HYPOTHESIS TESTING: FREQUENTIST APPROACH.


 Frank Nelson
 2 years ago
 Views:
Transcription
1 HYPOTHESIS TESTING: FREQUENTIST APPROACH. These notes summarize the lectures on (the frequentist approach to) hypothesis testing. You should be familiar with the standard hypothesis testing from previous stats classes. Here, we will explain where this approach comes from and develop new ideas (all within the context of the parametric setup). 1. SetUp The basic setup of (the NeymanPearson approach to) hypothesis testing is as follows. There are two hypotheses you are trying to decide between: the null (H 0 ) and the alternative (H A ). If a hypothesis fully determines the behaviour (pdf/pmf or other) of the random variables then it is called simple, otherwise it is known as composite. The hypothesis test will reject H 0 in favour of H A if a test statistic T = T (X) falls into a rejection region (RR). We therefore have that: accept H 0 reject H 0 H 0 true Type I error H A true Type II error The probability of a type I error is denoted as α and is also known as the significance level of a test. The probability of a type II error is denoted as β. The power of a test is 1 β; the probability of doing the correct thing under H A. Note that if H A is composite then both β and power depend on the particular member of H A which holds. In this case we will often plot the power function. Ideally we would have that α = β = 0. However, in practice we most often have that decreasing α drives up β and vice versa. 2. NeymanPearson The main idea behind NeymanPearson is to fix α in advance (choose α to be small) and then to find a test which yields a small value of β. The NeymanPearson lemma tells us that the in such a setup, the likelihood ratio test (LRT) is the most powerful of all the possible tests. This only works for two simple hypotheses. Date: November 25,
2 2 HYPOTHESIS TESTING: FREQUENTIST APPROACH. Thus, assume that H 0 and H A are both simple, and let f 0 (x) denote the pdf/pmf (likelihood) of the data under H 0 (and f A (x) under H A ). The LRT is the test which rejects if f 0 (x) f A (x) < c, where c is chosen in such a way so that P (reject) = α. Lemma 2.1 (NeymanPearson Lemma). Any other test with significance level α α has power less than or equal to that of the likelihood ratio test. First of all note that this is a very sensible thing to do (we reject H 0 if the data has a bigger likelihood under H A ). Thus, the basic idea is similar to that of maximum likelihood estimation. We next need to take the LRT test and translate it into something easier to handle. Example. ESP example (Bernoulli, sample size is 10). We have that P (T otal 6 H 0 ) = 0.02, and that P (T otal 5 H 0 ) = (for sample size 10), we therefore cannot choose α = 0.05 exactly. We will choose the rejection region to be {6, 7, 8, 9, 10}. In this case, the power function is given in Figure 1. The code used to generate Figure 1 in R was: x<rep(0,250) for(i in 1:250) { x[i]<1pbinom(5,10,0.25+i/250*0.75) } plot(x) What happens if we do n independent tests at the same time? Example. Population = exponential. Example. Population = normal, variance known. 3. Pvalues Performing an αlevel test is not very informative as to the amount of information for/against the alternative hypothesis. The quantity that does allow us to measure this is the pvalue. The pvalue is defined as the smallest value of α for which the null hypothesis will be rejected. Typically it is calculated as the probability of obtaining a test statistic as or more extreme than what was actually observed. Extreme is dictated by the form of the rejection region. For a specific example, in the ESP Bernoulli case, if
3 HYPOTHESIS TESTING: FREQUENTIST APPROACH. 3 x Index Figure 1. Power function in ESP example. we observed 5 total successes, then since we reject for T = total large, the pvalue is calculated as P (T 5 H 0 ). Imho, you should always report the pvalue for a hypothesis test in your research. Example. Under the null hypothesis show that the distribution of the pvalue is Uniform[0, 1].
4 4 HYPOTHESIS TESTING: FREQUENTIST APPROACH. 4. Generalized Likelihood Ratio Test (GLRT) The LRT is optimal for testing a simple hypothesis against a simple hypothesis. However, often we wish to compare simple vs. composite or two composite hypotheses. As the name implies, the generalized LRT is a generalization of the LRT which allows us to handle composite hypotheses. Although no optimality results exist for the generalized version, we do have some nice asymptotic results, and it is easily motivated as a natural extension of the LRT. The set up for the GLRT is as follows. Let f(x θ) denote the pdf/pmf of the data if the parameter θ (possibly multivariate) is known. Notice that f(x θ) is actually the likelihood. The null hypothesis specifies that θ Θ 0 and the alternative says that θ Θ A. We let Θ denote Θ 0 Θ A. The GLRT rejects the null if Λ = max θ Θ 0 f(x θ) max θ ΘA f(x θ) is small. Indeed, this is very reasonable. In practice however it is often easier to work with Λ = max θ Θ 0 f(x θ) max θ Θ f(x θ) and reject H 0 if this is small. Since Λ = min(λ, 1) both versions actually do the same thing. We take the latter as our official definition of the GLRT. Example. Twosided normal, unknown variance. Theorem 4.1. Under smoothness assumptions on the underlying pdf/pmf, the null distribution of 2 log Λ converges to a χ 2 distribution with degrees of freedom equal to dim Θ dim Θ 0 as the sample size tends to infinity. Since we reject for small values of Λ, we would reject for large values of 2 log Λ. Example. Compare this to the twosided normal, unknown variance. 5. Power In the previous sections we have really avoided the issue of power. The LRT chooses the test with the highest power for a fixed significance level, but what if this isn t good enough? In practice it often is the case that increasing sample size increases power. The following examples are designed to illustrate this. Example. Normal, variance known. Example. Normal, variance unknown. (To calculate power, approximate using normal!)
5 HYPOTHESIS TESTING: FREQUENTIST APPROACH Duality of Confidence Intervals and Hypothesis Tests. A confidence interval (or set, in general) can be obtained by inverting a hypothesis test and vice versa. Example. Normal with known variance. Theorem 6.1. Suppose that for every value θ 0 in Θ there is a test at level α of the hypothesis H 0 : θ = θ 0. Denote the acceptance region of the test as A(θ 0 ). Then the set C(X) = {θ : X A(θ)} is a 100(1 α)% confidence region for θ. In words, a 100(1 α)% confidence region for θ consists of all those values of θ 0 for which the hypothesis that θ = θ 0 will not be rejected at level α. Theorem 6.2. Suppose that C(X) is a 100(1 α)% confidence region for θ: that is, for every θ 0 P (θ 0 C(X) θ = θ 0 ) = 1 α. Then an acceptance region for a test at level α of the hypothesis H 0 : θ = θ 0 is A(θ 0 ) = {X θ 0 C(X)}. In words, this says that the hypothesis that θ = θ 0 is accepted if θ 0 lies in the confidence region. This duality works exactly for the ttest and ztests and associated confidence intervals. For other tests that are typically used (eg. testing a proportion or LRT for the Poisson, say) the typical tests do not invert exactly to the confidence interval and vice versa. This is not because duality fails in these cases, but because the test used is not an exact inversion of the confidence set. Example. Suppose a ttest rejects the twosided hypothesis test for µ = 0 at the 5% level. Would the 90% CI contain zero? References [R] Rice, J.; Mathematical Statistics and Data Analysis Duxbury Press, 2nd Edition, Prepared by Hanna Jankowski Department of Statistics, University of Washington Box , Seattle, WA U.S.A.
STAT 135 Lab 6 Duality of Hypothesis Testing and Confidence Intervals, GLRT, Pearson χ 2 Tests and QQ plots. March 8, 2015
STAT 135 Lab 6 Duality of Hypothesis Testing and Confidence Intervals, GLRT, Pearson χ 2 Tests and QQ plots March 8, 2015 The duality between CI and hypothesis testing The duality between CI and hypothesis
More informationPARAMETER ESTIMATION: BAYESIAN APPROACH. These notes summarize the lectures on Bayesian parameter estimation.
PARAMETER ESTIMATION: BAYESIAN APPROACH. These notes summarize the lectures on Bayesian parameter estimation.. Beta Distribution We ll start by learning about the Beta distribution, since we end up using
More informationLecture 21. Hypothesis Testing II
Lecture 21. Hypothesis Testing II December 7, 2011 In the previous lecture, we dened a few key concepts of hypothesis testing and introduced the framework for parametric hypothesis testing. In the parametric
More informationEconomics 520. Lecture Note 19: Hypothesis Testing via the NeymanPearson Lemma CB 8.1,
Economics 520 Lecture Note 9: Hypothesis Testing via the NeymanPearson Lemma CB 8., 8.3.8.3.3 Uniformly Most Powerful Tests and the NeymanPearson Lemma Let s return to the hypothesis testing problem
More informationMathematical Statistics
Mathematical Statistics MAS 713 Chapter 8 Previous lecture: 1 Bayesian Inference 2 Decision theory 3 Bayesian Vs. Frequentist 4 Loss functions 5 Conjugate priors Any questions? Mathematical Statistics
More informationStatistics  Lecture One. Outline. Charlotte Wickham 1. Basic ideas about estimation
Statistics  Lecture One Charlotte Wickham wickham@stat.berkeley.edu http://www.stat.berkeley.edu/~wickham/ Outline 1. Basic ideas about estimation 2. Method of Moments 3. Maximum Likelihood 4. Confidence
More informationHypothesis Testing. BS2 Statistical Inference, Lecture 11 Michaelmas Term Steffen Lauritzen, University of Oxford; November 15, 2004
Hypothesis Testing BS2 Statistical Inference, Lecture 11 Michaelmas Term 2004 Steffen Lauritzen, University of Oxford; November 15, 2004 Hypothesis testing We consider a family of densities F = {f(x; θ),
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 informationThe University of Hong Kong Department of Statistics and Actuarial Science STAT2802 Statistical Models Tutorial Solutions Solutions to Problems 7180
The University of Hong Kong Department of Statistics and Actuarial Science STAT2802 Statistical Models Tutorial Solutions Solutions to Problems 7180 71. Decide in each case whether the hypothesis is simple
More informationLECTURE 10: NEYMANPEARSON LEMMA AND ASYMPTOTIC TESTING. The last equality is provided so this can look like a more familiar parametric test.
Economics 52 Econometrics Professor N.M. Kiefer LECTURE 1: NEYMANPEARSON LEMMA AND ASYMPTOTIC TESTING NEYMANPEARSON LEMMA: Lesson: Good tests are based on the likelihood ratio. The proof is easy in the
More informationTopic 15: Simple Hypotheses
Topic 15: November 10, 2009 In the simplest setup for a statistical hypothesis, we consider two values θ 0, θ 1 in the parameter space. We write the test as H 0 : θ = θ 0 versus H 1 : θ = θ 1. H 0 is
More informationSTAT 135 Lab 5 Bootstrapping and Hypothesis Testing
STAT 135 Lab 5 Bootstrapping and Hypothesis Testing Rebecca Barter March 2, 2015 The Bootstrap Bootstrap Suppose that we are interested in estimating a parameter θ from some population with members x 1,...,
More informationHypothesis Testing Chap 10p460
Hypothesis Testing Chap 1p46 Elements of a statistical test p462  Null hypothesis  Alternative hypothesis  Test Statistic  Rejection region Rejection Region p462 The rejection region (RR) specifies
More informationSummary of Chapters 79
Summary of Chapters 79 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 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 likelihoodbased results of Chapter 8 give rise to several possible
More informationMath 494: Mathematical Statistics
Math 494: Mathematical Statistics Instructor: Jimin Ding jmding@wustl.edu Department of Mathematics Washington University in St. Louis Class materials are available on course website (www.math.wustl.edu/
More informationSTATS 200: Introduction to Statistical Inference. Lecture 29: Course review
STATS 200: Introduction to Statistical Inference Lecture 29: Course review Course review We started in Lecture 1 with a fundamental assumption: Data is a realization of a random process. The goal throughout
More informationexp{ (x i) 2 i=1 n i=1 (x i a) 2 (x i ) 2 = exp{ i=1 n i=1 n 2ax i a 2 i=1
4 Hypothesis testing 4. Simple hypotheses A computer tries to distinguish between two sources of signals. Both sources emit independent signals with normally distributed intensity, the signals of the first
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 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 informationReview. DS GA 1002 Statistical and Mathematical Models. Carlos FernandezGranda
Review DS GA 1002 Statistical and Mathematical Models http://www.cims.nyu.edu/~cfgranda/pages/dsga1002_fall16 Carlos FernandezGranda Probability and statistics Probability: Framework for dealing with
More informationLecture Testing Hypotheses: The NeymanPearson Paradigm
Math 408  Mathematical Statistics Lecture 2930. Testing Hypotheses: The NeymanPearson Paradigm April 1215, 2013 Konstantin Zuev (USC) Math 408, Lecture 2930 April 1215, 2013 1 / 12 Agenda Example:
More information4.5.1 The use of 2 log Λ when θ is scalar
4.5. ASYMPTOTIC FORM OF THE G.L.R.T. 97 4.5.1 The use of 2 log Λ when θ is scalar Suppose we wish to test the hypothesis NH : θ = θ where θ is a given value against the alternative AH : θ θ on the basis
More informationChapter 9: Hypothesis Testing Sections
Chapter 9: Hypothesis Testing Sections 9.1 Problems of Testing Hypotheses 9.2 Testing Simple Hypotheses 9.3 Uniformly Most Powerful Tests Skip: 9.4 TwoSided Alternatives 9.6 Comparing the Means of Two
More informationDefinition 3.1 A statistical hypothesis is a statement about the unknown values of the parameters of the population distribution.
Hypothesis Testing Definition 3.1 A statistical hypothesis is a statement about the unknown values of the parameters of the population distribution. Suppose the family of population distributions is indexed
More informationChapters 10. Hypothesis Testing
Chapters 10. Hypothesis Testing Some examples of hypothesis testing 1. Toss a coin 100 times and get 62 heads. Is this coin a fair coin? 2. Is the new treatment on blood pressure more effective than the
More informationTopic 10: Hypothesis Testing
Topic 10: Hypothesis Testing Course 003, 2016 Page 0 The Problem of Hypothesis Testing A statistical hypothesis is an assertion or conjecture about the probability distribution of one or more random variables.
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 informationStatistical Inference: Estimation and Confidence Intervals Hypothesis Testing
Statistical Inference: Estimation and Confidence Intervals Hypothesis Testing 1 In most statistics problems, we assume that the data have been generated from some unknown probability distribution. We desire
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 information14.30 Introduction to Statistical Methods in Economics Spring 2009
MIT OpenCourseWare http://ocw.mit.edu 4.0 Introduction to Statistical Methods in Economics Spring 009 For information about citing these materials or our Terms of Use, visit: http://ocw.mit.edu/terms.
More informationHypothesis Testing: The Generalized Likelihood Ratio Test
Hypothesis Testing: The Generalized Likelihood Ratio Test Consider testing the hypotheses H 0 : θ Θ 0 H 1 : θ Θ \ Θ 0 Definition: The Generalized Likelihood Ratio (GLR Let L(θ be a likelihood for a random
More informationOne sided tests. An example of a two sided alternative is what we ve been using for our two sample tests:
One sided tests So far all of our tests have been two sided. While this may be a bit easier to understand, this is often not the best way to do a hypothesis test. One simple thing that we can do to get
More informationLecture 21: October 19
36705: Intermediate Statistics Fall 2017 Lecturer: Siva Balakrishnan Lecture 21: October 19 21.1 Likelihood Ratio Test (LRT) To test composite versus composite hypotheses the general method is to use
More informationSTAT 830 Hypothesis Testing
STAT 830 Hypothesis Testing Richard Lockhart Simon Fraser University STAT 830 Fall 2018 Richard Lockhart (Simon Fraser University) STAT 830 Hypothesis Testing STAT 830 Fall 2018 1 / 30 Purposes of These
More informationLecture 12 November 3
STATS 300A: Theory of Statistics Fall 2015 Lecture 12 November 3 Lecturer: Lester Mackey Scribe: Jae Hyuck Park, Christian Fong Warning: These notes may contain factual and/or typographic errors. 12.1
More informationTopic 10: Hypothesis Testing
Topic 10: Hypothesis Testing Course 003, 2017 Page 0 The Problem of Hypothesis Testing A statistical hypothesis is an assertion or conjecture about the probability distribution of one or more random variables.
More informationRobustness and Distribution Assumptions
Chapter 1 Robustness and Distribution Assumptions 1.1 Introduction In statistics, one often works with model assumptions, i.e., one assumes that data follow a certain model. Then one makes use of methodology
More informationTesting Statistical Hypotheses
E.L. Lehmann Joseph P. Romano Testing Statistical Hypotheses Third Edition 4y Springer Preface vii I SmallSample Theory 1 1 The General Decision Problem 3 1.1 Statistical Inference and Statistical Decisions
More informationSTAT 830 Hypothesis Testing
STAT 830 Hypothesis Testing Hypothesis testing is a statistical problem where you must choose, on the basis of data X, between two alternatives. We formalize this as the problem of choosing between two
More informationTwo hours. To be supplied by the Examinations Office: Mathematical Formula Tables THE UNIVERSITY OF MANCHESTER. 21 June :45 11:45
Two hours MATH20802 To be supplied by the Examinations Office: Mathematical Formula Tables THE UNIVERSITY OF MANCHESTER STATISTICAL METHODS 21 June 2010 9:45 11:45 Answer any FOUR of the questions. Universityapproved
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 information8: Hypothesis Testing
Some definitions 8: Hypothesis Testing. Simple, compound, null and alternative hypotheses In test theory one distinguishes between simple hypotheses and compound hypotheses. A simple hypothesis Examples:
More informationHypothesis Test. The opposite of the null hypothesis, called an alternative hypothesis, becomes
NeymanPearson paradigm. Suppose that a researcher is interested in whether the new drug works. The process of determining whether the outcome of the experiment points to yes or no is called hypothesis
More informationPartitioning the Parameter Space. Topic 18 Composite Hypotheses
Topic 18 Composite Hypotheses Partitioning the Parameter Space 1 / 10 Outline Partitioning the Parameter Space 2 / 10 Partitioning the Parameter Space Simple hypotheses limit us to a decision between one
More informationF79SM STATISTICAL METHODS
F79SM STATISTICAL METHODS SUMMARY NOTES 9 Hypothesis testing 9.1 Introduction As before we have a random sample x of size n of a population r.v. X with pdf/pf f(x;θ). The distribution we assign to X is
More information7.2 OneSample Correlation ( = a) Introduction. Correlation analysis measures the strength and direction of association between
7.2 OneSample Correlation ( = a) Introduction Correlation analysis measures the strength and direction of association between variables. In this chapter we will test whether the population correlation
More informationINTERVAL ESTIMATION AND HYPOTHESES TESTING
INTERVAL ESTIMATION AND HYPOTHESES TESTING 1. IDEA An interval rather than a point estimate is often of interest. Confidence intervals are thus important in empirical work. To construct interval estimates,
More informationHypothesis Testing  Frequentist
Frequentist Hypothesis Testing  Frequentist Compare two hypotheses to see which one better explains the data. Or, alternatively, what is the best way to separate events into two classes, those originating
More informationHypothesis Testing One Sample Tests
STATISTICS Lecture no. 13 Department of Econometrics FEM UO Brno office 69a, tel. 973 442029 email:jiri.neubauer@unob.cz 12. 1. 2010 Tests on Mean of a Normal distribution Tests on Variance of a Normal
More informationVisual interpretation with normal approximation
Visual interpretation with normal approximation H 0 is true: H 1 is true: p =0.06 25 33 Reject H 0 α =0.05 (Type I error rate) Fail to reject H 0 β =0.6468 (Type II error rate) 30 Accept H 1 Visual interpretation
More informationHypothesis testing (cont d)
Hypothesis testing (cont d) Ulrich Heintz Brown University 4/12/2016 Ulrich Heintz  PHYS 1560 Lecture 11 1 Hypothesis testing Is our hypothesis about the fundamental physics correct? We will not be able
More informationhttp://www.math.uah.edu/stat/hypothesis/.xhtml 1 of 5 7/29/2009 3:14 PM Virtual Laboratories > 9. Hy pothesis Testing > 1 2 3 4 5 6 7 1. The Basic Statistical Model As usual, our starting point is a random
More informationBTRY 4090: Spring 2009 Theory of Statistics
BTRY 4090: Spring 2009 Theory of Statistics Guozhang Wang September 25, 2010 1 Review of Probability We begin with a real example of using probability to solve computationally intensive (or infeasible)
More informationTopic 3: Hypothesis Testing
CS 8850: Advanced Machine Learning Fall 07 Topic 3: Hypothesis Testing Instructor: Daniel L. PimentelAlarcón c Copyright 07 3. Introduction One of the simplest inference problems is that of deciding between
More informationFYST17 Lecture 8 Statistics and hypothesis testing. Thanks to T. Petersen, S. Maschiocci, G. Cowan, L. Lyons
FYST17 Lecture 8 Statistics and hypothesis testing Thanks to T. Petersen, S. Maschiocci, G. Cowan, L. Lyons 1 Plan for today: Introduction to concepts The Gaussian distribution Likelihood functions Hypothesis
More informationSTA 732: Inference. Notes 2. NeymanPearsonian Classical Hypothesis Testing B&D 4
STA 73: Inference Notes. NeymanPearsonian Classical Hypothesis Testing B&D 4 1 Testing as a rule Fisher s quantification of extremeness of observed evidence clearly lacked rigorous mathematical interpretation.
More informationTesting and Model Selection
Testing and Model Selection This is another digression on general statistics: see PE App C.8.4. The EViews output for least squares, probit and logit includes some statistics relevant to testing hypotheses
More informationTesting Statistical Hypotheses
E.L. Lehmann Joseph P. Romano, 02LEu1 ttd ~Lt~S Testing Statistical Hypotheses Third Edition With 6 Illustrations ~Springer 2 The Probability Background 28 2.1 Probability and Measure 28 2.2 Integration.........
More informationST495: Survival Analysis: Hypothesis testing and confidence intervals
ST495: Survival Analysis: Hypothesis testing and confidence intervals Eric B. Laber Department of Statistics, North Carolina State University April 3, 2014 I remember that one fateful day when Coach took
More informationSTAT 801: Mathematical Statistics. Hypothesis Testing
STAT 801: Mathematical Statistics Hypothesis Testing Hypothesis testing: a statistical problem where you must choose, on the basis o data X, between two alternatives. We ormalize this as the problem o
More informationPolitical Science 236 Hypothesis Testing: Review and Bootstrapping
Political Science 236 Hypothesis Testing: Review and Bootstrapping Rocío Titiunik Fall 2007 1 Hypothesis Testing Definition 1.1 Hypothesis. A hypothesis is a statement about a population parameter The
More informationHypothesis testing I.  In particular, we are talking about statistical hypotheses. [get everyone s finger length!] n =
Hypothesis testing I I. What is hypothesis testing? [Note we re temporarily bouncing around in the book a lot! Things will settle down again in a week or so]  Exactly what it says. We develop a hypothesis,
More informationStat 135, Fall 2006 A. Adhikari HOMEWORK 6 SOLUTIONS
Stat 135, Fall 2006 A. Adhikari HOMEWORK 6 SOLUTIONS 1a. Under the null hypothesis X has the binomial (100,.5) distribution with E(X) = 50 and SE(X) = 5. So P ( X 50 > 10) is (approximately) two tails
More informationLecture 8: Information Theory and Statistics
Lecture 8: Information Theory and Statistics Part II: Hypothesis Testing and IHsiang Wang Department of Electrical Engineering National Taiwan University ihwang@ntu.edu.tw December 23, 2015 1 / 50 IHsiang
More informationStatistics for Particle Physics. Kyle Cranmer. New York University. Kyle Cranmer (NYU) CERN Academic Training, Feb 25, 2009
Statistics for Particle Physics Kyle Cranmer New York University 91 Remaining Lectures Lecture 3:! Compound hypotheses, nuisance parameters, & similar tests! The NeymanConstruction (illustrated)! Inverted
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 informationSTA 2101/442 Assignment 3 1
STA 2101/442 Assignment 3 1 These questions are practice for the midterm and final exam, and are not to be handed in. 1. Suppose X 1,..., X n are a random sample from a distribution with mean µ and variance
More informationComposite Hypotheses and Generalized Likelihood Ratio Tests
Composite Hypotheses and Generalized Likelihood Ratio Tests Rebecca Willett, 06 In many real world problems, it is difficult to precisely specify probability distributions. Our models for data may involve
More informationTUTORIAL 8 SOLUTIONS #
TUTORIAL 8 SOLUTIONS #9.11.21 Suppose that a single observation X is taken from a uniform density on [0,θ], and consider testing H 0 : θ = 1 versus H 1 : θ =2. (a) Find a test that has significance level
More information4 Hypothesis testing. 4.1 Types of hypothesis and types of error 4 HYPOTHESIS TESTING 49
4 HYPOTHESIS TESTING 49 4 Hypothesis testing In sections 2 and 3 we considered the problem of estimating a single parameter of interest, θ. In this section we consider the related problem of testing whether
More informationLectures 5 & 6: Hypothesis Testing
Lectures 5 & 6: Hypothesis Testing in which you learn to apply the concept of statistical significance to OLS estimates, learn the concept of t values, how to use them in regression work and come across
More informationIntroduction 1. STA442/2101 Fall See last slide for copyright information. 1 / 33
Introduction 1 STA442/2101 Fall 2016 1 See last slide for copyright information. 1 / 33 Background Reading Optional Chapter 1 of Linear models with R Chapter 1 of Davison s Statistical models: Data, and
More informationH 2 : otherwise. that is simply the proportion of the sample points below level x. For any fixed point x the law of large numbers gives that
Lecture 28 28.1 KolmogorovSmirnov test. Suppose that we have an i.i.d. sample X 1,..., X n with some unknown distribution and we would like to test the hypothesis that is equal to a particular distribution
More informationLAB 2. HYPOTHESIS TESTING IN THE BIOLOGICAL SCIENCES Part 2
LAB 2. HYPOTHESIS TESTING IN THE BIOLOGICAL SCIENCES Part 2 Data Analysis: The mean egg masses (g) of the two different types of eggs may be exactly the same, in which case you may be tempted to accept
More information2. What are the tradeoffs among different measures of error (e.g. probability of false alarm, probability of miss, etc.)?
ECE 830 / CS 76 Spring 06 Instructors: R. Willett & R. Nowak Lecture 3: Likelihood ratio tests, NeymanPearson detectors, ROC curves, and sufficient statistics Executive summary In the last lecture we
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 informationDirection: This test is worth 250 points and each problem worth points. DO ANY SIX
Term Test 3 December 5, 2003 Name Math 52 Student Number Direction: This test is worth 250 points and each problem worth 4 points DO ANY SIX PROBLEMS You are required to complete this test within 50 minutes
More informationLecture 8: Information Theory and Statistics
Lecture 8: Information Theory and Statistics Part II: Hypothesis Testing and Estimation IHsiang Wang Department of Electrical Engineering National Taiwan University ihwang@ntu.edu.tw December 22, 2015
More informationInterval Estimation. Chapter 9
Chapter 9 Interval Estimation 9.1 Introduction Definition 9.1.1 An interval estimate of a realvalues parameter θ is any pair of functions, L(x 1,..., x n ) and U(x 1,..., x n ), of a sample that satisfy
More informationBasic Concepts of Inference
Basic Concepts of Inference Corresponds to Chapter 6 of Tamhane and Dunlop Slides prepared by Elizabeth Newton (MIT) with some slides by Jacqueline Telford (Johns Hopkins University) and Roy Welsch (MIT).
More informationHomework 7: Solutions. P3.1 from Lehmann, Romano, Testing Statistical Hypotheses.
Stat 300A Theory of Statistics Homework 7: Solutions Nikos Ignatiadis Due on November 28, 208 Solutions should be complete and concisely written. Please, use a separate sheet or set of sheets for each
More informationCherry Blossom run (1) The credit union Cherry Blossom Run is a 10 mile race that takes place every year in D.C. In 2009 there were participants
18.650 Statistics for Applications Chapter 5: Parametric hypothesis testing 1/37 Cherry Blossom run (1) The credit union Cherry Blossom Run is a 10 mile race that takes place every year in D.C. In 2009
More informationUsing R in Undergraduate and Graduate Probability and Mathematical Statistics Courses*
Using R in Undergraduate and Graduate Probability and Mathematical Statistics Courses* Amy G. Froelich Michael D. Larsen Iowa State University *The work presented in this talk was partially supported by
More informationTopic 17: Simple Hypotheses
Topic 17: November, 2011 1 Overview and Terminology Statistical hypothesis testing is designed to address the question: Do the data provide sufficient evidence to conclude that we must depart from our
More informationReview. December 4 th, Review
December 4 th, 2017 Att. Final exam: Course evaluation Friday, 12/14/2018, 10:30am 12:30pm Gore Hall 115 Overview Week 2 Week 4 Week 7 Week 10 Week 12 Chapter 6: Statistics and Sampling Distributions Chapter
More informationEconometrics. 4) Statistical inference
30C00200 Econometrics 4) Statistical inference Timo Kuosmanen Professor, Ph.D. http://nomepre.net/index.php/timokuosmanen Today s topics Confidence intervals of parameter estimates Student s tdistribution
More informationSTAT Chapter 8: Hypothesis Tests
STAT 515  Chapter 8: Hypothesis Tests CIs are possibly the most useful forms of inference because they give a range of reasonable values for a parameter. But sometimes we want to know whether one particular
More informationTests and Their Power
Tests and Their Power Ling Kiong Doong Department of Mathematics National University of Singapore 1. Introduction In Statistical Inference, the two main areas of study are estimation and testing of hypotheses.
More informationChapter 5: HYPOTHESIS TESTING
MATH411: Applied Statistics Dr. YU, Chi Wai Chapter 5: HYPOTHESIS TESTING 1 WHAT IS HYPOTHESIS TESTING? As its name indicates, it is about a test of hypothesis. To be more precise, we would first translate
More informationPsychology 282 Lecture #4 Outline Inferences in SLR
Psychology 282 Lecture #4 Outline Inferences in SLR Assumptions To this point we have not had to make any distributional assumptions. Principle of least squares requires no assumptions. Can use correlations
More informationStat 5101 Lecture Notes
Stat 5101 Lecture Notes Charles J. Geyer Copyright 1998, 1999, 2000, 2001 by Charles J. Geyer May 7, 2001 ii Stat 5101 (Geyer) Course Notes Contents 1 Random Variables and Change of Variables 1 1.1 Random
More informationParameter Estimation and Fitting to Data
Parameter Estimation and Fitting to Data Parameter estimation Maximum likelihood Least squares Goodnessoffit Examples Elton S. Smith, Jefferson Lab 1 Parameter estimation Properties of estimators 3 An
More informationLecture 4: Testing Stuff
Lecture 4: esting Stuff. esting Hypotheses usually has three steps a. First specify a Null Hypothesis, usually denoted, which describes a model of H 0 interest. Usually, we express H 0 as a restricted
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 informationEcon 325: Introduction to Empirical Economics
Econ 325: Introduction to Empirical Economics Chapter 9 Hypothesis Testing: Single Population Ch. 91 9.1 What is a Hypothesis? A hypothesis is a claim (assumption) about a population parameter: population
More informationStatistical Theory MT 2007 Problems 4: Solution sketches
Statistical Theory MT 007 Problems 4: Solution sketches 1. Consider a 1parameter exponential family model with density f(x θ) = f(x)g(θ)exp{cφ(θ)h(x)}, x X. Suppose that the prior distribution has the
More informationMath 152. Rumbos Fall Solutions to Exam #2
Math 152. Rumbos Fall 2009 1 Solutions to Exam #2 1. Define the following terms: (a) Significance level of a hypothesis test. Answer: The significance level, α, of a hypothesis test is the largest probability
More informationOnesample categorical data: approximate inference
Onesample categorical data: approximate inference Patrick Breheny October 6 Patrick Breheny Biostatistical Methods I (BIOS 5710) 1/25 Introduction It is relatively easy to think about the distribution
More informationLECTURE 5 HYPOTHESIS TESTING
October 25, 2016 LECTURE 5 HYPOTHESIS TESTING Basic concepts In this lecture we continue to discuss the normal classical linear regression defined by Assumptions A1A5. Let θ Θ R d be a parameter of interest.
More information