Asymptotics. Hypothesis Testing UMP. Asymptotic Tests and p-values
|
|
- Logan Manning
- 5 years ago
- Views:
Transcription
1 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 : If WX is a ubiased estimator of τθ, ϕt E[WX T] is a better ubiased estimator for a sufficiet statistic Uiqueess of MVUE : Theorem Best ubiased estimator is uique MVUE ad UE of zeros : Theorem 73 - Best ubiased estimator is ucorrelated with ay ubiased estimators of zero UMVE by complete sufficiet statistics : Theorem Ay fuctio of complete sufficiet statistic is the best ubiased estimator for its expected value How to get UMVUE Strategies to obtai best ubiased estimators: Coditio a simple ubiased estimator o complete sufficiet statistics Come up with a fuctio of sufficiet statistic whose expected value is τθ Hyu Mi Kag Biostatistics 6 - Lecture 6 Apil 3rd, 3 / 3 Bayesia Framework Bayesia Decisio Theory Prior distributio πθ Samplig distributio x θ f X x θ Joit distributio πθfx θ Margial distributio mx πθfx θdθ Posterior distributio πθ x f Xx θπθ mx Bayes Estimator is a posterior mea of θ : E[θ x] Loss Fuctio Lθ, ˆθ eg θ ˆθ Risk Fuctio is the average loss : Rθ, ˆθ E[Lθ, ˆθ θ] For squared error loss L θ ˆθ, the risk fuctio is MSE Bayes Risk is the average risk across all θ : E[Rθ, ˆθ πθ] Bayes Rule Estimator miimizes Bayes risk miimizes posterior expected loss Hyu Mi Kag Biostatistics 6 - Lecture 6 Apil 3rd, 3 3 / 3 Hyu Mi Kag Biostatistics 6 - Lecture 6 Apil 3rd, 3 4 / 3
2 Asymptotics Hypothesis Testig Cosistecy Usig law of large umbers, show variace ad bias coverges to zero, for ay cotiuous mappig fuctio τ Asymptotic Normality Usig cetral limit theorem, Slutsky Theorem, ad Delta Method Asymptotic Relative Efficiecy AREV, W σ W /σ V Asymptotically Efficiet ARE with CR-boud of ubiased estimator of τθ is Asymptotic Efficiecy of MLE Theorem MLE is always asymptotically efficiet uder regularity coditio Type I error PrX R θ whe θ Ω Type II error PrX R θ whe θ Ω c Power fuctio βθ PrX R θ βθ represets Type I error uder H, ad power -Type II error uder H Size α test sup θ Ω βθ α Level α test sup θ Ω βθ α LRT λx Lˆθ x Lˆθ x rejects H whe λx c log λx log c c LRT based o sufficiet statistics LRT based o full data ad sufficiet statistics are idetical Hyu Mi Kag Biostatistics 6 - Lecture 6 Apil 3rd, 3 5 / 3 Hyu Mi Kag Biostatistics 6 - Lecture 6 Apil 3rd, 3 6 / 3 UMP Asymptotic Tests ad p-values Ubiased Test βθ βθ for every θ Ω c ad θ Ω UMP Test βθ β θ for every θ Ω c ad β θ of every other test with a class of test C UMP level α Test UMP test i the class of all the level α test smallest Type II error give the upper boud of Type I error Neyma-Pearso For H : θ θ vs H : θ θ, a test with rejectio regio fx θ /fx θ > k is a UMP level α test for its size MLR gt θ /gt θ is a icreasig fuctio of t for every θ > θ Karli-Rabi If T is sufficiet ad has MLR, the test rejectig R {T : T > t } or R {T : T < t } is a UMP level α test for oe-sided composite hypothesis Hyu Mi Kag Biostatistics 6 - Lecture 6 Apil 3rd, 3 7 / 3 Asymptotic Distributio of LRT For testig, H : θ θ vs H : θ θ, d log λx χ uder regularity coditio Wald Test If W is a cosistet estimator of θ, ad S is a cosistet estimator of VarW, the Z W θ /S follows a stadard ormal distributio Two-sided test : Z > z α/ Oe-sided test : Z > z α/ or Z < z α/ p-value A p-value px is valid if, PrpX α θ α for every θ Ω ad α Costructig p-value Theorem 837 : If large WX value gives evidece that H is true, px sup θ Ω PrWX Wx θ is a valid p-value p-value give sufficiet statistics For a sufficiet statistic SX, px PrWX Wx SX Sx is also a valid p-value Hyu Mi Kag Biostatistics 6 - Lecture 6 Apil 3rd, 3 8 / 3
3 Iterval Estimatio Practice Problem cotiued from last week Coverage probability Prθ [LX, UX] Coverage coefficiet is α if if θ Ω Prθ [LX, UX] α Cofidece iterval [LX, UX] is α if if θ Ω Prθ [LX, UX] α Ivertig a level α test If Aθ is the acceptace regio of a level α test, the CX {θ : X Aθ} is a α cofidece set or iterval Problem Let fx θ be the logistic locatio pdf fx θ e x θ + e x θ < x <, < θ < a Show that this family has a MLR b Based o oe observatio X, fid the most powerful size α test of H : θ versus H : θ c Show that the test i part b is UMP size α for testig H : θ vs H : θ > Hyu Mi Kag Biostatistics 6 - Lecture 6 Apil 3rd, 3 9 / 3 Hyu Mi Kag Biostatistics 6 - Lecture 6 Apil 3rd, 3 / 3 Solutio for a Solutio for b For θ < θ, fx θ fx θ Let rx + e x θ / + e x θ e x θ +e x θ e x θ +e x θ e θ θ + e x θ + e x θ r x ex θ + e x θ + e x θ e x θ + e x θ ex θ e x θ + e x θ > x θ > x θ Therefore, the family of X has a MLR Hyu Mi Kag Biostatistics 6 - Lecture 6 Apil 3rd, 3 / 3 The UMP test rejects H if ad oly if fx + e x e fx + e x > k + e x + e x > k + e x e + e x > k X > x Because uder H, Fx θ ex +e, the rejectio regio of UMP level x α test satisfies Fx θ + e x α α x log α Hyu Mi Kag Biostatistics 6 - Lecture 6 Apil 3rd, 3 / 3
4 Solutio for c Practice Problem Because the family of X has a MLR, UMP size α for testig H : θ vs H : θ > should be a form of X > x PrX > x θ α Therefore, x log α α, which is idetical to the test defied i b Problem Suppose X,, X are iid radom samples with pdf f X x θ θ exp θx, where x, θ > a Show that b Show that distributio x X i x X i is a cosistet estimator for θ is asymptotically ormal ad derive its asymptotic c Derive the Wald asymptotic size α test for H : θ θ vs H : θ θ d Fid a asymptotic α cofidece iterval for θ by ivertig the above test You may use the fact that EX /θ ad VarX /θ Hyu Mi Kag Biostatistics 6 - Lecture 6 Apil 3rd, 3 3 / 3 Hyu Mi Kag Biostatistics 6 - Lecture 6 Apil 3rd, 3 4 / 3 Solutio a - Cosistecy Solutio b - Asymptotic Distributio Obtai EX /θ Derive yourself if ot give EX xfx θdx θx exp θxdx [ x exp θx] + exp θxdx [ + ] θ exp θx θ By LLN Law of Large Number, X P EX /θ 3 By Theorem of cotiuous map, / i X i /X P θ Obtai VarX /θ Derive if eeded, omitted here Apply CLTCetral Limit Theorem, X AN θ, θ 3 Apply Delta method Let gy /y, the g y /y Xi /X gx AN g/θ, [g /θ] θ AN θ, θ X θ N, θ Hyu Mi Kag Biostatistics 6 - Lecture 6 Apil 3rd, 3 5 / 3 Hyu Mi Kag Biostatistics 6 - Lecture 6 Apil 3rd, 3 6 / 3
5 Solutio c - Wald asymptotic size α test Solutio c - Wald Asymptotic size α test cot d Obtai a cosistet estimator of θ : i WX X i Obtai a costat estimator of VarW S AN X i X P VarX θ i i X i X i X i X P θ P θ θ, θ CLT Cotiuous Map Theorem Slutsky s Theorem 3 Costruct a two-sided asymptotic size α Wald test, whose rejectio regio is ZX WX θ S/ i X θ i i X i X X θ X i X z α/ i Hyu Mi Kag Biostatistics 6 - Lecture 6 Apil 3rd, 3 7 / 3 Hyu Mi Kag Biostatistics 6 - Lecture 6 Apil 3rd, 3 8 / 3 Solutio d - Asymptotic α cofidece iterval Practice Problem 3 The acceptace regio is A x : x θ x i x z α/ By ivertig the acceptace regio, the cofidece iterval is CX θ : X θ X i X z α/ which is equivalet to CX θ i i X z α/ i X i X, X + z α/ i X i X Problem The idepedet radom variables X,, X have the followig pdf fx θ, β βxβ θ β < x < θ, β > Fid the MLEs of β ad θ Whe β is a kow costat β, costruct a LRT testig H : θ θ vs H : θ < θ 3 Whe β is a kow costat β, fid the upper cofidece limit for θ with cofidece coefficiet α Hyu Mi Kag Biostatistics 6 - Lecture 6 Apil 3rd, 3 9 / 3 Hyu Mi Kag Biostatistics 6 - Lecture 6 Apil 3rd, 3 / 3
6 a - MLE b - LRT Lθ, β x β i x i β θ β Ix θ Because L is a decreasig fuctio of θ ad positive oly whe θ x ˆθ x lθ, β x log β + β log x i β log θ l β β + log x i log θ ˆβ log ˆθ log x i x log x i λx sup θ Ω Lˆθ x sup θ Ω Lˆθ x { θ < x Lθ x Lx x θ x θ < x x θ c x β θ β θ x c Hyu Mi Kag Biostatistics 6 - Lecture 6 Apil 3rd, 3 / 3 Hyu Mi Kag Biostatistics 6 - Lecture 6 Apil 3rd, 3 / 3 b - size α LRT c - Upper α cofidece limit x α Pr θ c β c α β c Therefore, the rejectio regio for size α LRT is is } R {x : x θ α β The acceptace regio of size α LRT is } Aθ {x : x > θ α β By isertig the acceptace regio, the α cofidece iterval becomes } CX {θ : X > θα β } {θ : θ < X α β Therefore, the upper α cofidece limit is X α β Hyu Mi Kag Biostatistics 6 - Lecture 6 Apil 3rd, 3 3 / 3 Hyu Mi Kag Biostatistics 6 - Lecture 6 Apil 3rd, 3 4 / 3
7 Practice Problem 4 a - MLE of θ Problem A radom sample X,, X is draw from a populatio N θ, θ where θ > a Fid the ˆθ, the MLE of θ b Fid the asymptotic distributio of ˆθ c Compute AREˆθ, X Determie whether ˆθ is asymptotically more efficiet tha X or ot You may use the followig fact: VarX 4θ 3 + θ Hyu Mi Kag Biostatistics 6 - Lecture 6 Apil 3rd, 3 5 / 3 [ Lθ x πθ / i exp x i θ ] θ lθ x logπ + log θ i x i θ θ logπ + x log θ i θ + x i θ l θ x x θ + i θ θ x i θ θ θ + θ x i ˆθ x i / x i ˆθ + ˆθ Hyu Mi Kag Biostatistics 6 - Lecture 6 Apil 3rd, 3 6 / 3 b - Asymptotic distributio of MLE b - Asymptotic distributio of MLE cot d By CLT, Let W X i, the W AN EX, VarX The asymptotic distributio of MLE ˆθ ˆθ AN θ, σ θ AN θ + θ, 4θ3 + θ for some fuctio σ θ ad we would like to fid σ θ usig the asymptotic distributio of W Let gy y + y, the g y y + ad gˆθ W The by the Delta Method, the asymptotic distributio of W ca be writte as W gˆθ AN gθ, g θ σ θ AN θ + θ, θ + σ θ AN θ + θ, 4θ3 + θ σ θ 4θ3 + θ θ + θ θ + θ + θ θ + Hyu Mi Kag Biostatistics 6 - Lecture 6 Apil 3rd, 3 7 / 3 Hyu Mi Kag Biostatistics 6 - Lecture 6 Apil 3rd, 3 8 / 3
8 b - Asymptotic distributio of MLE cot d c - ARE of MLE compared to X The asymptotic distributio of MLE ˆθ ˆθ AN θ, σ θ θ AN θ, θ + Note that you caot use CR-boud for the asymptotic variace of MLE because the regularity coditio does ot hold ope set criteria By CLT, the asymptotic distributio of X is X AN θ, θ The, AREˆθ, X is AREˆθ, X θ θ θ+ θ + + θ θ > Therefore, ˆθ is more efficiet estimator tha X Hyu Mi Kag Biostatistics 6 - Lecture 6 Apil 3rd, 3 9 / 3 Hyu Mi Kag Biostatistics 6 - Lecture 6 Apil 3rd, 3 3 / 3 Wrappig Up May thaks for your attetios ad feedbacks Please complete your teachig evaluatios, which will be very helpful for further improvemet i the ext year 3 Fial exam will be Thursday April 5th, 4:-6:pm 4 The last office hour will be held Wedesday April 4th, 4:-5:pm 5 The grade will be posted durig the weeked 6 Do t forget the materials we have leared, because they are the key topics for your cadidacy exam Hyu Mi Kag Biostatistics 6 - Lecture 6 Apil 3rd, 3 3 / 3
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 informationLast 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 informationSummary. 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 informationLast Lecture. Biostatistics Statistical Inference Lecture 16 Evaluation of Bayes Estimator. Recap - Example. Recap - Bayes Estimator
Last Lecture Biostatistics 60 - Statistical Iferece Lecture 16 Evaluatio of Bayes Estimator Hyu Mi Kag March 14th, 013 What is a Bayes Estimator? Is a Bayes Estimator the best ubiased estimator? Compared
More informationLast 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 informationIntroductory 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 informationEcon 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 informationLecture 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 informationDirection: 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 informationTopic 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 informationLECTURE 14 NOTES. A sequence of α-level tests {ϕ n (x)} is consistent if
LECTURE 14 NOTES 1. Asymptotic power of tests. Defiitio 1.1. A sequece of -level tests {ϕ x)} is cosistet if β θ) := E θ [ ϕ x) ] 1 as, for ay θ Θ 1. Just like cosistecy of a sequece of estimators, Defiitio
More informationLecture 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 informationProblem 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 informationTopic 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 informationTopic 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 informationLecture 33: Bootstrap
Lecture 33: ootstrap Motivatio To evaluate ad compare differet estimators, we eed cosistet estimators of variaces or asymptotic variaces of estimators. This is also importat for hypothesis testig ad cofidece
More informationLecture 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 informationMATH 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 information2. 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 informationSTATISTICAL 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 informationStat410 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 informationStatistical 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 informationStat 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 informationIntroduction to Econometrics (3 rd Updated Edition) Solutions to Odd- Numbered End- of- Chapter Exercises: Chapter 3
Itroductio to Ecoometrics (3 rd Updated Editio) by James H. Stock ad Mark W. Watso Solutios to Odd- Numbered Ed- of- Chapter Exercises: Chapter 3 (This versio August 17, 014) 015 Pearso Educatio, Ic. Stock/Watso
More informationDirection: 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 informationExam II Review. CEE 3710 November 15, /16/2017. EXAM II Friday, November 17, in class. Open book and open notes.
Exam II Review CEE 3710 November 15, 017 EXAM II Friday, November 17, i class. Ope book ad ope otes. Focus o material covered i Homeworks #5 #8, Note Packets #10 19 1 Exam II Topics **Will emphasize material
More informationRandom 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 informationUnbiased 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 informationThis 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 informationStatistical 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 information5. 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 informationLecture 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 informationLecture 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 informationThis 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 information1.010 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 informationStatistical 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 informationMATH 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 informationResampling 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( 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 informationEstimation 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 informationx = 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 informationParameter, Statistic and Random Samples
Parameter, Statistic ad Radom Samples A parameter is a umber that describes the populatio. It is a fixed umber, but i practice we do ot kow its value. A statistic is a fuctio of the sample data, i.e.,
More informationof 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 informationFirst 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 informationProbability 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 informationLecture 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 informationLecture 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 informationEfficient GMM LECTURE 12 GMM II
DECEMBER 1 010 LECTURE 1 II Efficiet The estimator depeds o the choice of the weight matrix A. The efficiet estimator is the oe that has the smallest asymptotic variace amog all estimators defied by differet
More informationIIT 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 informationEECS564 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 informationExponential 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 informationAn Introduction to Asymptotic Theory
A Itroductio to Asymptotic Theory Pig Yu School of Ecoomics ad Fiace The Uiversity of Hog Kog Pig Yu (HKU) Asymptotic Theory 1 / 20 Five Weapos i Asymptotic Theory Five Weapos i Asymptotic Theory Pig Yu
More information10-701/ 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 informationMOST PEOPLE WOULD RATHER LIVE WITH A PROBLEM THEY CAN'T SOLVE, THAN ACCEPT A SOLUTION THEY CAN'T UNDERSTAND.
XI-1 (1074) MOST PEOPLE WOULD RATHER LIVE WITH A PROBLEM THEY CAN'T SOLVE, THAN ACCEPT A SOLUTION THEY CAN'T UNDERSTAND. R. E. D. WOOLSEY AND H. S. SWANSON XI-2 (1075) STATISTICAL DECISION MAKING Advaced
More informationComposite Hypotheses
Composite Hypotheses March 25-27, 28 For a composite hypothesis, the parameter space Θ is divided ito two disjoit regios, Θ ad Θ 1. The test is writte H : Θ versus H 1 : Θ 1 with H is called the ull hypothesis
More informationSample Size Determination (Two or More Samples)
Sample Sie Determiatio (Two or More Samples) STATGRAPHICS Rev. 963 Summary... Data Iput... Aalysis Summary... 5 Power Curve... 5 Calculatios... 6 Summary This procedure determies a suitable sample sie
More informationSimulation. 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 informationLarge 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 informationStatistical inference: example 1. Inferential Statistics
Statistical iferece: example 1 Iferetial Statistics POPULATION SAMPLE A clothig store chai regularly buys from a supplier large quatities of a certai piece of clothig. Each item ca be classified either
More informationStat 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 informationChapter 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 informationKurskod: 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 informationFrequentist 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 informationTAMS24: 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 informationHomework 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 informationSTAT Homework 7 - Solutions
STAT-36700 Homework 7 - Solutios Fall 208 October 28, 208 This cotais solutios for Homework 7. Please ote that we have icluded several additioal commets ad approaches to the problems to give you better
More informationSDS 321: Introduction to Probability and Statistics
SDS 321: Itroductio to Probability ad Statistics Lecture 23: Cotiuous radom variables- Iequalities, CLT Puramrita Sarkar Departmet of Statistics ad Data Sciece The Uiversity of Texas at Austi www.cs.cmu.edu/
More informationProperties 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 information1. 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 informationSince 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 informationLecture 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 informationMBACATÓLICA. Quantitative Methods. Faculdade de Ciências Económicas e Empresariais UNIVERSIDADE CATÓLICA PORTUGUESA 9. SAMPLING DISTRIBUTIONS
MBACATÓLICA Quatitative Methods Miguel Gouveia Mauel Leite Moteiro Faculdade de Ciêcias Ecoómicas e Empresariais UNIVERSIDADE CATÓLICA PORTUGUESA 9. SAMPLING DISTRIBUTIONS MBACatólica 006/07 Métodos Quatitativos
More informationLogit regression Logit regression
Logit regressio Logit regressio models the probability of Y= as the cumulative stadard logistic distributio fuctio, evaluated at z = β 0 + β X: Pr(Y = X) = F(β 0 + β X) F is the cumulative logistic distributio
More informationSolutions: 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 informationPoint 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 informationNovember 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 informationLet 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 informationLecture 8: Convergence of transformations and law of large numbers
Lecture 8: Covergece of trasformatios ad law of large umbers Trasformatio ad covergece Trasformatio is a importat tool i statistics. If X coverges to X i some sese, we ofte eed to check whether g(x ) coverges
More informationEXAMINATIONS 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 informationMathematical 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 informationMachine 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 informationThe variance of a sum of independent variables is the sum of their variances, since covariances are zero. Therefore. V (xi )= n n 2 σ2 = σ2.
SAMPLE STATISTICS A radom sample x 1,x,,x from a distributio f(x) is a set of idepedetly ad idetically variables with x i f(x) for all i Their joit pdf is f(x 1,x,,x )=f(x 1 )f(x ) f(x )= f(x i ) The sample
More informationSTA 4032 Final Exam Formula Sheet
Chapter 2. Probability STA 4032 Fial Eam Formula Sheet Some Baic Probability Formula: (1) P (A B) = P (A) + P (B) P (A B). (2) P (A ) = 1 P (A) ( A i the complemet of A). (3) If S i a fiite ample pace
More informationLECTURE 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 informationBayesian 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 informationLecture 2: Monte Carlo Simulation
STAT/Q SCI 43: Itroductio to Resamplig ethods Sprig 27 Istructor: Ye-Chi Che Lecture 2: ote Carlo Simulatio 2 ote Carlo Itegratio Assume we wat to evaluate the followig itegratio: e x3 dx What ca we do?
More informationMath 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 informationEFFECTIVE WLLN, SLLN, AND CLT IN STATISTICAL MODELS
EFFECTIVE WLLN, SLLN, AND CLT IN STATISTICAL MODELS Ryszard Zieliński Ist Math Polish Acad Sc POBox 21, 00-956 Warszawa 10, Polad e-mail: rziel@impagovpl ABSTRACT Weak laws of large umbers (W LLN), strog
More information6. 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 information2 1. The r.s., of size n2, from population 2 will be. 2 and 2. 2) The two populations are independent. This implies that all of the n1 n2
Chapter 8 Comparig Two Treatmets Iferece about Two Populatio Meas We wat to compare the meas of two populatios to see whether they differ. There are two situatios to cosider, as show i the followig examples:
More informationTopic 18: Composite Hypotheses
Toc 18: November, 211 Simple hypotheses limit us to a decisio betwee oe of two possible states of ature. This limitatio does ot allow us, uder the procedures of hypothesis testig to address the basic questio:
More informationECONOMETRIC 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 informationHYPOTHESIS TESTS FOR ONE POPULATION MEAN WORKSHEET MTH 1210, FALL 2018
HYPOTHESIS TESTS FOR ONE POPULATION MEAN WORKSHEET MTH 1210, FALL 2018 We are resposible for 2 types of hypothesis tests that produce ifereces about the ukow populatio mea, µ, each of which has 3 possible
More informationSlide Set 13 Linear Model with Endogenous Regressors and the GMM estimator
Slide Set 13 Liear Model with Edogeous Regressors ad the GMM estimator Pietro Coretto pcoretto@uisa.it Ecoometrics Master i Ecoomics ad Fiace (MEF) Uiversità degli Studi di Napoli Federico II Versio: Friday
More informationProbability 2 - Notes 10. Lemma. If X is a random variable and g(x) 0 for all x in the support of f X, then P(g(X) 1) E[g(X)].
Probability 2 - Notes 0 Some Useful Iequalities. Lemma. If X is a radom variable ad g(x 0 for all x i the support of f X, the P(g(X E[g(X]. Proof. (cotiuous case P(g(X Corollaries x:g(x f X (xdx x:g(x
More informationHypothesis Testing. Evaluation of Performance of Learned h. Issues. Trade-off Between Bias and Variance
Hypothesis Testig Empirically evaluatig accuracy of hypotheses: importat activity i ML. Three questios: Give observed accuracy over a sample set, how well does this estimate apply over additioal samples?
More informationLinear 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 information22S:194 Statistical Inference II Homework Assignments. Luke Tierney
S:194 Statistical Iferece II Homework Assigmets Luke Sprig 003 Assigmet 1 Problem 6.3 Problem 6.6 Due Friday, Jauary 31, 003. Problem 6.9 Problem 6.10 Due Friday, Jauary 31, 003. Problem 6.14 Problem 6.0
More informationTesting Statistical Hypotheses for Compare. Means with Vague Data
Iteratioal Mathematical Forum 5 o. 3 65-6 Testig Statistical Hypotheses for Compare Meas with Vague Data E. Baloui Jamkhaeh ad A. adi Ghara Departmet of Statistics Islamic Azad iversity Ghaemshahr Brach
More informationCommon Large/Small Sample Tests 1/55
Commo Large/Small Sample Tests 1/55 Test of Hypothesis for the Mea (σ Kow) Covert sample result ( x) to a z value Hypothesis Tests for µ Cosider the test H :μ = μ H 1 :μ > μ σ Kow (Assume the populatio
More information