A NEW METHOD FOR CONSTRUCTING APPROXIMATE CONFIDENCE INTERVALS FOR M-ESTU1ATES. Dennis D. Boos
|
|
- Malcolm Matthews
- 5 years ago
- Views:
Transcription
1 .- A NEW METHOD FOR CONSTRUCTING APPROXIMATE CONFIDENCE INTERVALS FOR M-ESTU1ATES by Deis D. Boos Departmet of Statistics North Carolia State Uiversity Istitute of Statistics Mimeo Series #1198 September, 1978
2 e A New Method for Costructig Approximate Cofidece Itervals for M-estimates by Deis D. Boos The empirical fuctio used to defie M-estimates of locatio is similar to a distributio fuctio whe ~ is odecreasig. This similarity allows approximate cofidece itervals to be costructed from the "percetiles" of the defiig fuctio. KEY WORDS: M-estimates; Cofidece itervals; Quatiles; t statistic. "e 1
3 1. INTRODUCTION Let Xl,,X be a sample from a distributio F ad defie the locatio "parameter" 6 to be the solutio of 00! ~(x-6)df(x) = 0. _00 (1.1) A M-estimate for 6 is the solutio e of the empirical aalogue to (1.1) 1 [~(x.-e) = 0. i=l ~ (1. 2) '-e Asymptotic properties of A 6 are well-kow ad the Priceto study Adrews, et al (1972) suggests that r ( -&) approaches ormality fairly quickly. Huber (1970), Gross (1976, 1977), ad Shorack (1976) have co- structed approximate cofidece itervals for 6 based o studetizatio of r (8-6) by estimates of the asymptotic stadard deviatio. I this paper a ew method of costructig approximate cofidece itervals for 6 is proposed for the special class of mootoe odecreasig, right cotiuous ~ fuctios. The method exploits the fact that A (c) F = --~~(Xi-c) is like a distributio fuctio (Le., odecreasig ad right cotiuous). I particular, the edpoits of the proposed cofidece iterval are "percetiles" of A F. Sectio 2 gives a motivatig example ad Sectio 3 provides the basic ideas ad method. I Sectio 4 Mote Carlo results ad comparisos with other results are metioed. Sectio 5 shows how to exted to the regressio situatio ad Sectio 6 is a short summary. 2-
4 e 2. MOTIVATION FROM QUANTILE ESTIMATION Let quatile -1 F (p) = if{x:f(x) ~ p}. Cosider estimatio of the pth -1 F (p),o<p<l, from a sample havig distributio F. If F is the usual empirical df, the = P(a<F (F- 1 (p»<b) usig the fact that all df's G satisfy G (t).::.x iff t<g(x). For idepedet X. ~ -1 the statistic F (F (p» is biomial -1 (,F(F (p»). Thus the ormal approximatio to the biomial ad the assumptio -1 F(F (p» '" p lead us to choose '-e for a I p(l-p) -a = b = p + ~ - za/2 approximate (l-a) cofidece iterval for -1 F (p) (z a is the 100 (l-a)th percetile of the stadard ormal.) Although exact oparametric procedures exist for iid samples from cotiuous distributios, the above method geeralizes to quatile estimatio i more complicated situatios, e.g., stratified samplig from fiite populatios. The importat poit for the preset discussio is that M-estimatio ca use the same idea with F replaced by A F. 3. APPROXIMATE CONFIDENCE INTERVALS FOR e Let ~(t) be odecreasig, right cotiuous, ad strictly positive (egative) for large positive (egative) values of t. Two families of such ~ are "Hubers" ~(x) = max(-k, mi(k,x» ad "vth power" ~(x) = IxIVsg(x),0<v.::.1. For df's G defie 3
5 e 00 AG(C) = - f ~(x-c)dg(x) _00 -oo<c<oo ad t s (if A G (x), -oo<x<oo sup AG(X)). -oo<x<oo The parameter ad estimate are defied by Similar to the case of df's it follows that ad thus 8 = A;l(O) ad -l- AG (t)<x p(af-l(a) < 8 < A;l(b)) P(a ~ A F (8) < b) where is a reasoable estimate of The statistic of iterest, r A F (8) T = ~,--_ o I 1 2: ~(x.-e) i=l 1. has a form very close to a t statistic based o the rv's ~(X.-8). 1. the X. are symmetric about 8 ad ~(x) = -~(-x), the we expect T 1. to be close to a t distributio with -l degrees of freedom. Choosig -I alo = t a/2 = I b/o ' our proposed approximate (l-a) cofidece iterval is If 4
6 e (3.1) Uder suitable regularity coditios, the asymptotic width of (3.1) is comparable to methods based o studetizig ~ (6-8), i.e., It is ofte desirable that locatio estimates satisfy ""- For M-estimates the usual procedure is to replace ~(x) where ~ is a suitable scale estimate, or solve simultaeous equatios as i Huber's Proposal 2. The above methods carry through exactly ad the aalogous statistic of iterest is 1:. ~ ~(Xi-8 \ i=l & I! I -l 2 x.- 8 l: ~._1._ i=l \ A cr (3.2) 4. COMPARISONS AND MONTE CARLO RESULTS For small samples the form (3.2) is more appealig tha ~ (6-8)/8, where A S is a estimate of the asymptotic stadard deviatio 5
7 e ~ar IC(X 1 )]\ for the followig reaso. Although 6-e' is approximated by - 1 EIC(X.), Boos (1977) shows that this approximatio is at best 0 (3. ~ p sice ~-6--1EIC(Xi8 has a limit distributio. Thus, proximity of I(e-6)/S to a t distributio depeds o the t-1ike statistic -~ "- 2C (X.) /S ~ ad the approximatio of "- 6-6 by I fact Gross (1976) prefers to avoid use of the t distributio. O the other had Shorack (1976) seems to get very good t approximatios for certai Hampe1s. I order to spot cheek the performace of the approximate cofidece itervals based o (3.2), a small Mote Carlo study was performed. I Table 1 is foud the empirical error probabilities ad I times the expected cofidece iterval legths (ECIL) for 10,000 Mote Carlo "samples" --e geerated by the M~Gi11 "Super-Duper" radom umber geerator. A differet set of 10,000 samples was used for each distributio - ormal, logistic, D-EXP = double expoetial, T3 = t distributio with 3 degrees of freedom, slash a stadard ormal deviate divided by a idepedet uiform (0,1) deviate, ad for each sample size, = 10 ad = 20. Oly crude Mote Carlo techiques were used, so cosiderable error may exist i the 3rd decimal of the empirical probabilities ad i the 2d decimal of the ECIL. This is exemplified by the mea whose exact error probability we kow to be.05 for the ormal. SQRT is the M-estimator based o ~ *(x) = Ixl sg(x~ ad Hk = 1.0, 1.5 are Hubers with k = 1.0, 1.5 usig a ormalized iterquarti1e rage as a estimate of scale. Hk* = 1.5 is Huber's Proposal 2 with k = 1.5. For both = 10 ad = 20 the true levels are geerally coservative, but Hk = 1.5 ad Hk* = 1.5 are fairly close to.05 except for the slash distributio ad each has reasoably short ECIL. It is mildly surprisig that the mea is so 6
8 .- Table l. Empirical Error Probabilities ad Expected 95-Percet Cofidece Iterval Legths (multiplied by ~) - 10,. 20 Estimator Normal Logistic D-Exp T3 Slash Normal Logistic D-Exp T3 Slash a Empirical Error Probabilities Mea SQRT Hk"l.O Hk Hk*"l b. Expected 95-Percet Cofidece Iterval Legths (multiplied by.~) Mea SQRT Hk..l.O Hk"l Hk*
9 close to.05 from ormal to T3, though the ECIL are expectedly large for e heavy tails. SQRT seems to perform worst over all. Table 2 represets Mote Carlo estimates of the percetiles of T cr. The percetiles ted to be larger tha those of a t distributio for the ormal ad geerally smaller for the heavier-tailed distributios. For = 20 ad a =.05 all estimates except for SQRT ad the mea evaluated at the slash distributio are very close to t. = 1.73 (we should ote 05 that the method of calculatig the estimated percetiles resulted i cosiderable error i the secod decimal place). 5. REGRESSION..- X. ~ P Lc..6.+U. j=l ~J J ~ The Huber (1973) regressio model is where E (U.) = 0 ad the c.. are kow coefficiets. Let ~ ~J (8,... l,8,a) p be solutios of p L W(X.- L c. 8 )c.. = 0 i=l cr ~ k=l ~ k k ~J j l,p, 1 2 p (-p) L W(X.- L c. k 8 k ) = s. i=l cr ~ k=l ~ Defie Q (t),r p p... = - E ~A(X.- E c_ k 8 k -c.. t)c. i=l cr ~ k=l ~ ~r ~r k~r r = l,p. 8
10 The 8 = Q-I (0) ad r,r p(q-i (a) < 8,r r < Q-I (b» = P(a < Q (8) < b).,r -,r r By Taylor expasio i 6 r 8 we fid r p = - L WA(X.- L c. k 8 k + c. (6-8)c. i=l 0 ~ k=l ~ ~r r r ~r PI, p A * 2 = - L WA(X.- L c. k 8 )c. + ~ k L WA(X.- L c. 8 +c 8 )c. k (8-8 ) i=l 0 ~ i=l ~ ~r 0 i=l 0 ~ k=l ~ k i r ~r r r The first term i the above expressio is 0 ad 8*...L-> 0 if -e e --p-> 8 r r Thus, uder suitable regularity coditios, asymptotically ormal with mea 0 ad variace -~ ~ (8) is,r r A approximate cofidece iterval for 8 r is IQ-I (-t /20 I~), Q-I (t /20 I)], I,r a,r a where The advatage of this method over the usual methods is ot clear (The simplicity of the locatio model is goe!). Note though, that use of W' is ot required ad that for least absolute value regressio the above method circumvets a estimate of f ~-l (~)J 9
11 e 6. SUMMARY AND CONCLUSIONS A ew procedure for costructig cofidece itervals for a locatio parameter has bee proposed which exploits the mootoicity of a class of ~ fuctios. The distributioal problem is reduced to cosideratio of a t-1ike statistic ad Mote Carlo results verify that "Hubers" perform fairly well over a rage of distributios ad for samples of size = 10 ad =
12 REFERENCES Adrews, D. F., et al. (1972), Robust Estimatio of Locatio: Survey ad Advaces, Priceto, N. J.: Priceto Uiversity Press. Boos, Deis D. (1977), "Limitig Secod Order Distributios for First Order Fuctioa1s, with Applicatio to L- ad M-Statistics," Istitute of Statistics Mimeo Series #1152, North Carolia State Uiversity, Raleigh, N. C. Gross, Ala M. (1976), "Cofidece Iterval Robustess with Log-Tailed Symmetric Distributios," Joural of the America Statistical Associatio 3 71, (1977), "Cofidece Itervals for Bisquare Regressio Estimates," Joural of the America Statistical Associatio 3 72, Huber, Peter J. (1970), "Studetizig Robust Estimates," i Noparametric Techiques i Statistical Iferece, ed. Mada L. Puri, Cambridge: Cambridge Uiversity Press, (1973), "Robust Regressio: Asymptotics, Cojectures, ad Mote Carlo," Aals of Statistics, 1, Shorack, Gale R. (1976), "Robust Studetizatio of Locatio Estimates," Statistica Neerladica, 30,
Resampling Methods. X (1/2), i.e., Pr (X i m) = 1/2. We order the data: X (1) X (2) X (n). Define the sample median: ( n.
Jauary 1, 2019 Resamplig Methods Motivatio We have so may estimators with the property θ θ d N 0, σ 2 We ca also write θ a N θ, σ 2 /, where a meas approximately distributed as Oce we have a cosistet estimator
More informationConfidence Interval for Standard Deviation of Normal Distribution with Known Coefficients of Variation
Cofidece Iterval for tadard Deviatio of Normal Distributio with Kow Coefficiets of Variatio uparat Niwitpog Departmet of Applied tatistics, Faculty of Applied ciece Kig Mogkut s Uiversity of Techology
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 informationChapter 6 Sampling Distributions
Chapter 6 Samplig Distributios 1 I most experimets, we have more tha oe measuremet for ay give variable, each measuremet beig associated with oe radomly selected a member of a populatio. Hece we eed to
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 information1 Introduction to reducing variance in Monte Carlo simulations
Copyright c 010 by Karl Sigma 1 Itroductio to reducig variace i Mote Carlo simulatios 11 Review of cofidece itervals for estimatig a mea I statistics, we estimate a ukow mea µ = E(X) of a distributio by
More informationInterval Estimation (Confidence Interval = C.I.): An interval estimate of some population parameter is an interval of the form (, ),
Cofidece Iterval Estimatio Problems Suppose we have a populatio with some ukow parameter(s). Example: Normal(,) ad are parameters. We eed to draw coclusios (make ifereces) about the ukow parameters. We
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 informationApproximate Confidence Interval for the Reciprocal of a Normal Mean with a Known Coefficient of Variation
Metodološki zvezki, Vol. 13, No., 016, 117-130 Approximate Cofidece Iterval for the Reciprocal of a Normal Mea with a Kow Coefficiet of Variatio Wararit Paichkitkosolkul 1 Abstract A approximate cofidece
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 informationA new distribution-free quantile estimator
Biometrika (1982), 69, 3, pp. 635-40 Prited i Great Britai 635 A ew distributio-free quatile estimator BY FRANK E. HARRELL Cliical Biostatistics, Duke Uiversity Medical Ceter, Durham, North Carolia, U.S.A.
More informationExpectation and Variance of a random variable
Chapter 11 Expectatio ad Variace of a radom variable The aim of this lecture is to defie ad itroduce mathematical Expectatio ad variace of a fuctio of discrete & cotiuous radom variables ad the distributio
More informationFACULTY OF MATHEMATICAL STUDIES MATHEMATICS FOR PART I ENGINEERING. Lectures
FACULTY OF MATHEMATICAL STUDIES MATHEMATICS FOR PART I ENGINEERING Lectures MODULE 5 STATISTICS II. Mea ad stadard error of sample data. Biomial distributio. Normal distributio 4. Samplig 5. Cofidece itervals
More information(6) Fundamental Sampling Distribution and Data Discription
34 Stat Lecture Notes (6) Fudametal Samplig Distributio ad Data Discriptio ( Book*: Chapter 8,pg5) Probability& Statistics for Egieers & Scietists By Walpole, Myers, Myers, Ye 8.1 Radom Samplig: Populatio:
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 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 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 informationGoodness-Of-Fit For The Generalized Exponential Distribution. Abstract
Goodess-Of-Fit For The Geeralized Expoetial Distributio By Amal S. Hassa stitute of Statistical Studies & Research Cairo Uiversity Abstract Recetly a ew distributio called geeralized expoetial or expoetiated
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 informationChapter 3. Strong convergence. 3.1 Definition of almost sure convergence
Chapter 3 Strog covergece As poited out i the Chapter 2, there are multiple ways to defie the otio of covergece of a sequece of radom variables. That chapter defied covergece i probability, covergece i
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 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 informationA statistical method to determine sample size to estimate characteristic value of soil parameters
A statistical method to determie sample size to estimate characteristic value of soil parameters Y. Hojo, B. Setiawa 2 ad M. Suzuki 3 Abstract Sample size is a importat factor to be cosidered i determiig
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 informationDepartment of Mathematics
Departmet of Mathematics Ma 3/103 KC Border Itroductio to Probability ad Statistics Witer 2017 Lecture 19: Estimatio II Relevat textbook passages: Larse Marx [1]: Sectios 5.2 5.7 19.1 The method of momets
More informationStatistics 511 Additional Materials
Cofidece Itervals o mu Statistics 511 Additioal Materials This topic officially moves us from probability to statistics. We begi to discuss makig ifereces about the populatio. Oe way to differetiate probability
More information(7 One- and Two-Sample Estimation Problem )
34 Stat Lecture Notes (7 Oe- ad Two-Sample Estimatio Problem ) ( Book*: Chapter 8,pg65) Probability& Statistics for Egieers & Scietists By Walpole, Myers, Myers, Ye Estimatio 1 ) ( ˆ S P i i Poit estimate:
More informationStatisticians use the word population to refer the total number of (potential) observations under consideration
6 Samplig Distributios Statisticias use the word populatio to refer the total umber of (potetial) observatios uder cosideratio The populatio is just the set of all possible outcomes i our sample space
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 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 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 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 informationBinomial Distribution
0.0 0.5 1.0 1.5 2.0 2.5 3.0 0 1 2 3 4 5 6 7 0.0 0.5 1.0 1.5 2.0 2.5 3.0 Overview Example: coi tossed three times Defiitio Formula Recall that a r.v. is discrete if there are either a fiite umber of possible
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 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 informationEstimation of a population proportion March 23,
1 Social Studies 201 Notes for March 23, 2005 Estimatio of a populatio proportio Sectio 8.5, p. 521. For the most part, we have dealt with meas ad stadard deviatios this semester. This sectio of the otes
More information1 of 7 7/16/2009 6:06 AM Virtual Laboratories > 6. Radom Samples > 1 2 3 4 5 6 7 6. Order Statistics Defiitios Suppose agai that we have a basic radom experimet, ad that X is a real-valued radom variable
More information17. Joint distributions of extreme order statistics Lehmann 5.1; Ferguson 15
17. Joit distributios of extreme order statistics Lehma 5.1; Ferguso 15 I Example 10., we derived the asymptotic distributio of the maximum from a radom sample from a uiform distributio. We did this usig
More 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 informationBIOS 4110: Introduction to Biostatistics. Breheny. Lab #9
BIOS 4110: Itroductio to Biostatistics Brehey Lab #9 The Cetral Limit Theorem is very importat i the realm of statistics, ad today's lab will explore the applicatio of it i both categorical ad cotiuous
More informationConfidence intervals summary Conservative and approximate confidence intervals for a binomial p Examples. MATH1005 Statistics. Lecture 24. M.
MATH1005 Statistics Lecture 24 M. Stewart School of Mathematics ad Statistics Uiversity of Sydey Outlie Cofidece itervals summary Coservative ad approximate cofidece itervals for a biomial p The aïve iterval
More informationMath 113 Exam 3 Practice
Math Exam Practice Exam 4 will cover.-., 0. ad 0.. Note that eve though. was tested i exam, questios from that sectios may also be o this exam. For practice problems o., refer to the last review. This
More informationESTIMATION AND PREDICTION BASED ON K-RECORD VALUES FROM NORMAL DISTRIBUTION
STATISTICA, ao LXXIII,. 4, 013 ESTIMATION AND PREDICTION BASED ON K-RECORD VALUES FROM NORMAL DISTRIBUTION Maoj Chacko Departmet of Statistics, Uiversity of Kerala, Trivadrum- 695581, Kerala, Idia M. Shy
More informationApril 18, 2017 CONFIDENCE INTERVALS AND HYPOTHESIS TESTING, UNDERGRADUATE MATH 526 STYLE
April 18, 2017 CONFIDENCE INTERVALS AND HYPOTHESIS TESTING, UNDERGRADUATE MATH 526 STYLE TERRY SOO Abstract These otes are adapted from whe I taught Math 526 ad meat to give a quick itroductio to cofidece
More informationProbability and statistics: basic terms
Probability ad statistics: basic terms M. Veeraraghava August 203 A radom variable is a rule that assigs a umerical value to each possible outcome of a experimet. Outcomes of a experimet form the sample
More informationDiscrete Mathematics for CS Spring 2008 David Wagner Note 22
CS 70 Discrete Mathematics for CS Sprig 2008 David Wager Note 22 I.I.D. Radom Variables Estimatig the bias of a coi Questio: We wat to estimate the proportio p of Democrats i the US populatio, by takig
More 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 information1 Inferential Methods for Correlation and Regression Analysis
1 Iferetial Methods for Correlatio ad Regressio Aalysis I the chapter o Correlatio ad Regressio Aalysis tools for describig bivariate cotiuous data were itroduced. The sample Pearso Correlatio Coefficiet
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 informationChapter 8: Estimating with Confidence
Chapter 8: Estimatig with Cofidece Sectio 8.2 The Practice of Statistics, 4 th editio For AP* STARNES, YATES, MOORE Chapter 8 Estimatig with Cofidece 8.1 Cofidece Itervals: The Basics 8.2 8.3 Estimatig
More informationIt should be unbiased, or approximately unbiased. Variance of the variance estimator should be small. That is, the variance estimator is stable.
Chapter 10 Variace Estimatio 10.1 Itroductio Variace estimatio is a importat practical problem i survey samplig. Variace estimates are used i two purposes. Oe is the aalytic purpose such as costructig
More information7-1. Chapter 4. Part I. Sampling Distributions and Confidence Intervals
7-1 Chapter 4 Part I. Samplig Distributios ad Cofidece Itervals 1 7- Sectio 1. Samplig Distributio 7-3 Usig Statistics Statistical Iferece: Predict ad forecast values of populatio parameters... Test hypotheses
More informationA Note on Box-Cox Quantile Regression Estimation of the Parameters of the Generalized Pareto Distribution
A Note o Box-Cox Quatile Regressio Estimatio of the Parameters of the Geeralized Pareto Distributio JM va Zyl Abstract: Makig use of the quatile equatio, Box-Cox regressio ad Laplace distributed disturbaces,
More informationSome Properties of the Exact and Score Methods for Binomial Proportion and Sample Size Calculation
Some Properties of the Exact ad Score Methods for Biomial Proportio ad Sample Size Calculatio K. KRISHNAMOORTHY AND JIE PENG Departmet of Mathematics, Uiversity of Louisiaa at Lafayette Lafayette, LA 70504-1010,
More informationTHE DATA-BASED CHOICE OF BANDWIDTH FOR KERNEL QUANTILE ESTIMATOR OF VAR
Iteratioal Joural of Iovative Maagemet, Iformatio & Productio ISME Iteratioal c2013 ISSN 2185-5439 Volume 4, Number 1, Jue 2013 PP. 17-24 THE DATA-BASED CHOICE OF BANDWIDTH FOR KERNEL QUANTILE ESTIMATOR
More informationA quick activity - Central Limit Theorem and Proportions. Lecture 21: Testing Proportions. Results from the GSS. Statistics and the General Population
A quick activity - Cetral Limit Theorem ad Proportios Lecture 21: Testig Proportios Statistics 10 Coli Rudel Flip a coi 30 times this is goig to get loud! Record the umber of heads you obtaied ad calculate
More information4. Partial Sums and the Central Limit Theorem
1 of 10 7/16/2009 6:05 AM Virtual Laboratories > 6. Radom Samples > 1 2 3 4 5 6 7 4. Partial Sums ad the Cetral Limit Theorem The cetral limit theorem ad the law of large umbers are the two fudametal theorems
More informationComparing Two Populations. Topic 15 - Two Sample Inference I. Comparing Two Means. Comparing Two Pop Means. Background Reading
Topic 15 - Two Sample Iferece I STAT 511 Professor Bruce Craig Comparig Two Populatios Research ofte ivolves the compariso of two or more samples from differet populatios Graphical summaries provide visual
More informationMATH/STAT 352: Lecture 15
MATH/STAT 352: Lecture 15 Sectios 5.2 ad 5.3. Large sample CI for a proportio ad small sample CI for a mea. 1 5.2: Cofidece Iterval for a Proportio Estimatig proportio of successes i a biomial experimet
More informationA goodness-of-fit test based on the empirical characteristic function and a comparison of tests for normality
A goodess-of-fit test based o the empirical characteristic fuctio ad a compariso of tests for ormality J. Marti va Zyl Departmet of Mathematical Statistics ad Actuarial Sciece, Uiversity of the Free State,
More information[ ] ( ) ( ) [ ] ( ) 1 [ ] [ ] Sums of Random Variables Y = a 1 X 1 + a 2 X 2 + +a n X n The expected value of Y is:
PROBABILITY FUNCTIONS A radom variable X has a probabilit associated with each of its possible values. The probabilit is termed a discrete probabilit if X ca assume ol discrete values, or X = x, x, x 3,,
More informationModule 1 Fundamentals in statistics
Normal Distributio Repeated observatios that differ because of experimetal error ofte vary about some cetral value i a roughly symmetrical distributio i which small deviatios occur much more frequetly
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 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 informationMa 530 Introduction to Power Series
Ma 530 Itroductio to Power Series Please ote that there is material o power series at Visual Calculus. Some of this material was used as part of the presetatio of the topics that follow. What is a Power
More informationThe standard deviation of the mean
Physics 6C Fall 20 The stadard deviatio of the mea These otes provide some clarificatio o the distictio betwee the stadard deviatio ad the stadard deviatio of the mea.. The sample mea ad variace Cosider
More informationInferential Statistics. Inference Process. Inferential Statistics and Probability a Holistic Approach. Inference Process.
Iferetial Statistics ad Probability a Holistic Approach Iferece Process Chapter 8 Poit Estimatio ad Cofidece Itervals This Course Material by Maurice Geraghty is licesed uder a Creative Commos Attributio-ShareAlike
More informationDS 100: Principles and Techniques of Data Science Date: April 13, Discussion #10
DS 00: Priciples ad Techiques of Data Sciece Date: April 3, 208 Name: Hypothesis Testig Discussio #0. Defie these terms below as they relate to hypothesis testig. a) Data Geeratio Model: Solutio: A set
More informationTests of Hypotheses Based on a Single Sample (Devore Chapter Eight)
Tests of Hypotheses Based o a Sigle Sample Devore Chapter Eight MATH-252-01: Probability ad Statistics II Sprig 2018 Cotets 1 Hypothesis Tests illustrated with z-tests 1 1.1 Overview of Hypothesis Testig..........
More informationSummary. 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 informationEconomics Spring 2015
1 Ecoomics 400 -- Sprig 015 /17/015 pp. 30-38; Ch. 7.1.4-7. New Stata Assigmet ad ew MyStatlab assigmet, both due Feb 4th Midterm Exam Thursday Feb 6th, Chapters 1-7 of Groeber text ad all relevat lectures
More informationDouble Stage Shrinkage Estimator of Two Parameters. Generalized Exponential Distribution
Iteratioal Mathematical Forum, Vol., 3, o. 3, 3-53 HIKARI Ltd, www.m-hikari.com http://dx.doi.org/.9/imf.3.335 Double Stage Shrikage Estimator of Two Parameters Geeralized Expoetial Distributio Alaa M.
More informationPOWER COMPARISON OF EMPIRICAL LIKELIHOOD RATIO TESTS: SMALL SAMPLE PROPERTIES THROUGH MONTE CARLO STUDIES*
Kobe Uiversity Ecoomic Review 50(2004) 3 POWER COMPARISON OF EMPIRICAL LIKELIHOOD RATIO TESTS: SMALL SAMPLE PROPERTIES THROUGH MONTE CARLO STUDIES* By HISASHI TANIZAKI There are various kids of oparametric
More informationStatistical Intervals for a Single Sample
3/5/06 Applied Statistics ad Probability for Egieers Sixth Editio Douglas C. Motgomery George C. Ruger Chapter 8 Statistical Itervals for a Sigle Sample 8 CHAPTER OUTLINE 8- Cofidece Iterval o the Mea
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 informationBull. Korean Math. Soc. 36 (1999), No. 3, pp. 451{457 THE STRONG CONSISTENCY OF NONLINEAR REGRESSION QUANTILES ESTIMATORS Seung Hoe Choi and Hae Kyung
Bull. Korea Math. Soc. 36 (999), No. 3, pp. 45{457 THE STRONG CONSISTENCY OF NONLINEAR REGRESSION QUANTILES ESTIMATORS Abstract. This paper provides suciet coditios which esure the strog cosistecy of regressio
More informationChapter 2 The Monte Carlo Method
Chapter 2 The Mote Carlo Method The Mote Carlo Method stads for a broad class of computatioal algorithms that rely o radom sampligs. It is ofte used i physical ad mathematical problems ad is most useful
More informationConfidence Intervals รศ.ดร. อน นต ผลเพ ม Assoc.Prof. Anan Phonphoem, Ph.D. Intelligent Wireless Network Group (IWING Lab)
Cofidece Itervals รศ.ดร. อน นต ผลเพ ม Assoc.Prof. Aa Phophoem, Ph.D. aa.p@ku.ac.th Itelliget Wireless Network Group (IWING Lab) http://iwig.cpe.ku.ac.th Computer Egieerig Departmet Kasetsart Uiversity,
More informationOutput Analysis and Run-Length Control
IEOR E4703: Mote Carlo Simulatio Columbia Uiversity c 2017 by Marti Haugh Output Aalysis ad Ru-Legth Cotrol I these otes we describe how the Cetral Limit Theorem ca be used to costruct approximate (1 α%
More 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 information4.5 Multiple Imputation
45 ultiple Imputatio Itroductio Assume a parametric model: y fy x; θ We are iterested i makig iferece about θ I Bayesia approach, we wat to make iferece about θ from fθ x, y = πθfy x, θ πθfy x, θdθ where
More informationOverview. p 2. Chapter 9. Pooled Estimate of. q = 1 p. Notation for Two Proportions. Inferences about Two Proportions. Assumptions
Chapter 9 Slide Ifereces from Two Samples 9- Overview 9- Ifereces about Two Proportios 9- Ifereces about Two Meas: Idepedet Samples 9-4 Ifereces about Matched Pairs 9-5 Comparig Variatio i Two Samples
More informationChapter 6 Part 5. Confidence Intervals t distribution chi square distribution. October 23, 2008
Chapter 6 Part 5 Cofidece Itervals t distributio chi square distributio October 23, 2008 The will be o help sessio o Moday, October 27. Goal: To clearly uderstad the lik betwee probability ad cofidece
More informationIntroduction to Extreme Value Theory Laurens de Haan, ISM Japan, Erasmus University Rotterdam, NL University of Lisbon, PT
Itroductio to Extreme Value Theory Laures de Haa, ISM Japa, 202 Itroductio to Extreme Value Theory Laures de Haa Erasmus Uiversity Rotterdam, NL Uiversity of Lisbo, PT Itroductio to Extreme Value Theory
More informationChapter 8: STATISTICAL INTERVALS FOR A SINGLE SAMPLE. Part 3: Summary of CI for µ Confidence Interval for a Population Proportion p
Chapter 8: STATISTICAL INTERVALS FOR A SINGLE SAMPLE Part 3: Summary of CI for µ Cofidece Iterval for a Populatio Proportio p Sectio 8-4 Summary for creatig a 100(1-α)% CI for µ: Whe σ 2 is kow ad paret
More informationThere is no straightforward approach for choosing the warmup period l.
B. Maddah INDE 504 Discrete-Evet Simulatio Output Aalysis () Statistical Aalysis for Steady-State Parameters I a otermiatig simulatio, the iterest is i estimatig the log ru steady state measures of performace.
More informationSTAC51: Categorical data Analysis
STAC51: Categorical data Aalysis Mahida Samarakoo Jauary 28, 2016 Mahida Samarakoo STAC51: Categorical data Aalysis 1 / 35 Table of cotets Iferece for Proportios 1 Iferece for Proportios Mahida Samarakoo
More informationConfidence interval for the two-parameter exponentiated Gumbel distribution based on record values
Iteratioal Joural of Applied Operatioal Research Vol. 4 No. 1 pp. 61-68 Witer 2014 Joural homepage: www.ijorlu.ir Cofidece iterval for the two-parameter expoetiated Gumbel distributio based o record values
More informationON BARTLETT CORRECTABILITY OF EMPIRICAL LIKELIHOOD IN GENERALIZED POWER DIVERGENCE FAMILY. Lorenzo Camponovo and Taisuke Otsu.
ON BARTLETT CORRECTABILITY OF EMPIRICAL LIKELIHOOD IN GENERALIZED POWER DIVERGENCE FAMILY By Lorezo Campoovo ad Taisuke Otsu October 011 COWLES FOUNDATION DISCUSSION PAPER NO. 185 COWLES FOUNDATION FOR
More informationAccess to the published version may require journal subscription. Published with permission from: Elsevier.
This is a author produced versio of a paper published i Statistics ad Probability Letters. This paper has bee peer-reviewed, it does ot iclude the joural pagiatio. Citatio for the published paper: Forkma,
More information( µ /σ)ζ/(ζ+1) µ /σ ( µ /σ)ζ/(ζ 1)
A eective CI for the mea with samples of size 1 ad Melaie Wall James Boe ad Richard Tweedie 1 Abstract It is couterituitive that with a sample of oly oe value from a ormal distributio oe ca costruct a
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 informationA proposed discrete distribution for the statistical modeling of
It. Statistical Ist.: Proc. 58th World Statistical Cogress, 0, Dubli (Sessio CPS047) p.5059 A proposed discrete distributio for the statistical modelig of Likert data Kidd, Marti Cetre for Statistical
More informationOrthogonal Gaussian Filters for Signal Processing
Orthogoal Gaussia Filters for Sigal Processig Mark Mackezie ad Kiet Tieu Mechaical Egieerig Uiversity of Wollogog.S.W. Australia Abstract A Gaussia filter usig the Hermite orthoormal series of fuctios
More informationElement sampling: Part 2
Chapter 4 Elemet samplig: Part 2 4.1 Itroductio We ow cosider uequal probability samplig desigs which is very popular i practice. I the uequal probability samplig, we ca improve the efficiecy of the resultig
More informationM-Estimators in Regression Models
M-Estimators i Regressio Models MuthukrishaR Departmet of Statistics, Bharathiar Uiversity Coimbatore-641 046, Tamiladu, Idia E-mail: muthukrisha70@rediffmailcom RadhaM Departmet of Statistics, Bharathiar
More informationCEE 522 Autumn Uncertainty Concepts for Geotechnical Engineering
CEE 5 Autum 005 Ucertaity Cocepts for Geotechical Egieerig Basic Termiology Set A set is a collectio of (mutually exclusive) objects or evets. The sample space is the (collectively exhaustive) collectio
More information11 Correlation and Regression
11 Correlatio Regressio 11.1 Multivariate Data Ofte we look at data where several variables are recorded for the same idividuals or samplig uits. For example, at a coastal weather statio, we might record
More informationStochastic Simulation
Stochastic Simulatio 1 Itroductio Readig Assigmet: Read Chapter 1 of text. We shall itroduce may of the key issues to be discussed i this course via a couple of model problems. Model Problem 1 (Jackso
More informationConfidence Intervals for the Population Proportion p
Cofidece Itervals for the Populatio Proportio p The cocept of cofidece itervals for the populatio proportio p is the same as the oe for, the samplig distributio of the mea, x. The structure is idetical:
More information