC Border Itroductio to Probability ad Statistics Witer 2017 Lecture 19: Estimatio II Relevat textbook passages: Larse Marx [1]: Sectios 5.2 5.7 19.1 The method of momets

More information

Large Sample Theory. Convergence. Central Limit Theorems Asymptotic Distribution Delta Method. Convergence in Probability Convergence in Distribution

Large Sample Theory. Convergence. Central Limit Theorems Asymptotic Distribution Delta Method. Convergence in Probability Convergence in Distribution Large Sample Theory Covergece Covergece i Probability Covergece i Distributio Cetral Limit Theorems Asymptotic Distributio Delta Method Covergece i Probability A sequece of radom scalars {z } = (z 1,z,

More information

f(x i ; ) L(x; p) = i=1 To estimate the value of that maximizes L or equivalently ln L we will set =0, for i =1, 2,...,m p x i (1 p) 1 x i i=1

f(x i ; ) L(x; p) = i=1 To estimate the value of that maximizes L or equivalently ln L we will set =0, for i =1, 2,...,m p x i (1 p) 1 x i i=1 Parameter Estimatio Samples from a probability distributio F () are: [,,..., ] T.Theprobabilitydistributio has a parameter vector [,,..., m ] T. Estimator: Statistic used to estimate ukow. Estimate: Observed

More information

Chapter 13: Tests of Hypothesis Section 13.1 Introduction

Chapter 13: Tests of Hypothesis Section 13.1 Introduction Chapter 13: Tests of Hypothesis Sectio 13.1 Itroductio RECAP: Chapter 1 discussed the Likelihood Ratio Method as a geeral approach to fid good test procedures. Testig for the Normal Mea Example, discussed

More information

Exercise 4.3 Use the Continuity Theorem to prove the Cramér-Wold Theorem, Theorem. (1) φ a X(1).

Exercise 4.3 Use the Continuity Theorem to prove the Cramér-Wold Theorem, Theorem. (1) φ a X(1). Assigmet 7 Exercise 4.3 Use the Cotiuity Theorem to prove the Cramér-Wold Theorem, Theorem 4.12. Hit: a X d a X implies that φ a X (1) φ a X(1). Sketch of solutio: As we poited out i class, the oly tricky

More information

2.2. Central limit theorem.

2.2. Central limit theorem. 36.. Cetral limit theorem. The most ideal case of the CLT is that the radom variables are iid with fiite variace. Although it is a special case of the more geeral Lideberg-Feller CLT, it is most stadard

More information