TAMS24: Notations and Formulas
|
|
- Millicent Ross
- 5 years ago
- Views:
Transcription
1 TAMS4: Notatios ad Formulas Basic otatios ad defiitios X: radom variable stokastiska variabel Mea Vätevärde: µ = X = by Xiagfeg Yag kpx k, if X is discrete, xf Xxdx, if X is cotiuous Variace Varias: = V X = X µ = X X Stadard deviatio Stadardavvikelse: = DX = V X Populatio X Radom sample slumpmässigt stickprov: X,, X are idepedet ad have the same distributio as the populatio X Before observe/measure, X,, X are radom variables, ad after observe/measure, we use x,, x which are umbers ot radom variables Sample mea Stickprovsmedelvärde: Before observe/measure, X = X i, ad after observe/measure, x = x i Sample variace Stickprovsvarias: Before observe/measure, S = X i X, ad after observe/measure, s = x i x Sample stadard deviatio Stickprovsstadardavvikelse: Before observe/measure, S = S, ad after observe/measure, s = s c i X i = c i X i, V c i X i = c i V X i, if X,, X are idepedet oberoede If X Nµ,, the X µ N, If X,, X are idepedet ad X i Nµ i, i, the d + c i X i Nd + c i µ i, c i i For a populatio X with a ukow parameter θ, ad a radom sample X,, X } : stimator Stickprovsvariabel: ˆΘ = gx,, X, a radom variable / stimate Puktskattig: ˆθ = gx,, x, a umber Ubiased Vätevärdesriktig: ˆΘ = θ ffective ffektiv: Two estimators ˆΘ ad ˆΘ are ubiased, we say that ˆΘ is more effective tha ˆΘ if V ˆΘ < V ˆΘ Biomial distributio X BiN, p : there are N idepedet ad idetical trials, each trial has a probability of success p, ad X = the umber of successes i these N trials The radom variable X BiN, p has a probability fuctio saolikhetsfuktio N pk = P X = k = p k p N k k xpoetial distributio X xp/µ : whe we cosider the waitig time/lifetime The radom variable X xp/µ has a desity fuctio täthetsfuktio Poit estimatio fx = µ e x/µ, x Method of momets Mometmetode: # of equatios depeds o # of ukow parameters, X = x, X = x i, X 3 = x 3 i, Cosistet Kosistet: A estimator ˆΘ = gx,, X is cosistet if lim P ˆΘ θ > ε =, for ay costat ε > This is called covergece i probability Theorem: If ˆΘ = θ ad lim V ˆΘ =, the ˆΘ is cosistet Least square method mista-kvadrat-metode: The least square estimate ˆθ is the oe miimizig Qθ = x i X Maximum-likelihood method Maximum-likelihood-metode: The maximum-likelihood estimate ˆθ is the oe maximizig the likelihood fuctio Lθ = fx i θ, if X is cotiuous, px i θ, if X is discrete Remark o ML: I geeral, it is easier/better to maximize l Lθ Remark o ML: If there are several radom samples say m from differet populatios with a same ukow parameter θ, the the maximum-likelihood estimate ˆθ is the oe maximizig the likelihood fuctio defied as Lθ = L θ L m θ, where L i θ is the likelihood fuctio from the i-th populatio /
2 stimates of populatio variace : If there is oly oe populatio with a ukow mea, the method of momets ad maximum-likelihood method, i geeral, give a estimate of as follows = x i x NOT ubiased A adjusted or corrected estimate would be the sample variace s = x i x ubiased If there are m differet populatios with ukow meas ad a same variace, the a adjusted or corrected ML estimate is s = s + + m s m + + m ubiased where i is the sample size of the i-th populatio, ad s i is the sample variace of the i-th populatio Stadard error medelfelet of a estimator ˆΘ: is a estimate of the stadard deviatio D ˆΘ 3 Iterval estimatio Oe sample X,, X } from Nµ, Two samples X,, X } from Nµ, Y,, Y } from Nµ, Nµ, ad Nµ, are idepedet x λ α/, if is kow fact / I µ = N, x t α/ s, if is ukow fact s/ t s I = χ α, s fact S χ χ α Ukow ca be estimated by the sample variace s = x i x x ȳ λ α/ +, if ad are kow fact X Ȳ µ µ N, + x ȳ t α/ + s +, if = = is ukow X Ȳ µ µ fact I µ µ = S t + + s x ȳ t α/ f + s, if both are ukow fact X Ȳ µ µ S + S tf degrees of freedom f = s / + s / I = + s χ α, + s, if + χ α = = + fact + S χ + Ukow ca be estimated by the samples variace s = s + s + s /+s / Remark: The idea of usig fact to fid cofidece itervals is very importat There are a lot more differet cofidece itervals besides above For istace, we cosider two idepedet samples: X,, X } from Nµ, ad Y,, Y } from Nµ, I this case, we ca easily prove that c X + c Ȳ N c µ + c µ, If is kow, the fact c X+c Ȳ c µ +c µ c + c If is ukow, the fact c X+c Ȳ c µ +c µ S c + c N, So we ca fid I cµ +c µ c + c 3 Cofidece itervals from ormal approximatios t + So we ca fid I cµ +c µ ˆp ˆp X BiN, p : I p = ˆp λ α/ N, fact ˆP p N, ˆP ˆP N we require that N ˆp > ad N ˆp ˆp > N X HypN,, p : I p = ˆp λ α/ N ˆP ˆp ˆp, fact X P oµ : I µ = x λ α/ x, fact X µ X N, p N N ˆP ˆP we require that x > 5 X xp µ : I µ = x x + λ, α/ λ, fact X µ α/ µ/ N,, x X µ I µ = x λ α/, fact X/ N, we require that 3 N, Remark: Agai there are more cofidece itervals besides above For istace, we cosider two idepedet samples: X from BiN, p ad Y from BiN, p, with ukow p ad p As we kow p p ˆP N p, ad ˆP p p N p,, so ˆP ˆP N p p, p p + p p Therefore, fact is ˆP ˆP p p N,, ˆP ˆP + ˆP ˆP I p p = ˆp ˆp λ α/ ˆp ˆp + ˆp ˆp m samples: The ukow = = m = ca be estimated by s = s ++m s m ++ m 3/ 3 Cofidece itervals from the ratio of two populatio variaces 4/
3 Suppose there are two idepedet samples X,, X } from Nµ,, ad Y,, Y } from Nµ, The S Thus χ ad S χ, therefore S / S F,, / s I / = s F α,, s s fact F α, 33 Large sample size 3, populatio may be completely ukow If there is o iformatio about the populatios, the we ca apply Cetral Limit Theorem usually with a large sample 3 to get a approximated ormal distributios Here are two examples: xample : Let X,, X }, 3, be a radom sample from a populatio, the o matter what distributio the populatio is X µ s/ N, xample : Let X,, X }, 3, be a radom sample from a populatio, ad Y,, Y }, 3, be a radom sample from aother populatio which is idepedet from the first populatio, the o matter what distributios the populatios are 4 Hypothesis testig X Ȳ µ µ s + s N, 4 Oe sample ad the geeral theory of hypothesis testig Suppose there is a radom sample X,, X } from a populatio X with a ukow parameter θ, H : θ = θ vs H : θ < θ, or θ > θ, or θ θ H is true H is false ad θ = θ reject H type I error or sigificace level α power hθ do t reject H α type II error βθ = hθ Regardig the p-value: For otatioal simplicity, we employ reject H if ad oly if p-value < α TS := test statistic ad C := critical regio reject H if TS C reject H if ad oly if p-value < α 4 Hypothesis testig for populatio meas Oe sample: X,, X } from Nµ, Null hypothesis H : µ = µ H : µ < µ : TS = x µ is kow: H : µ > µ : TS = x µ / N, H : µ µ : TS = x µ H : µ < µ : TS = x µ is ukow: H : µ > µ : TS = x µ s/ t H : µ µ : TS = x µ /, C =, λ α, p-value = P N, TS /, C = λ α, +, p-value = P N, TS /, C =, λ α/ λ α/, +, p-value = P N, TS s/, C =, t α, p-value = P t TS s/, C = t α, +, p-value = P t TS s/, C =, t α/ t α/, +, p-value = P t TS Two samples: X,, X } from Nµ, Y,, Y } from Nµ, Null hypothesis H : µ = µ H : µ < µ : TS = x ȳ, C =, λ α, + p-value = P N, TS, are kow: H : µ > µ : TS = x ȳ, C = λ α, +, X Ȳ µ µ + N, + p-value = P N, TS H : µ µ : TS = x ȳ, C =, λ α/ λ α/, +, + p-value = P N, TS X Ȳ µ µ S + H : µ < µ : TS = x ȳ s, C =, t α +, + p-value = P t + TS x ȳ s + H = is ukow: : µ > µ : TS =, C = t α +, +, t + p-value = P t + TS H : µ µ : TS = x ȳ s, C =, t + α/ + p-value = P t + TS both ukow: similarly as i the tree of cofidece itervals t α/ +, +, 5/ 6/
4 43 Hypothesis testig for populatio variaces X,, X } from Nµ, S χ H : = H : < H : > H : s : TS =, C =, χ α, p-value = P χ TS s : TS =, C = χ α, +, p-value = P χ TS s : TS =, C =, χ α χ α, +, p-value = P χ TS or P χ TS H : < : TS = s /s, C =, F α,, p-value = P F, TS X,, X } from Nµ, H : Y,, Y } from Nµ, > : TS = s /s, C = F α,, +, p-value = P F, TS S / F S, H : / : TS = s /s, C =, F α, H : = F α,, +, p-value = P F, TS or P F, TS X ad Y are ucorrelated: if covx, Y = A importat theorem: Suppose that a radom vector X has a mea µ X ad a covariace matrix C X Defie a ew radom vector Y = AX + b, for some matrix A ad vector b The µ Y = Aµ X + b, C Y = AC X A Stadard ormal vectors: X i } are idepedet ad X i N,, X X X =, thus µ X =, C X =, desity f Xx = π e X Geeral ormal vectors: Y = AX + b, where X is a stadard ormal vector, ad µ Y = b, C Y = AA, desity f Y y = π detc Y e 6 Simple ad multiple Liear regressios y µ y C y µy Y x x 44 Large sample size 3, populatio may be completely ukow If there is o iformatio about the populatios, the we ca apply Cetral Limit Theorem usually with a large sample 3 The idea is exactly the same as the oe used i cofidece itervals Oe example is: a sample X,, X }, 3, from some populatio which is ukow with a mea µ ad stadard deviatio Null hypothesis H : µ = µ The it follows from CLT that s/ N,, therefore H : µ < µ : TS = x µ H : µ > µ : TS = x µ s/, C =, λ α, p-value = P N, TS s/, C = λ α, +, p-value = P N, TS H : µ µ : TS = x µ s/, C =, λ α/ λ α/, +, p-value = P N, TS 5 Multi-dimesio radom variables or radom vectors Covariace Kovarias of X, Y : X,Y = covx, Y = X µ X Y µ Y, covx, X = V X Correlatio coefficiet Korrelatio of X, Y : ρ X,Y = covx,y = X,Y V X V Y X Y A rule: for real costats a, a i, b ad b j, m m cova + a i X i, b + b j Y j = a i b j covx i, Y j j= 7/ j= Simple liear regressio: Y j = β + β x j + ε j, ε j N,, j =,, Multiple liear regressio: Y j = β + β x j + β x j + + β k x jk + ε j, ε j N,, j =,, Both Simple liear regressio ad Multiple liear regressio ca be writte as vector forms: Y x x k Y Y = Xβ + ε : Y =, X = x x k β, β =, ε N, I β Y x x k k Y Nµ Y, C Y, where µ Y = Xβ ad C Y = I stimate of the coefficiet β: ˆβ = X X X y stimator of the coefficiet β: ˆB = X X X Y N β, X X stimated lie is: ˆµ j = ˆβ + ˆβ x j + ˆβ x j + + ˆβ k x jk Aalysis of variace: SS T OT = SS R = SS = y j ȳ, j= ˆµ j ȳ, j= y j ˆµ j, j= SS T OT j= = Y j Ȳ χ, if β = = β k = j= ˆµ j Ȳ SS R = SS = χ k, if β = = β k = j= Y j ˆµ j χ k 8/
5 SS T OT = SS R + SS, ad R = SS R SS T OT is estimated as ˆ = S = SS k For the Hypothesis testig: H : β = = β k = vs H : at least oe β j, SS R/k SS / k F k, k TS = SS R/k SS / k C = F α k, k, + We kow ˆB = X X X Y N β, X X, thus if we deote h h h k X X h h h k =, h k h k h kk ad we wat to test H : β k+ = = β k+p = vs H : at least oe β k+i, SS SS /p F p, k p / k p SS TS = SS SS SS /p / k p C = F α p, k p, + Variable selectio If we have a respose variable y with possibly may predictors x,, x k, the how to choose appropriate x s some x s are useful to Y, ad some are ot: Step : corrx,, x k, y, choose a maximal correlatio say x i, Y = β + β i x i + ε, test if β i =? Step : do regressio Y = β + β i x i + β x + ε for =,, i, i +,, k, choose a miimal SS say x j, Y = β + β i x i + β j x j + ε, test if β j =? Step 3: repeat Step util the last test for β = is ot rejected the ˆB j Nβ j, h jj ad ˆB j β j N, But is geerally ukow, therefore h jj ˆB j β j S h jj t k, Cofidece iterval of β j is: I βj = ˆβ j t α/ k s h jj s h jj is sometimes deoted as d ˆβ j or se ˆβ j 7 Basic χ -test Suppose we wat to test H : H : X distributio with or without ukow parameters X distributio with or without ukow parameters Hypothesis testig H : β j = vs H : β j has TS = ˆβ j s h jj C =, t α/ k t α/ k, + Rewrite simple ad multiple liear regressios as follows: Y = β + β x + + β k x k + ε, ε N,, the model µ = Y = β + β x + + β k x k, the mea ˆµ = ˆβ + ˆβ x + + ˆβ k x k, the estimated lie For a give/fixed x =, x,, x k, the scalar ˆµ is a estimate of ukow µ ad Y The we ca talk about accuracy of this estimate i terms of cofidece itervals ad predictio itervals Cofidece iterval of µ: I µ = ˆµ t α/ k s x X X x Predictio iterval of Y : I Y = ˆµ t α/ k s + x X X x Suppose we have two models: Model : Y = β + β x + + β k x k + ε Model : Y = β + β x + + β k x k + β k+ x k+ + + β k+p x k+p + ε, fact is : k N i p i p i χ k #of ukow parameters The TS = k N i p i p i C = χ αk #of ukow parameters, + Homogeeity test Suppose we have a data with r rows ad k colums, H : differet rows have a same patter i terms of colums quivaletly, The H : differet rows have differet patters i terms of colums H : H : rows ad colums are idepedet rows ad colums are ot idepedet fact is : k r N ij p ij j= p ij χ r k TS = k r N ij p ij j= p ij C = χ αr k, +, where p ij = p i q j are the theoretical probabilities 9/ /
Kurskod: TAMS24 (Statistisk teori)/provkod: TEN :00-12:00. English Version. 1 (3 points) 2 (3 points)
Kurskod: TAMS4 Statistisk teori)/provkod: TEN 08-08-4 08:00-:00 Examiator: Zhexia Liu Tel: 070089508) You are permitted to brig: a calculator, ad formel -och tabellsamlig i matematisk statistik Scores
More information(all terms are scalars).the minimization is clearer in sum notation:
7 Multiple liear regressio: with predictors) Depedet data set: y i i = 1, oe predictad, predictors x i,k i = 1,, k = 1, ' The forecast equatio is ŷ i = b + Use matrix otatio: k =1 b k x ik Y = y 1 y 1
More informationEnglish Version P (1 X < 1.5) P (X 1) = c[x3 /3 + x 2 ] 1.5. = c c[x 3 /3 + x 2 ] 2 1
Kurskod: TAMS28 Matematisk statistik/provkod: TEN 208-08-20 08:00-2:00 Examiator: Zhexia Liu Tel: 0700895208 You are permitted to brig: a calculator, ad formel -och tabellsamlig i matematisk statistik
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 informationSTATISTICAL INFERENCE
STATISTICAL INFERENCE POPULATION AND SAMPLE Populatio = all elemets of iterest Characterized by a distributio F with some parameter θ Sample = the data X 1,..., X, selected subset of the populatio = sample
More 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 informationKurskod: TAMS11 Provkod: TENB 21 March 2015, 14:00-18:00. English Version (no Swedish Version)
Kurskod: TAMS Provkod: TENB 2 March 205, 4:00-8:00 Examier: Xiagfeg Yag (Tel: 070 2234765). Please aswer i ENGLISH if you ca. a. You are allowed to use: a calculator; formel -och tabellsamlig i matematisk
More information1.010 Uncertainty in Engineering Fall 2008
MIT OpeCourseWare http://ocw.mit.edu.00 Ucertaity i Egieerig Fall 2008 For iformatio about citig these materials or our Terms of Use, visit: http://ocw.mit.edu.terms. .00 - Brief Notes # 9 Poit ad Iterval
More 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 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 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 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 informationThe variance of a sum of independent variables is the sum of their variances, since covariances are zero. Therefore. V (xi )= n n 2 σ2 = σ2.
SAMPLE STATISTICS A radom sample x 1,x,,x from a distributio f(x) is a set of idepedetly ad idetically variables with x i f(x) for all i Their joit pdf is f(x 1,x,,x )=f(x 1 )f(x ) f(x )= f(x i ) The sample
More 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 informationLecture 3. Properties of Summary Statistics: Sampling Distribution
Lecture 3 Properties of Summary Statistics: Samplig Distributio Mai Theme How ca we use math to justify that our umerical summaries from the sample are good summaries of the populatio? Lecture Summary
More informationGoodness-of-Fit Tests and Categorical Data Analysis (Devore Chapter Fourteen)
Goodess-of-Fit Tests ad Categorical Data Aalysis (Devore Chapter Fourtee) MATH-252-01: Probability ad Statistics II Sprig 2019 Cotets 1 Chi-Squared Tests with Kow Probabilities 1 1.1 Chi-Squared Testig................
More informationFinal Review. Fall 2013 Prof. Yao Xie, H. Milton Stewart School of Industrial Systems & Engineering Georgia Tech
Fial Review Fall 2013 Prof. Yao Xie, yao.xie@isye.gatech.edu H. Milto Stewart School of Idustrial Systems & Egieerig Georgia Tech 1 Radom samplig model radom samples populatio radom samples: x 1,..., x
More informationLinear regression. Daniel Hsu (COMS 4771) (y i x T i β)2 2πσ. 2 2σ 2. 1 n. (x T i β y i ) 2. 1 ˆβ arg min. β R n d
Liear regressio Daiel Hsu (COMS 477) Maximum likelihood estimatio Oe of the simplest liear regressio models is the followig: (X, Y ),..., (X, Y ), (X, Y ) are iid radom pairs takig values i R d R, ad Y
More informationResampling Methods. X (1/2), i.e., Pr (X i m) = 1/2. We order the data: X (1) X (2) X (n). Define the sample median: ( n.
Jauary 1, 2019 Resamplig Methods Motivatio We have so may estimators with the property θ θ d N 0, σ 2 We ca also write θ a N θ, σ 2 /, where a meas approximately distributed as Oce we have a cosistet estimator
More 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 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 informationMA Advanced Econometrics: Properties of Least Squares Estimators
MA Advaced Ecoometrics: Properties of Least Squares Estimators Karl Whela School of Ecoomics, UCD February 5, 20 Karl Whela UCD Least Squares Estimators February 5, 20 / 5 Part I Least Squares: Some Fiite-Sample
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 informationOpen book and notes. 120 minutes. Cover page and six pages of exam. No calculators.
IE 330 Seat # Ope book ad otes 120 miutes Cover page ad six pages of exam No calculators Score Fial Exam (example) Schmeiser Ope book ad otes No calculator 120 miutes 1 True or false (for each, 2 poits
More informationMathematical Statistics - MS
Paper Specific Istructios. The examiatio is of hours duratio. There are a total of 60 questios carryig 00 marks. The etire paper is divided ito three sectios, A, B ad C. All sectios are compulsory. Questios
More informationStatistics 20: Final Exam Solutions Summer Session 2007
1. 20 poits Testig for Diabetes. Statistics 20: Fial Exam Solutios Summer Sessio 2007 (a) 3 poits Give estimates for the sesitivity of Test I ad of Test II. Solutio: 156 patiets out of total 223 patiets
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 informationProbability 2 - Notes 10. Lemma. If X is a random variable and g(x) 0 for all x in the support of f X, then P(g(X) 1) E[g(X)].
Probability 2 - Notes 0 Some Useful Iequalities. Lemma. If X is a radom variable ad g(x 0 for all x i the support of f X, the P(g(X E[g(X]. Proof. (cotiuous case P(g(X Corollaries x:g(x f X (xdx x:g(x
More informationStat 200 -Testing Summary Page 1
Stat 00 -Testig Summary Page 1 Mathematicias are like Frechme; whatever you say to them, they traslate it ito their ow laguage ad forthwith it is somethig etirely differet Goethe 1 Large Sample Cofidece
More informationStat 319 Theory of Statistics (2) Exercises
Kig Saud Uiversity College of Sciece Statistics ad Operatios Research Departmet Stat 39 Theory of Statistics () Exercises Refereces:. Itroductio to Mathematical Statistics, Sixth Editio, by R. Hogg, J.
More informationLECTURE 8: ASYMPTOTICS I
LECTURE 8: ASYMPTOTICS I We are iterested i the properties of estimators as. Cosider a sequece of radom variables {, X 1}. N. M. Kiefer, Corell Uiversity, Ecoomics 60 1 Defiitio: (Weak covergece) A sequece
More informationAsymptotic Results for the Linear Regression Model
Asymptotic Results for the Liear Regressio Model C. Fli November 29, 2000 1. Asymptotic Results uder Classical Assumptios The followig results apply to the liear regressio model y = Xβ + ε, where X is
More informationCommon Large/Small Sample Tests 1/55
Commo Large/Small Sample Tests 1/55 Test of Hypothesis for the Mea (σ Kow) Covert sample result ( x) to a z value Hypothesis Tests for µ Cosider the test H :μ = μ H 1 :μ > μ σ Kow (Assume the populatio
More informationIntroductory statistics
CM9S: Machie Learig for Bioiformatics Lecture - 03/3/06 Itroductory statistics Lecturer: Sriram Sakararama Scribe: Sriram Sakararama We will provide a overview of statistical iferece focussig o the key
More informationImportant Formulas. Expectation: E (X) = Σ [X P(X)] = n p q σ = n p q. P(X) = n! X1! X 2! X 3! X k! p X. Chapter 6 The Normal Distribution.
Importat Formulas Chapter 3 Data Descriptio Mea for idividual data: X = _ ΣX Mea for grouped data: X= _ Σf X m Stadard deviatio for a sample: _ s = Σ(X _ X ) or s = 1 (Σ X ) (Σ X ) ( 1) Stadard deviatio
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 informationLecture Note 8 Point Estimators and Point Estimation Methods. MIT Spring 2006 Herman Bennett
Lecture Note 8 Poit Estimators ad Poit Estimatio Methods MIT 14.30 Sprig 2006 Herma Beett Give a parameter with ukow value, the goal of poit estimatio is to use a sample to compute a umber that represets
More informationClass 23. Daniel B. Rowe, Ph.D. Department of Mathematics, Statistics, and Computer Science. Marquette University MATH 1700
Class 23 Daiel B. Rowe, Ph.D. Departmet of Mathematics, Statistics, ad Computer Sciece Copyright 2017 by D.B. Rowe 1 Ageda: Recap Chapter 9.1 Lecture Chapter 9.2 Review Exam 6 Problem Solvig Sessio. 2
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 informationSome Basic Probability Concepts. 2.1 Experiments, Outcomes and Random Variables
Some Basic Probability Cocepts 2. Experimets, Outcomes ad Radom Variables A radom variable is a variable whose value is ukow util it is observed. The value of a radom variable results from a experimet;
More informationof the matrix is =-85, so it is not positive definite. Thus, the first
BOSTON COLLEGE Departmet of Ecoomics EC771: Ecoometrics Sprig 4 Prof. Baum, Ms. Uysal Solutio Key for Problem Set 1 1. Are the followig quadratic forms positive for all values of x? (a) y = x 1 8x 1 x
More 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 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 informationStatistical Theory MT 2009 Problems 1: Solution sketches
Statistical Theory MT 009 Problems : Solutio sketches. Which of the followig desities are withi a expoetial family? Explai your reasoig. (a) Let 0 < θ < ad put f(x, θ) = ( θ)θ x ; x = 0,,,... (b) (c) where
More informationLecture 2: Monte Carlo Simulation
STAT/Q SCI 43: Itroductio to Resamplig ethods Sprig 27 Istructor: Ye-Chi Che Lecture 2: ote Carlo Simulatio 2 ote Carlo Itegratio Assume we wat to evaluate the followig itegratio: e x3 dx What ca we do?
More informationLecture 22: Review for Exam 2. 1 Basic Model Assumptions (without Gaussian Noise)
Lecture 22: Review for Exam 2 Basic Model Assumptios (without Gaussia Noise) We model oe cotiuous respose variable Y, as a liear fuctio of p umerical predictors, plus oise: Y = β 0 + β X +... β p X p +
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 informationThis exam contains 19 pages (including this cover page) and 10 questions. A Formulae sheet is provided with the exam.
Probability ad Statistics FS 07 Secod Sessio Exam 09.0.08 Time Limit: 80 Miutes Name: Studet ID: This exam cotais 9 pages (icludig this cover page) ad 0 questios. A Formulae sheet is provided with the
More informationLet us give one more example of MLE. Example 3. The uniform distribution U[0, θ] on the interval [0, θ] has p.d.f.
Lecture 5 Let us give oe more example of MLE. Example 3. The uiform distributio U[0, ] o the iterval [0, ] has p.d.f. { 1 f(x =, 0 x, 0, otherwise The likelihood fuctio ϕ( = f(x i = 1 I(X 1,..., X [0,
More informationChapter 1 Simple Linear Regression (part 6: matrix version)
Chapter Simple Liear Regressio (part 6: matrix versio) Overview Simple liear regressio model: respose variable Y, a sigle idepedet variable X Y β 0 + β X + ε Multiple liear regressio model: respose Y,
More informationSample Size Estimation in the Proportional Hazards Model for K-sample or Regression Settings Scott S. Emerson, M.D., Ph.D.
ample ie Estimatio i the Proportioal Haards Model for K-sample or Regressio ettigs cott. Emerso, M.D., Ph.D. ample ie Formula for a Normally Distributed tatistic uppose a statistic is kow to be ormally
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 informationLecture 11 Simple Linear Regression
Lecture 11 Simple Liear Regressio Fall 2013 Prof. Yao Xie, yao.xie@isye.gatech.edu H. Milto Stewart School of Idustrial Systems & Egieerig Georgia Tech Midterm 2 mea: 91.2 media: 93.75 std: 6.5 2 Meddicorp
More informationFinal 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 informationStat410 Probability and Statistics II (F16)
Some Basic Cocepts of Statistical Iferece (Sec 5.) Suppose we have a rv X that has a pdf/pmf deoted by f(x; θ) or p(x; θ), where θ is called the parameter. I previous lectures, we focus o probability problems
More informationRule of probability. Let A and B be two events (sets of elementary events). 11. If P (AB) = P (A)P (B), then A and B are independent.
Percetile: the αth percetile of a populatio is the value x 0, such that P (X x 0 ) α% For example the 5th is the x 0, such that P (X x 0 ) 5% 05 Rule of probability Let A ad B be two evets (sets of elemetary
More informationWorksheet 23 ( ) Introduction to Simple Linear Regression (continued)
Worksheet 3 ( 11.5-11.8) Itroductio to Simple Liear Regressio (cotiued) This worksheet is a cotiuatio of Discussio Sheet 3; please complete that discussio sheet first if you have ot already doe so. This
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 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 informationStatistical Theory MT 2008 Problems 1: Solution sketches
Statistical Theory MT 008 Problems : Solutio sketches. Which of the followig desities are withi a expoetial family? Explai your reasoig. a) Let 0 < θ < ad put fx, θ) = θ)θ x ; x = 0,,,... b) c) where α
More informationTABLES AND FORMULAS FOR MOORE Basic Practice of Statistics
TABLES AND FORMULAS FOR MOORE Basic Practice of Statistics Explorig Data: Distributios Look for overall patter (shape, ceter, spread) ad deviatios (outliers). Mea (use a calculator): x = x 1 + x 2 + +
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 information( θ. sup θ Θ f X (x θ) = L. sup Pr (Λ (X) < c) = α. x : Λ (x) = sup θ H 0. sup θ Θ f X (x θ) = ) < c. NH : θ 1 = θ 2 against AH : θ 1 θ 2
82 CHAPTER 4. MAXIMUM IKEIHOOD ESTIMATION Defiitio: et X be a radom sample with joit p.m/d.f. f X x θ. The geeralised likelihood ratio test g.l.r.t. of the NH : θ H 0 agaist the alterative AH : θ H 1,
More informationSDS 321: Introduction to Probability and Statistics
SDS 321: Itroductio to Probability ad Statistics Lecture 23: Cotiuous radom variables- Iequalities, CLT Puramrita Sarkar Departmet of Statistics ad Data Sciece The Uiversity of Texas at Austi www.cs.cmu.edu/
More 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 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 informationProbability and Statistics
ICME Refresher Course: robability ad Statistics Staford Uiversity robability ad Statistics Luyag Che September 20, 2016 1 Basic robability Theory 11 robability Spaces A probability space is a triple (Ω,
More 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 informationQuestions and Answers on Maximum Likelihood
Questios ad Aswers o Maximum Likelihood L. Magee Fall, 2008 1. Give: a observatio-specific log likelihood fuctio l i (θ) = l f(y i x i, θ) the log likelihood fuctio l(θ y, X) = l i(θ) a data set (x i,
More informationST 305: Exam 3 ( ) = P(A)P(B A) ( ) = P(A) + P(B) ( ) = 1 P( A) ( ) = P(A) P(B) σ X 2 = σ a+bx. σ ˆp. σ X +Y. σ X Y. σ X. σ Y. σ n.
ST 305: Exam 3 By hadig i this completed exam, I state that I have either give or received assistace from aother perso durig the exam period. I have used o resources other tha the exam itself ad the basic
More informationTMA4245 Statistics. Corrected 30 May and 4 June Norwegian University of Science and Technology Department of Mathematical Sciences.
Norwegia Uiversity of Sciece ad Techology Departmet of Mathematical Scieces Corrected 3 May ad 4 Jue Solutios TMA445 Statistics Saturday 6 May 9: 3: Problem Sow desity a The probability is.9.5 6x x dx
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 informationSTATISTICAL PROPERTIES OF LEAST SQUARES ESTIMATORS. Comments:
Recall: STATISTICAL PROPERTIES OF LEAST SQUARES ESTIMATORS Commets:. So far we have estimates of the parameters! 0 ad!, but have o idea how good these estimates are. Assumptio: E(Y x)! 0 +! x (liear coditioal
More informationECONOMETRIC THEORY. MODULE XIII Lecture - 34 Asymptotic Theory and Stochastic Regressors
ECONOMETRIC THEORY MODULE XIII Lecture - 34 Asymptotic Theory ad Stochastic Regressors Dr. Shalabh Departmet of Mathematics ad Statistics Idia Istitute of Techology Kapur Asymptotic theory The asymptotic
More informationChapters 5 and 13: REGRESSION AND CORRELATION. Univariate data: x, Bivariate data (x,y).
Chapters 5 ad 13: REGREION AND CORRELATION (ectios 5.5 ad 13.5 are omitted) Uivariate data: x, Bivariate data (x,y). Example: x: umber of years studets studied paish y: score o a proficiecy test For each
More informationAsymptotics. Hypothesis Testing UMP. Asymptotic Tests and p-values
of the secod half Biostatistics 6 - Statistical Iferece Lecture 6 Fial Exam & Practice Problems for the Fial Hyu Mi Kag Apil 3rd, 3 Hyu Mi Kag Biostatistics 6 - Lecture 6 Apil 3rd, 3 / 3 Rao-Blackwell
More 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 informationTABLES AND FORMULAS FOR MOORE Basic Practice of Statistics
TABLES AND FORMULAS FOR MOORE Basic Practice of Statistics Explorig Data: Distributios Look for overall patter (shape, ceter, spread) ad deviatios (outliers). Mea (use a calculator): x = x 1 + x 2 + +
More informationEcon 325/327 Notes on Sample Mean, Sample Proportion, Central Limit Theorem, Chi-square Distribution, Student s t distribution 1.
Eco 325/327 Notes o Sample Mea, Sample Proportio, Cetral Limit Theorem, Chi-square Distributio, Studet s t distributio 1 Sample Mea By Hiro Kasahara We cosider a radom sample from a populatio. Defiitio
More informationEECS564 Estimation, Filtering, and Detection Hwk 2 Solns. Winter p θ (z) = (2θz + 1 θ), 0 z 1
EECS564 Estimatio, Filterig, ad Detectio Hwk 2 Sols. Witer 25 4. Let Z be a sigle observatio havig desity fuctio where. p (z) = (2z + ), z (a) Assumig that is a oradom parameter, fid ad plot the maximum
More informationLinear Regression Models
Liear Regressio Models Dr. Joh Mellor-Crummey Departmet of Computer Sciece Rice Uiversity johmc@cs.rice.edu COMP 528 Lecture 9 15 February 2005 Goals for Today Uderstad how to Use scatter diagrams to ispect
More informationMatrix Representation of Data in Experiment
Matrix Represetatio of Data i Experimet Cosider a very simple model for resposes y ij : y ij i ij, i 1,; j 1,,..., (ote that for simplicity we are assumig the two () groups are of equal sample size ) Y
More informationHypothesis Testing. Evaluation of Performance of Learned h. Issues. Trade-off Between Bias and Variance
Hypothesis Testig Empirically evaluatig accuracy of hypotheses: importat activity i ML. Three questios: Give observed accuracy over a sample set, how well does this estimate apply over additioal samples?
More informationUnbiased Estimation. February 7-12, 2008
Ubiased Estimatio February 7-2, 2008 We begi with a sample X = (X,..., X ) of radom variables chose accordig to oe of a family of probabilities P θ where θ is elemet from the parameter space Θ. For radom
More informationEfficient GMM LECTURE 12 GMM II
DECEMBER 1 010 LECTURE 1 II Efficiet The estimator depeds o the choice of the weight matrix A. The efficiet estimator is the oe that has the smallest asymptotic variace amog all estimators defied by differet
More informationf(x i ; ) L(x; p) = i=1 To estimate the value of that maximizes L or equivalently ln L we will set =0, for i =1, 2,...,m p x i (1 p) 1 x i i=1
Parameter Estimatio Samples from a probability distributio F () are: [,,..., ] T.Theprobabilitydistributio has a parameter vector [,,..., m ] T. Estimator: Statistic used to estimate ukow. Estimate: Observed
More informationSection 14. Simple linear regression.
Sectio 14 Simple liear regressio. Let us look at the cigarette dataset from [1] (available to dowload from joural s website) ad []. The cigarette dataset cotais measuremets of tar, icotie, weight ad carbo
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 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 informationMaximum Likelihood Estimation
Chapter 9 Maximum Likelihood Estimatio 9.1 The Likelihood Fuctio The maximum likelihood estimator is the most widely used estimatio method. This chapter discusses the most importat cocepts behid maximum
More informationChapter 13: Tests of Hypothesis Section 13.1 Introduction
Chapter 13: Tests of Hypothesis Sectio 13.1 Itroductio RECAP: Chapter 1 discussed the Likelihood Ratio Method as a geeral approach to fid good test procedures. Testig for the Normal Mea Example, discussed
More informationSimple Linear Regression
Simple Liear Regressio 1. Model ad Parameter Estimatio (a) Suppose our data cosist of a collectio of pairs (x i, y i ), where x i is a observed value of variable X ad y i is the correspodig observatio
More informationIIT JAM Mathematical Statistics (MS) 2006 SECTION A
IIT JAM Mathematical Statistics (MS) 6 SECTION A. If a > for ad lim a / L >, the which of the followig series is ot coverget? (a) (b) (c) (d) (d) = = a = a = a a + / a lim a a / + = lim a / a / + = lim
More informationAgenda: Recap. Lecture. Chapter 12. Homework. Chapt 12 #1, 2, 3 SAS Problems 3 & 4 by hand. Marquette University MATH 4740/MSCS 5740
Ageda: Recap. Lecture. Chapter Homework. Chapt #,, 3 SAS Problems 3 & 4 by had. Copyright 06 by D.B. Rowe Recap. 6: Statistical Iferece: Procedures for μ -μ 6. Statistical Iferece Cocerig μ -μ Recall yes
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 informationMBACATÓLICA. Quantitative Methods. Faculdade de Ciências Económicas e Empresariais UNIVERSIDADE CATÓLICA PORTUGUESA 9. SAMPLING DISTRIBUTIONS
MBACATÓLICA Quatitative Methods Miguel Gouveia Mauel Leite Moteiro Faculdade de Ciêcias Ecoómicas e Empresariais UNIVERSIDADE CATÓLICA PORTUGUESA 9. SAMPLING DISTRIBUTIONS MBACatólica 006/07 Métodos Quatitativos
More 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 informationLecture Notes 15 Hypothesis Testing (Chapter 10)
1 Itroductio Lecture Notes 15 Hypothesis Testig Chapter 10) Let X 1,..., X p θ x). Suppose we we wat to kow if θ = θ 0 or ot, where θ 0 is a specific value of θ. For example, if we are flippig a coi, we
More information32 estimating the cumulative distribution function
32 estimatig the cumulative distributio fuctio 4.6 types of cofidece itervals/bads Let F be a class of distributio fuctios F ad let θ be some quatity of iterest, such as the mea of F or the whole fuctio
More informationSample Size Determination (Two or More Samples)
Sample Sie Determiatio (Two or More Samples) STATGRAPHICS Rev. 963 Summary... Data Iput... Aalysis Summary... 5 Power Curve... 5 Calculatios... 6 Summary This procedure determies a suitable sample sie
More information