Final Examination Statistics 200C. T. Ferguson June 11, 2009
|
|
- Spencer Roy Johnston
- 5 years ago
- Views:
Transcription
1 Final Examination Statistics 00C T. Ferguson June, 009. (a) Define: X n converges in probability to X. (b) Define: X m converges in quadratic mean to X. (c) Show that if X n converges in quadratic mean to X, thenx n converges in probability to X.. Let X,X,... be independent random variables with X k having a uniform distribution over the interval (0,k). Let S n = n k= X k. (a) Find E(S n )andvar(s n ). (b) Show that (S n E(S n ))/ Var(S n ) converges in law to the standard normal distribution by checking the Lindeberg condition. 3. Let X,...,X n be a sequence of i.i.d. random variables taking values in the set {red,white,blue} with P(X j =red)=p(x j =white)=p(x j =blue)=/3. Let S n denote the total number of times that the sequence (red,white,blue) appears in that order from left to right in the sequence. That is, if Z j =I(X j =red,x j+ =white,x j+ =blue), then S n = n j= Z j. (a) Find E(S n )andvar(s n ). (b) What is the limiting distribution of (S n ES N )/ n? 4. Let X,...,X n be a sample from the Pareto distribution with distribution function F (x) =( (/x))i(x >) and density f(x) =(/x )I(x >). (a) Find the first quartile and the median of the distribution. (b) Find the joint asymptotic distribution of the sample first quartile and the sample median. (c) Let M n denote the sample maximum. Find the limiting distribution function of M n /b n for suitable choice of b n to get a non-degenerate limit. 5. Let X,...,X n be a sample from the logistic distribution with density f(x θ) = e (x θ) ( + e (x θ) ) (a) Find the likelihood equation for solving for the maximum liklihood estimate. (b) This density is symmetric about θ, soθ is the median. What is the asymptotic distribution of the sample median as a estimate of θ? (c) Show how to improve the median to a fully efficient estimate using the likelihood equation. (Note: Fisher information is I(θ) =/3.)
2 6. (a) Let (X,Y ),...,(X n,y n ) be( a sample from ) a bivariate distribution with mean σ (µ x,µ y ), and covariance matrix, Σ = x σ xy. As an estimate of θ = µ x /µ y,one σ xy may use simply θn = X n/y n. Assume µ y 0, and find the asymptotic distribution of θn. (b) Specialize to the case where the distribution of Y i is exponential with mean, and where the conditional distribution of X i given Y i = y i is normal with mean, θy i,and variance,. First note that E(X i )=θ and E(Y i ) =, so that µ x /µ y = θ. FindΣand the asymptotic distribution of θn in this case. (c) In case (b), find the maximum likelihood estimate of θ. What is the asymptotic efficiency of θn found in (b). 7. Let X,...,X n be a sample from the exponential distribution with mean θ for some θ>0, (density f(x θ) =θ e x/θ I(x >0)), and let Y,...,Y n be an independent sample from f(y µ) forsomeµ>0. (a) Find the likelihood ratio test statistic for testing the hypothesis H 0 : µ =θ based on X,...,X n,andy,...,y n. (b) What is its asymptotic distribution for large n under H 0? 8. A sample of size n is taken in a multinomial experiment with c cells denoted (i, j), i =,...,cand j =,...,c.letp ij denote the probability of cell (i, j), and let n ij denote the number falling in cell (i, j), so that p ij =and n ij = n. (a) Let H denote the hypothesis of symmetry, that p ij = p ji for all i and j. Findthe chi-square test of H against all alternatives? How many degrees of freedom does it have? (b) Let H 0 denote the hypothesis of symmetry and independence, namely that p ij = π i π j for some probability vector, (π,...,π c ). Find the chi-square test of H 0 against all alternatives. How many degrees of freedom? (c) What, then, is the chi-square test of H 0 against H, and how many degrees of freedom does it have? σ y
3 Solutions to the Final Examination, Stat 00C, Spring (a) X n converges to X in probability if for all ɛ>0, P( X n X >ɛ) 0as n. (b) X n converges to X in quadratic mean if E( X n X ) 0asn. (c) This is done using a Chebychev argument. Let ɛ>0. Since E(X n X) E{(X n X) I( X n X >ɛ)} ɛ P( X n X >ɛ), we see that if E( X n X ) 0, then P( X n X >ɛ) 0.. (a) E(S n )= n k= E(X k)= n (k/) = n(n +)/4. Bn =Var(S n)= n k= Var(X k)= n (k /) = n(n + )(n +)/7. (b) Let Z k = X k (k/) so that EZ k =0and Z k <k/a.s.then B n E{Z ki(z k >ɛ B n)} B n B n E{ZkI((k/) >ɛ Bn)} E{Zk I((n/) >ɛ Bn )} =I((n/) >ɛ Bn ) This goes to zero as n since B n goes to infinity at rate n (a) The Z j form a -dependent stationary sequence. E(S n )= n E(Z j )= (n )E(Z ) (n )/7. Var(S n )=(n )Var(Z )+(n 3)Cov(Z,Z )+(n 4)Cov(Z,Z 3 )=(n )(6/7 ) (n 3)/7 (n 4)/7. (b) (S n ES n )/ n L N (0, /7 ). 4. (a) The first quartile is x.5 =4/3 andthemedianisx.5 =. (b) f(x.5 )=f(4/3) = (3/4),andf(x.5 )=f() = (/).So ( ) ( ) ( ) X n/4 4/3 L 0 6/7 8/9 n N (, ) X n/ 0 8/9 4 (c) P(M n /b n x) = P(M n b n x) = F (b n x) n = ( (/(b n x))) n I(x > /b n ) lim n exp{ (n/b n x)}i(x >/b n )=exp{ /x}i(x >0), if we take b n = n. 5. (a) The log likelihood function is l n (θ) = (X j θ) log( + exp{ (X j θ)}) The likelihood equation is l n =0,or n exp{x j θ} + =0. 3
4 (b) Let θ n denote the sample median. Then, since f(θ θ) =/4, L n( θ n θ) N (0, (/4)/f(θ, θ) )=N(0, 4). (c) We may use ˆθ n = θ n l n ( θ n ) ln ( θ n )ormoresimply ˆθ n = θ n + I ( θ n ) n l n ( θ n )= θ n +3[ n exp{x j θ n } + ]. 6. (a) By the Central Limit Theorem, L n((x n,y n ) (µ x,µ y )) N ((0, 0), Σ). Then we apply Cramér s Theorem with g(x, y) =x/y and ġ(x, y) =(/y, x/y ), so that ġ(µ x,µ y )=(/µ y )(, θ). We find n(θ n θ) L N (0, µ y = N (0, ( )( ) σ (, θ) x σ xy σ xy σy ) θ µ y (σ x θσ xy + θ σ y )). (b) For the exponential distribution of Y, µ y = and σy =. Then E(X) = E(E(X Y )) = E(θY )=θ. Similarly, E(X ) = E(E(X Y )) = E( + θ Y )=+θ, σx =+θ,e(xy ) = E(E(XY Y )) = E(Y E(X Y )) = E(θY )=θ, andσ xy = θ. So in this case, n(θn L θ) N (0, ). (c) In case (b), the joint density of (X, Y )isf(x, y θ) =(π) / e y e (x θy) / for y>0. We find ( / θ)logf(x, y θ) =(x θy)y. From this we can see that the maximum likelihood estimate of θ is ˆθ n =( X i Y i / Yi ). From ( / θ) f(x, y θ) = y, we see that Fisher information is I(θ) = E(Y ) =. Therefore, L n(ˆθ n θ) N (0, /). The asymptotic efficiency of θn (relative to ˆθ n )is only 50%. 7. (a) L n (θ, µ) =θ n µ n exp{ (/θ) n X i (/µ) n Y i}. The maximum likelihood estimates under the general hypothesis are ˆθ n = X n and ˆµ n = Y n. Under H 0,they are θ n =(/)X n +(/4)Y n and µ n = θ n. the likelihood ratio test statistic is Λ n = L n( θ n, µ n ) L n (ˆθ n, ˆµ n ) n θ n n e n = ˆθ n n ˆµ n n e = n [ X n Y n (((/)X n +(/4)Y n ) (b) Under H 0, log(λ n ) converges in law to a chisquare distribution with degree of freedom. 8. (a) Under H, the maximum likelihood estimates of the p ij are ˆp ii = n ii /n and for i j, ˆp ij =(n ij + n ji )/n. Thereare(c ) + (c ) + +=c(c )/ restrictions going from the general hypothesis to H. So the chi-square test of H rejects H if χ (ˆp) 4 ] n
5 is greater than the appropriate cutoff point for a chi-square distribution with c(c )/ (b) Under H 0, the likelihood is proportional to i j (π iπ j ) n ij =( i πn i. i )( j πn.j j )= i πn i.+n.i i. So the maximum likelihood estimates are π i =(n i. + n.i )/(n). There are c parameters estimated so the chi-square test of H 0 rejects H 0 if χ ( p) is greater than the appropriate cutoff point for a chi-square distribution with (c ) (c ) = c(c ) (c) The chi-square test of H 0 within H, rejects H 0 if χ ( p) χ (ˆp) is greater than the appropriate cutoff point for a chisquare distribution with c(c ) (c(c )/) = c(c )/ 5
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 informationMaster s Written Examination
Master s Written Examination Option: Statistics and Probability Spring 016 Full points may be obtained for correct answers to eight questions. Each numbered question which may have several parts is worth
More informationExercises and Answers to Chapter 1
Exercises and Answers to Chapter The continuous type of random variable X has the following density function: a x, if < x < a, f (x), otherwise. Answer the following questions. () Find a. () Obtain mean
More informationStatement: With my signature I confirm that the solutions are the product of my own work. Name: Signature:.
MATHEMATICAL STATISTICS Homework assignment Instructions Please turn in the homework with this cover page. You do not need to edit the solutions. Just make sure the handwriting is legible. You may discuss
More informationAsymptotic Statistics-III. Changliang Zou
Asymptotic Statistics-III Changliang Zou The multivariate central limit theorem Theorem (Multivariate CLT for iid case) Let X i be iid random p-vectors with mean µ and and covariance matrix Σ. Then n (
More informationCentral Limit Theorem ( 5.3)
Central Limit Theorem ( 5.3) Let X 1, X 2,... be a sequence of independent random variables, each having n mean µ and variance σ 2. Then the distribution of the partial sum S n = X i i=1 becomes approximately
More informationSpring 2012 Math 541B Exam 1
Spring 2012 Math 541B Exam 1 1. A sample of size n is drawn without replacement from an urn containing N balls, m of which are red and N m are black; the balls are otherwise indistinguishable. Let X denote
More informationChapter 3: Maximum Likelihood Theory
Chapter 3: Maximum Likelihood Theory Florian Pelgrin HEC September-December, 2010 Florian Pelgrin (HEC) Maximum Likelihood Theory September-December, 2010 1 / 40 1 Introduction Example 2 Maximum likelihood
More informationFinal Exam. 1. (6 points) True/False. Please read the statements carefully, as no partial credit will be given.
1. (6 points) True/False. Please read the statements carefully, as no partial credit will be given. (a) If X and Y are independent, Corr(X, Y ) = 0. (b) (c) (d) (e) A consistent estimator must be asymptotically
More informationRecall that in order to prove Theorem 8.8, we argued that under certain regularity conditions, the following facts are true under H 0 : 1 n
Chapter 9 Hypothesis Testing 9.1 Wald, Rao, and Likelihood Ratio Tests Suppose we wish to test H 0 : θ = θ 0 against H 1 : θ θ 0. The likelihood-based results of Chapter 8 give rise to several possible
More informationLecture 28: Asymptotic confidence sets
Lecture 28: Asymptotic confidence sets 1 α asymptotic confidence sets Similar to testing hypotheses, in many situations it is difficult to find a confidence set with a given confidence coefficient or level
More informationStatistics Ph.D. Qualifying Exam: Part I October 18, 2003
Statistics Ph.D. Qualifying Exam: Part I October 18, 2003 Student Name: 1. Answer 8 out of 12 problems. Mark the problems you selected in the following table. 1 2 3 4 5 6 7 8 9 10 11 12 2. Write your answer
More informationSTAT 512 sp 2018 Summary Sheet
STAT 5 sp 08 Summary Sheet Karl B. Gregory Spring 08. Transformations of a random variable Let X be a rv with support X and let g be a function mapping X to Y with inverse mapping g (A = {x X : g(x A}
More informationProbability Theory and Statistics. Peter Jochumzen
Probability Theory and Statistics Peter Jochumzen April 18, 2016 Contents 1 Probability Theory And Statistics 3 1.1 Experiment, Outcome and Event................................ 3 1.2 Probability............................................
More informationSummary of Chapters 7-9
Summary of Chapters 7-9 Chapter 7. Interval Estimation 7.2. Confidence Intervals for Difference of Two Means Let X 1,, X n and Y 1, Y 2,, Y m be two independent random samples of sizes n and m from two
More informationStatistics GIDP Ph.D. Qualifying Exam Theory Jan 11, 2016, 9:00am-1:00pm
Statistics GIDP Ph.D. Qualifying Exam Theory Jan, 06, 9:00am-:00pm Instructions: Provide answers on the supplied pads of paper; write on only one side of each sheet. Complete exactly 5 of the 6 problems.
More informationQualifying Exam in Probability and Statistics. https://www.soa.org/files/edu/edu-exam-p-sample-quest.pdf
Part : Sample Problems for the Elementary Section of Qualifying Exam in Probability and Statistics https://www.soa.org/files/edu/edu-exam-p-sample-quest.pdf Part 2: Sample Problems for the Advanced Section
More informationFall 2017 STAT 532 Homework Peter Hoff. 1. Let P be a probability measure on a collection of sets A.
1. Let P be a probability measure on a collection of sets A. (a) For each n N, let H n be a set in A such that H n H n+1. Show that P (H n ) monotonically converges to P ( k=1 H k) as n. (b) For each n
More information2017 Financial Mathematics Orientation - Statistics
2017 Financial Mathematics Orientation - Statistics Written by Long Wang Edited by Joshua Agterberg August 21, 2018 Contents 1 Preliminaries 5 1.1 Samples and Population............................. 5
More informationf(x θ)dx with respect to θ. Assuming certain smoothness conditions concern differentiating under the integral the integral sign, we first obtain
0.1. INTRODUCTION 1 0.1 Introduction R. A. Fisher, a pioneer in the development of mathematical statistics, introduced a measure of the amount of information contained in an observaton from f(x θ). Fisher
More informationAPPM/MATH 4/5520 Solutions to Exam I Review Problems. f X 1,X 2. 2e x 1 x 2. = x 2
APPM/MATH 4/5520 Solutions to Exam I Review Problems. (a) f X (x ) f X,X 2 (x,x 2 )dx 2 x 2e x x 2 dx 2 2e 2x x was below x 2, but when marginalizing out x 2, we ran it over all values from 0 to and so
More informationTheory of Statistics.
Theory of Statistics. Homework V February 5, 00. MT 8.7.c When σ is known, ˆµ = X is an unbiased estimator for µ. If you can show that its variance attains the Cramer-Rao lower bound, then no other unbiased
More informationA Very Brief Summary of Statistical Inference, and Examples
A Very Brief Summary of Statistical Inference, and Examples Trinity Term 2008 Prof. Gesine Reinert 1 Data x = x 1, x 2,..., x n, realisations of random variables X 1, X 2,..., X n with distribution (model)
More informationA Very Brief Summary of Statistical Inference, and Examples
A Very Brief Summary of Statistical Inference, and Examples Trinity Term 2009 Prof. Gesine Reinert Our standard situation is that we have data x = x 1, x 2,..., x n, which we view as realisations of random
More informationFirst Year Examination Department of Statistics, University of Florida
First Year Examination Department of Statistics, University of Florida August 19, 010, 8:00 am - 1:00 noon Instructions: 1. You have four hours to answer questions in this examination.. You must show your
More informationUnbiased Estimation. Binomial problem shows general phenomenon. An estimator can be good for some values of θ and bad for others.
Unbiased Estimation Binomial problem shows general phenomenon. An estimator can be good for some values of θ and bad for others. To compare ˆθ and θ, two estimators of θ: Say ˆθ is better than θ if it
More informationThe Delta Method and Applications
Chapter 5 The Delta Method and Applications 5.1 Local linear approximations Suppose that a particular random sequence converges in distribution to a particular constant. The idea of using a first-order
More informationy = 1 N y i = 1 N y. E(W )=a 1 µ 1 + a 2 µ a n µ n (1.2) and its variance is var(w )=a 2 1σ a 2 2σ a 2 nσ 2 n. (1.
The probability density function of a continuous random variable Y (or the probability mass function if Y is discrete) is referred to simply as a probability distribution and denoted by f(y; θ) where θ
More informationSTT 843 Key to Homework 1 Spring 2018
STT 843 Key to Homework Spring 208 Due date: Feb 4, 208 42 (a Because σ = 2, σ 22 = and ρ 2 = 05, we have σ 2 = ρ 2 σ σ22 = 2/2 Then, the mean and covariance of the bivariate normal is µ = ( 0 2 and Σ
More informationIntroduction to Computational Finance and Financial Econometrics Probability Review - Part 2
You can t see this text! Introduction to Computational Finance and Financial Econometrics Probability Review - Part 2 Eric Zivot Spring 2015 Eric Zivot (Copyright 2015) Probability Review - Part 2 1 /
More informationPrinciples of Statistics
Part II Year 2018 2017 2016 2015 2014 2013 2012 2011 2010 2009 2008 2007 2006 2005 2018 81 Paper 4, Section II 28K Let g : R R be an unknown function, twice continuously differentiable with g (x) M for
More informationMcGill University. Faculty of Science. Department of Mathematics and Statistics. Part A Examination. Statistics: Theory Paper
McGill University Faculty of Science Department of Mathematics and Statistics Part A Examination Statistics: Theory Paper Date: 10th May 2015 Instructions Time: 1pm-5pm Answer only two questions from Section
More informationMAS113 Introduction to Probability and Statistics. Proofs of theorems
MAS113 Introduction to Probability and Statistics Proofs of theorems Theorem 1 De Morgan s Laws) See MAS110 Theorem 2 M1 By definition, B and A \ B are disjoint, and their union is A So, because m is a
More informationStatistics STAT:5100 (22S:193), Fall Sample Final Exam B
Statistics STAT:5 (22S:93), Fall 25 Sample Final Exam B Please write your answers in the exam books provided.. Let X, Y, and Y 2 be independent random variables with X N(µ X, σ 2 X ) and Y i N(µ Y, σ 2
More informationLecture 2: Basic Concepts and Simple Comparative Experiments Montgomery: Chapter 2
Lecture 2: Basic Concepts and Simple Comparative Experiments Montgomery: Chapter 2 Fall, 2013 Page 1 Random Variable and Probability Distribution Discrete random variable Y : Finite possible values {y
More informationThis paper is not to be removed from the Examination Halls
~~ST104B ZA d0 This paper is not to be removed from the Examination Halls UNIVERSITY OF LONDON ST104B ZB BSc degrees and Diplomas for Graduates in Economics, Management, Finance and the Social Sciences,
More information5 Operations on Multiple Random Variables
EE360 Random Signal analysis Chapter 5: Operations on Multiple Random Variables 5 Operations on Multiple Random Variables Expected value of a function of r.v. s Two r.v. s: ḡ = E[g(X, Y )] = g(x, y)f X,Y
More informationIIT JAM : MATHEMATICAL STATISTICS (MS) 2013
IIT JAM : MATHEMATICAL STATISTICS (MS 2013 Question Paper with Answer Keys Ctanujit Classes Of Mathematics, Statistics & Economics Visit our website for more: www.ctanujit.in IMPORTANT NOTE FOR CANDIDATES
More informationSTA 2201/442 Assignment 2
STA 2201/442 Assignment 2 1. This is about how to simulate from a continuous univariate distribution. Let the random variable X have a continuous distribution with density f X (x) and cumulative distribution
More informationMaster s Written Examination
Master s Written Examination Option: Statistics and Probability Spring 05 Full points may be obtained for correct answers to eight questions Each numbered question (which may have several parts) is worth
More informationECON 4160, Autumn term Lecture 1
ECON 4160, Autumn term 2017. Lecture 1 a) Maximum Likelihood based inference. b) The bivariate normal model Ragnar Nymoen University of Oslo 24 August 2017 1 / 54 Principles of inference I Ordinary least
More informationUniversität Potsdam Institut für Informatik Lehrstuhl Maschinelles Lernen. Hypothesis testing. Anna Wegloop Niels Landwehr/Tobias Scheffer
Universität Potsdam Institut für Informatik Lehrstuhl Maschinelles Lernen Hypothesis testing Anna Wegloop iels Landwehr/Tobias Scheffer Why do a statistical test? input computer model output Outlook ull-hypothesis
More information5.1 Consistency of least squares estimates. We begin with a few consistency results that stand on their own and do not depend on normality.
88 Chapter 5 Distribution Theory In this chapter, we summarize the distributions related to the normal distribution that occur in linear models. Before turning to this general problem that assumes normal
More informationChapter 2. Review of basic Statistical methods 1 Distribution, conditional distribution and moments
Chapter 2. Review of basic Statistical methods 1 Distribution, conditional distribution and moments We consider two kinds of random variables: discrete and continuous random variables. For discrete random
More informationMathematics Ph.D. Qualifying Examination Stat Probability, January 2018
Mathematics Ph.D. Qualifying Examination Stat 52800 Probability, January 2018 NOTE: Answers all questions completely. Justify every step. Time allowed: 3 hours. 1. Let X 1,..., X n be a random sample from
More informationLecture 10: Generalized likelihood ratio test
Stat 200: Introduction to Statistical Inference Autumn 2018/19 Lecture 10: Generalized likelihood ratio test Lecturer: Art B. Owen October 25 Disclaimer: These notes have not been subjected to the usual
More informationTest Problems for Probability Theory ,
1 Test Problems for Probability Theory 01-06-16, 010-1-14 1. Write down the following probability density functions and compute their moment generating functions. (a) Binomial distribution with mean 30
More informationChapter 7. Confidence Sets Lecture 30: Pivotal quantities and confidence sets
Chapter 7. Confidence Sets Lecture 30: Pivotal quantities and confidence sets Confidence sets X: a sample from a population P P. θ = θ(p): a functional from P to Θ R k for a fixed integer k. C(X): a confidence
More informationInformation in a Two-Stage Adaptive Optimal Design
Information in a Two-Stage Adaptive Optimal Design Department of Statistics, University of Missouri Designed Experiments: Recent Advances in Methods and Applications DEMA 2011 Isaac Newton Institute for
More informationLecture 2: Review of Probability
Lecture 2: Review of Probability Zheng Tian Contents 1 Random Variables and Probability Distributions 2 1.1 Defining probabilities and random variables..................... 2 1.2 Probability distributions................................
More informationSummary of Chapter 7 (Sections ) and Chapter 8 (Section 8.1)
Summary of Chapter 7 (Sections 7.2-7.5) and Chapter 8 (Section 8.1) Chapter 7. Tests of Statistical Hypotheses 7.2. Tests about One Mean (1) Test about One Mean Case 1: σ is known. Assume that X N(µ, σ
More informationChapter 3. Point Estimation. 3.1 Introduction
Chapter 3 Point Estimation Let (Ω, A, P θ ), P θ P = {P θ θ Θ}be probability space, X 1, X 2,..., X n : (Ω, A) (IR k, B k ) random variables (X, B X ) sample space γ : Θ IR k measurable function, i.e.
More information[y i α βx i ] 2 (2) Q = i=1
Least squares fits This section has no probability in it. There are no random variables. We are given n points (x i, y i ) and want to find the equation of the line that best fits them. We take the equation
More informationThis does not cover everything on the final. Look at the posted practice problems for other topics.
Class 7: Review Problems for Final Exam 8.5 Spring 7 This does not cover everything on the final. Look at the posted practice problems for other topics. To save time in class: set up, but do not carry
More informationSpring 2012 Math 541A Exam 1. X i, S 2 = 1 n. n 1. X i I(X i < c), T n =
Spring 2012 Math 541A Exam 1 1. (a) Let Z i be independent N(0, 1), i = 1, 2,, n. Are Z = 1 n n Z i and S 2 Z = 1 n 1 n (Z i Z) 2 independent? Prove your claim. (b) Let X 1, X 2,, X n be independent identically
More informationProbability & Statistics - FALL 2008 FINAL EXAM
550.3 Probability & Statistics - FALL 008 FINAL EXAM NAME. An urn contains white marbles and 8 red marbles. A marble is drawn at random from the urn 00 times with replacement. Which of the following is
More informationMATH c UNIVERSITY OF LEEDS Examination for the Module MATH2715 (January 2015) STATISTICAL METHODS. Time allowed: 2 hours
MATH2750 This question paper consists of 8 printed pages, each of which is identified by the reference MATH275. All calculators must carry an approval sticker issued by the School of Mathematics. c UNIVERSITY
More informationInstitute of Actuaries of India
Institute of Actuaries of India Subject CT3 Probability & Mathematical Statistics May 2011 Examinations INDICATIVE SOLUTION Introduction The indicative solution has been written by the Examiners with the
More informationLoglikelihood and Confidence Intervals
Stat 504, Lecture 2 1 Loglikelihood and Confidence Intervals The loglikelihood function is defined to be the natural logarithm of the likelihood function, l(θ ; x) = log L(θ ; x). For a variety of reasons,
More informationsimple if it completely specifies the density of x
3. Hypothesis Testing Pure significance tests Data x = (x 1,..., x n ) from f(x, θ) Hypothesis H 0 : restricts f(x, θ) Are the data consistent with H 0? H 0 is called the null hypothesis simple if it completely
More informationNotes, March 4, 2013, R. Dudley Maximum likelihood estimation: actual or supposed
18.466 Notes, March 4, 2013, R. Dudley Maximum likelihood estimation: actual or supposed 1. MLEs in exponential families Let f(x,θ) for x X and θ Θ be a likelihood function, that is, for present purposes,
More informationLecture 13: Subsampling vs Bootstrap. Dimitris N. Politis, Joseph P. Romano, Michael Wolf
Lecture 13: 2011 Bootstrap ) R n x n, θ P)) = τ n ˆθn θ P) Example: ˆθn = X n, τ n = n, θ = EX = µ P) ˆθ = min X n, τ n = n, θ P) = sup{x : F x) 0} ) Define: J n P), the distribution of τ n ˆθ n θ P) under
More informationElements of statistics (MATH0487-1)
Elements of statistics (MATH0487-1) Prof. Dr. Dr. K. Van Steen University of Liège, Belgium November 12, 2012 Introduction to Statistics Basic Probability Revisited Sampling Exploratory Data Analysis -
More informationMAS223 Statistical Inference and Modelling Exercises
MAS223 Statistical Inference and Modelling Exercises The exercises are grouped into sections, corresponding to chapters of the lecture notes Within each section exercises are divided into warm-up questions,
More informationBivariate Paired Numerical Data
Bivariate Paired Numerical Data Pearson s correlation, Spearman s ρ and Kendall s τ, tests of independence University of California, San Diego Instructor: Ery Arias-Castro http://math.ucsd.edu/~eariasca/teaching.html
More informationTesting Hypothesis. Maura Mezzetti. Department of Economics and Finance Università Tor Vergata
Maura Department of Economics and Finance Università Tor Vergata Hypothesis Testing Outline It is a mistake to confound strangeness with mystery Sherlock Holmes A Study in Scarlet Outline 1 The Power Function
More informationStatistics Ph.D. Qualifying Exam
Department of Statistics Carnegie Mellon University May 7 2008 Statistics Ph.D. Qualifying Exam You are not expected to solve all five problems. Complete solutions to few problems will be preferred to
More informationECE531 Lecture 10b: Maximum Likelihood Estimation
ECE531 Lecture 10b: Maximum Likelihood Estimation D. Richard Brown III Worcester Polytechnic Institute 05-Apr-2011 Worcester Polytechnic Institute D. Richard Brown III 05-Apr-2011 1 / 23 Introduction So
More informationChapter 2: Fundamentals of Statistics Lecture 15: Models and statistics
Chapter 2: Fundamentals of Statistics Lecture 15: Models and statistics Data from one or a series of random experiments are collected. Planning experiments and collecting data (not discussed here). Analysis:
More informationAsymptotic Statistics-VI. Changliang Zou
Asymptotic Statistics-VI Changliang Zou Kolmogorov-Smirnov distance Example (Kolmogorov-Smirnov confidence intervals) We know given α (0, 1), there is a well-defined d = d α,n such that, for any continuous
More informationQualifying Exam in Probability and Statistics. https://www.soa.org/files/edu/edu-exam-p-sample-quest.pdf
Part 1: Sample Problems for the Elementary Section of Qualifying Exam in Probability and Statistics https://www.soa.org/files/edu/edu-exam-p-sample-quest.pdf Part 2: Sample Problems for the Advanced Section
More informationEC212: Introduction to Econometrics Review Materials (Wooldridge, Appendix)
1 EC212: Introduction to Econometrics Review Materials (Wooldridge, Appendix) Taisuke Otsu London School of Economics Summer 2018 A.1. Summation operator (Wooldridge, App. A.1) 2 3 Summation operator For
More informationRandom Variables. Random variables. A numerically valued map X of an outcome ω from a sample space Ω to the real line R
In probabilistic models, a random variable is a variable whose possible values are numerical outcomes of a random phenomenon. As a function or a map, it maps from an element (or an outcome) of a sample
More informationStatistical Methods for Handling Incomplete Data Chapter 2: Likelihood-based approach
Statistical Methods for Handling Incomplete Data Chapter 2: Likelihood-based approach Jae-Kwang Kim Department of Statistics, Iowa State University Outline 1 Introduction 2 Observed likelihood 3 Mean Score
More informationTesting Algebraic Hypotheses
Testing Algebraic Hypotheses Mathias Drton Department of Statistics University of Chicago 1 / 18 Example: Factor analysis Multivariate normal model based on conditional independence given hidden variable:
More informationGraduate Econometrics I: Asymptotic Theory
Graduate Econometrics I: Asymptotic Theory Yves Dominicy Université libre de Bruxelles Solvay Brussels School of Economics and Management ECARES Yves Dominicy Graduate Econometrics I: Asymptotic Theory
More informationLinear Models and Estimation by Least Squares
Linear Models and Estimation by Least Squares Jin-Lung Lin 1 Introduction Causal relation investigation lies in the heart of economics. Effect (Dependent variable) cause (Independent variable) Example:
More informationNotes on Random Vectors and Multivariate Normal
MATH 590 Spring 06 Notes on Random Vectors and Multivariate Normal Properties of Random Vectors If X,, X n are random variables, then X = X,, X n ) is a random vector, with the cumulative distribution
More information18.440: Lecture 26 Conditional expectation
18.440: Lecture 26 Conditional expectation Scott Sheffield MIT 1 Outline Conditional probability distributions Conditional expectation Interpretation and examples 2 Outline Conditional probability distributions
More informationChapter 2. Discrete Distributions
Chapter. Discrete Distributions Objectives ˆ Basic Concepts & Epectations ˆ Binomial, Poisson, Geometric, Negative Binomial, and Hypergeometric Distributions ˆ Introduction to the Maimum Likelihood Estimation
More informationStat 704 Data Analysis I Probability Review
1 / 39 Stat 704 Data Analysis I Probability Review Dr. Yen-Yi Ho Department of Statistics, University of South Carolina A.3 Random Variables 2 / 39 def n: A random variable is defined as a function that
More informationt x 1 e t dt, and simplify the answer when possible (for example, when r is a positive even number). In particular, confirm that EX 4 = 3.
Mathematical Statistics: Homewor problems General guideline. While woring outside the classroom, use any help you want, including people, computer algebra systems, Internet, and solution manuals, but mae
More informationMaster s Written Examination - Solution
Master s Written Examination - Solution Spring 204 Problem Stat 40 Suppose X and X 2 have the joint pdf f X,X 2 (x, x 2 ) = 2e (x +x 2 ), 0 < x < x 2
More informationHT Introduction. P(X i = x i ) = e λ λ x i
MODS STATISTICS Introduction. HT 2012 Simon Myers, Department of Statistics (and The Wellcome Trust Centre for Human Genetics) myers@stats.ox.ac.uk We will be concerned with the mathematical framework
More informationLinear Methods for Prediction
Chapter 5 Linear Methods for Prediction 5.1 Introduction We now revisit the classification problem and focus on linear methods. Since our prediction Ĝ(x) will always take values in the discrete set G we
More informationIntroduction to Normal Distribution
Introduction to Normal Distribution Nathaniel E. Helwig Assistant Professor of Psychology and Statistics University of Minnesota (Twin Cities) Updated 17-Jan-2017 Nathaniel E. Helwig (U of Minnesota) Introduction
More informationStatistics Ph.D. Qualifying Exam: Part II November 3, 2001
Statistics Ph.D. Qualifying Exam: Part II November 3, 2001 Student Name: 1. Answer 8 out of 12 problems. Mark the problems you selected in the following table. 1 2 3 4 5 6 7 8 9 10 11 12 2. Write your
More informationMathematical statistics
October 18 th, 2018 Lecture 16: Midterm review Countdown to mid-term exam: 7 days Week 1 Chapter 1: Probability review Week 2 Week 4 Week 7 Chapter 6: Statistics Chapter 7: Point Estimation Chapter 8:
More informationMath 181B Homework 1 Solution
Math 181B Homework 1 Solution 1. Write down the likelihood: L(λ = n λ X i e λ X i! (a One-sided test: H 0 : λ = 1 vs H 1 : λ = 0.1 The likelihood ratio: where LR = L(1 L(0.1 = 1 X i e n 1 = λ n X i e nλ
More informationIntroduction to Estimation Methods for Time Series models Lecture 2
Introduction to Estimation Methods for Time Series models Lecture 2 Fulvio Corsi SNS Pisa Fulvio Corsi Introduction to Estimation () Methods for Time Series models Lecture 2 SNS Pisa 1 / 21 Estimators:
More informationCh 2: Simple Linear Regression
Ch 2: Simple Linear Regression 1. Simple Linear Regression Model A simple regression model with a single regressor x is y = β 0 + β 1 x + ɛ, where we assume that the error ɛ is independent random component
More informationStatistics Ph.D. Qualifying Exam: Part II November 9, 2002
Statistics Ph.D. Qualifying Exam: Part II November 9, 2002 Student Name: 1. Answer 8 out of 12 problems. Mark the problems you selected in the following table. 1 2 3 4 5 6 7 8 9 10 11 12 2. Write your
More informationUnbiased Estimation. Binomial problem shows general phenomenon. An estimator can be good for some values of θ and bad for others.
Unbiased Estimation Binomial problem shows general phenomenon. An estimator can be good for some values of θ and bad for others. To compare ˆθ and θ, two estimators of θ: Say ˆθ is better than θ if it
More informationPreliminary Statistics. Lecture 3: Probability Models and Distributions
Preliminary Statistics Lecture 3: Probability Models and Distributions Rory Macqueen (rm43@soas.ac.uk), September 2015 Outline Revision of Lecture 2 Probability Density Functions Cumulative Distribution
More informationEstimation theory. Parametric estimation. Properties of estimators. Minimum variance estimator. Cramer-Rao bound. Maximum likelihood estimators
Estimation theory Parametric estimation Properties of estimators Minimum variance estimator Cramer-Rao bound Maximum likelihood estimators Confidence intervals Bayesian estimation 1 Random Variables Let
More informationCourse information: Instructor: Tim Hanson, Leconte 219C, phone Office hours: Tuesday/Thursday 11-12, Wednesday 10-12, and by appointment.
Course information: Instructor: Tim Hanson, Leconte 219C, phone 777-3859. Office hours: Tuesday/Thursday 11-12, Wednesday 10-12, and by appointment. Text: Applied Linear Statistical Models (5th Edition),
More informationMathematical statistics
October 4 th, 2018 Lecture 12: Information Where are we? Week 1 Week 2 Week 4 Week 7 Week 10 Week 14 Probability reviews Chapter 6: Statistics and Sampling Distributions Chapter 7: Point Estimation Chapter
More informationHypothesis Testing. 1 Definitions of test statistics. CB: chapter 8; section 10.3
Hypothesis Testing CB: chapter 8; section 0.3 Hypothesis: statement about an unknown population parameter Examples: The average age of males in Sweden is 7. (statement about population mean) The lowest
More informationLecture 25: Review. Statistics 104. April 23, Colin Rundel
Lecture 25: Review Statistics 104 Colin Rundel April 23, 2012 Joint CDF F (x, y) = P [X x, Y y] = P [(X, Y ) lies south-west of the point (x, y)] Y (x,y) X Statistics 104 (Colin Rundel) Lecture 25 April
More informationSome Assorted Formulae. Some confidence intervals: σ n. x ± z α/2. x ± t n 1;α/2 n. ˆp(1 ˆp) ˆp ± z α/2 n. χ 2 n 1;1 α/2. n 1;α/2
STA 248 H1S MIDTERM TEST February 26, 2008 SURNAME: SOLUTIONS GIVEN NAME: STUDENT NUMBER: INSTRUCTIONS: Time: 1 hour and 50 minutes Aids allowed: calculator Tables of the standard normal, t and chi-square
More information