# Asymptotic normality of the Nadaraya-Watson estimator for non-stationary functional data and applications to telecommunications.

Save this PDF as:

Size: px
Start display at page:

## Transcription

5 Asymptotic ormality of the Nadaraya-Watso estimator for o-statioary data 5 H 4 There exist a fuctio h : R + R +, such that lim x + hx = 0 ad a fuctio g : R + R m R +, with gj, t < for all fixed J > 0 ad sup t R m gj, t = g J <, such that [ ] [ ] [ ] E e isb C, t,x,j E e isb, t,x,j E e isc, t,x,j gj, thdb, C, for all disjoit sets B, C L, all N, J > 0 ad t R m. Here, represets the scalar product o R m. H 5 There exist sequeces γ J i, j, k ad γi, j, k, i, j {,...,m}, k L such that for all J > 0 we have { } lim E X i,,j 0 X j,,j k = γ J i, j, k, ad for i, j {,...,m} ad k L. We have the two followig propositios proved i Sectio 5. lim J γj i, j, k = γi, j, k 4 Propositio. If X = X k k N belogs to BN, the for ay asymptotically measurable family { A i : i =,...,m } i N, we have as teds to where for i, j {,...,m} S A, X,,...,S A m, X m, w = N m 0, Σ, Σi, j = γi, j,0fi, j,0 + k {γi, j, kfi, j, k + γj, i, kfj, i, k} ad Fi, j, k = lim card{a i A j k}. Propositio 2. If X = X k k Z d belogs to BZd, the for ay asymptotically measurable family { A i : i =,...,m } i Z d, we have as teds to S A, X,,...,S A m, X m, w = N m 0, Σ, where for i, j {,...,m} ad Σi, j = k L γi, j, kfi, j, k card{a i A j k} Fi, j, k = lim 2 + d.

7 Asymptotic ormality of the Nadaraya-Watso estimator for o-statioary data 7 with u x K u = K, u D. 6 ad ψh is give i assumptio A. We recall that we use the covetio 0/0 = 0. h I Subsectio 3., we give the assumptios uder which the asymptotic ormality of φ x is obtaied ad i Subsectio 3.2 we give our results. 3. Assumptios We make the followig assumptios o the distributio of X, Y. A There exist positive fuctios ψ, ψ,...,ψ m defied o R + D, fuctios c,...,c k defied o D ad a subset of {,...,m} such that for all h > 0 P [ ϕξ, z k x h] = c k xψ k h, x, ψ k h, x where lim h 0 ψh, x = if k, ad lim ψ k h, x h 0 ψh, x = 0 if k C. I the followig, to simplify the otatio, we replace ψh, x by ψh, but the quatity ψh still depeds of x. A 2 The fuctios u ψ k u, x are differetiable o R +, with differetial fuctio deoted ψ k u, x ad satisfy lim h 0 h ψ k h, x where the d k s are fuctios defied o D. 0 Kuψ k uh, xdu = d kx, A 3 The process Z is regular ad satisfies for k {,...,m} as teds to that ψh {Zi =z k } P PZ i = z k = 0, 7 where P = meas covergece i probability. For k {,...,m}, deote by p k the limit p k = lim PZ i = z k. Let the R 2m -value radom process X = X,,..., X 2m, defied for i N as follows: for l {,...,m} X l, i = ψh K ϕξ i, z l φϕξ i, z l + ε i ψh E [K ϕξ i, z l φϕξ i, z l ],

9 Asymptotic ormality of the Nadaraya-Watso estimator for o-statioary data Asymptotic ormality of φ x The asymptotic ormality of φ x is obtaied i three stages: asymptotic ormality of the field X i Propositio 4, of g x Eg x, f x Ef x i Propositio 5 ad the of φ x i Theorem. Theorem 2 gives the asymptotic ormality of φ x φx which allows to costruct asymptotic cofidece iterval for φx. Propositio 4. For l {,...,m}, deote A l = {k N, Z k = z l } ad A l+m = A l. If Z is asymptotically measurable, the uder Assumptio A 4, coditioally o Z, we have, as teds to S A, X,,..., S A 2m, X 2m, w = N 2m 0, Σ, where for i, j {,...,2m} Σi, j = γi, j,0fi, j,0 + k N {γi, j, kfi, j, k + γj, i, kfj, i, k}, with γi, j, k, k N, defied by 4 for the field X ad Fi, j, k = lim card{a i A j k}. Propositio 5. Uder Assumptios A 3 ad A 4, we have, as teds to, that w ψh g x Eg x, f x Ef x = N 2 0, A, where with a x = A = Σi, j, a 2 x = i,j= a x a 2 x a 2 x a 3 x 2 i= j=m+ Σi, j, a 3 x = 2 i,j=m+ Σi, j. Theorem. Uder Assumptios A A 8, we have as teds to ψh φ x Eg x w = N 0, σ 2 x, Ef x where Theorem 2. We suppose that B the fuctio φ satisfies with β > 0 ad L a positive costat, σ 2 x = a x 2a 2 xφx + a 3 xφ 2 x f 2. 8 x φu φv L u v β,

13 Asymptotic ormality of the Nadaraya-Watso estimator for o-statioary data Predictio delay Measured delay Upper boud cofidece Lower boud cofidece Delayms Sample umber Figure 3: Estimated delays for simulated data observatios betwee 230 ad Relative error Sample umber Figure 4: Relative error for simulated data

14 Asymptotic ormality of the Nadaraya-Watso estimator for o-statioary data Predictio delay Measured delay Upper boud cofidece Lower boud cofidece 200 Delayms Sample umber Figure 5: Estimated RTT for real data Relative error Sample umber Figure 6: Relative error for real data

15 Asymptotic ormality of the Nadaraya-Watso estimator for o-statioary data 5 5 Proof of the propositios ad theorems 5. Proof of Propositios ad 2 This result is obtaied by meas of Bershtei method cf. [8], [22] ad it is a geeralizatio of the theorem 4.5 o triagular arrays for R-valued radom fields obtaied by Tablar 2006 [23], p 86 to R m -valued radom fields. To prove that the vectorial field S A, X,,..., S A m, X m, w = N m 0, Σ, it is sufficiet to prove that for all λ R m w λ, S = N0, λ t Σλ where S = S A, X,,..., S A m, X m, ad λ t = λ,...,λ m. We have that S, λ = = λ i S A i, X i, = i= l /2 k D i= i= l /2λ i λ i X i, k {k A i } Let us cosider a real valued field X λ where X i, k λ = λ ix i, k if k A i. The S, λ = S D, X λ. As the field X λ verifies the hypotheses i the work of [23] we ca apply the theorem for a real valued radom field i order to obtai the result of propositios ad Proof of Propositio 4 Sice Z is asymptotically measurable, coditioally to Z, the subset family A,...,A 2m is a asymptotically measurable family i N. The applyig Propositio to X which belogs to BN, we deduce the result. 5.3 Proof of Propositio 5 Before provig Propositio 5, here we prove the followig lema. Lemma. Let k {,...,m}. Uder the Assumptios A, A 2, A 3, A 5, A 7 ad A 8, we have the followig limit lim k A i X i, k ψh E [K ϕξ, z k ] = d k xc k x {k }, 2 for β > 0, lim ψh E [ ϕξ, z k x h β K ϕξ, z k ] d k xc k x, 3 that the estimator f x satisfies lim Ef x = fx,

16 Asymptotic ormality of the Nadaraya-Watso estimator for o-statioary data 6 4 ad the estimator g x satisfies lim Eg x = φxfx. Proof. The results ad 2 of this lemma come from the fact that uder Assumptio A, the desity of ϕξ, z k x is the fuctio u c k xψ k u, x. The uder Assumptio A 7 ad [ ϕξ, z k x E h E [K ϕξ, z k ] = h c k x β K ϕξ, z k ] 0 = h c k x Kuψ k uh, xdu ad it is sufficiet to use Assumptios A 2 ad A 8 to coclude. Here we prove 3 ad 4. We have for i {,...,}, EK X i = E{EK X i Z i } = 0 Kuu β ψ k uh, xdu E{K ϕξ i, z k }PZ i = z k. k= As ϕξ i, z k i=,..., is idetically distributed, we obtai that, Ef x = ψh E K ϕξ, z k PZ i = z k. 9 k= Now applyig A 3 ad the assertio of this lemma i 9, we obtai 3. Similar calculus give that Eg x = ψh E φϕξ, z k K ϕξ, z k PZ i = z k. The we have where R = k= Eg x = φxef x + R, ψh E {φϕξ, z k φx} K ϕξ, z k PZ i = z k k= sup u: x u h φu φx Ef x ad we obtai 4 usig the cotiuity of fuctio φ Assumptio A 5 ad usig that lim Ef x = fx. Coditioally to Z, we have ψh g x Eg x Z, f x Ef x Z = S A j, X,j, j= 2 j=m+ S A j, X,j.

17 Asymptotic ormality of the Nadaraya-Watso estimator for o-statioary data 7 Usig Propositio 4, we obtai that, coditioally to Z, S A j, X,j, 2 S A j, X,j = w N 2 0, A. j= j=m+ Sice Z is regular, the matrix A is ot radom ad the we have w ψh g x Eg x Z, f x Ef x Z = N 2 0, A. Now g x Eg x, f x Ef x = g x Eg x Z, f x Ef x Z We have Ef x Z Ef x = 2 j= + Eg x Z Eg x, Ef x Z Ef x. EK ξ, z j ψh carda j P[Z i = j] Uder Assumptio A 3 ad usig assertio of Lemma, we have that ψh Ef x Z Ef x. coverges i probability to 0 as teds to. Usig Assumptio A 3, ψh Eg x Z Eg x coverges i probability to 0 as teds to. Usig the lemma of Slutsky, we obtai the result of the propositio. 5.4 Proof of Theorem We have where ad φ x Eg x Ef x = Q x B xf x E f x, f x B x = Eg x Ef x φx Q x = g x Eg x φxf x Ef x. Usig 3 y 4 of Lemma, we deduce that lim B x = 0. The usig A 6, we have that f x coverges i probability to fx > 0 ad the it implies that B xf x E f x 0 f x coverges i probability to 0. Fially, usig A 6, 0 ad Propositio 5, applyig Slutsky Lemma, we deduce the result of the propositio.

18 Asymptotic ormality of the Nadaraya-Watso estimator for o-statioary data Proof of Theorem 2 Theorem 2 is obtaied usig Theorem ad the followig lemma. Lemma 2. Uder Assumptios B B 4 ad Assumptios A, A 2, A 6, A 7 ad A 8, we have that Eg x lim ψh Ef x φx = 0. Proof. We have the followig decompositio Eg x Ef x φx = Ef x [Eg x fxφx] + φx Ef x [fx Ef x]. Here we study first the quatity Ef x fx. As i Lemma, we have Ef x = ψh E K ϕξ, z k PZ i = z k ad the Ef x fx = + k= k= ψh E K ϕξ, z k d k xc k x {k } PZ i = z k d k xc k x {k } PZ i = z k p k. k= Uder the Assumptios B 2, B 3 ad B 4 imply that ψh Ef x fx = 0. 2 lim Here we study the term Eg x fxφx. It satisfies Eg x fxφx φx Ef x fx + Eg x Ef xφx. We have Eg x Ef xφx = E [K ϕξ, z k φϕξ, z k φx] PZ i = z k ψh k= [ Lh β ψh E K ϕξ, z k ϕξ, z k x ] β h PZ i = z k k= L h β + o, 3 where the secod lie comes from Assumptio B ad the third is a cosequece of assertio 2 of Lemma. From 2, 3 ad Assumptio B 3, we deduce that ψh Eg x fxφx = 0. 4 lim Assertio of Lemma, 2 ad 4 applyig i imply the propositio.

### Properties of MLE: consistency, asymptotic normality. Fisher information.

Lecture 3 Properties of MLE: cosistecy, asymptotic ormality. Fisher iformatio. I this sectio we will try to uderstad why MLEs are good. Let us recall two facts from probability that we be used ofte throughout

### I. Chi-squared Distributions

1 M 358K Supplemet to Chapter 23: CHI-SQUARED DISTRIBUTIONS, T-DISTRIBUTIONS, AND DEGREES OF FREEDOM To uderstad t-distributios, we first eed to look at aother family of distributios, the chi-squared distributios.

### Output Analysis (2, Chapters 10 &11 Law)

B. Maddah ENMG 6 Simulatio 05/0/07 Output Aalysis (, Chapters 10 &11 Law) Comparig alterative system cofiguratio Sice the output of a simulatio is radom, the comparig differet systems via simulatio should

### 3. Covariance and Correlation

Virtual Laboratories > 3. Expected Value > 1 2 3 4 5 6 3. Covariace ad Correlatio Recall that by takig the expected value of various trasformatios of a radom variable, we ca measure may iterestig characteristics

### Chapter 7 Methods of Finding Estimators

Chapter 7 for BST 695: Special Topics i Statistical Theory. Kui Zhag, 011 Chapter 7 Methods of Fidig Estimators Sectio 7.1 Itroductio Defiitio 7.1.1 A poit estimator is ay fuctio W( X) W( X1, X,, X ) of

### 7. Sample Covariance and Correlation

1 of 8 7/16/2009 6:06 AM Virtual Laboratories > 6. Radom Samples > 1 2 3 4 5 6 7 7. Sample Covariace ad Correlatio The Bivariate Model Suppose agai that we have a basic radom experimet, ad that X ad Y

### Asymptotic Growth of Functions

CMPS Itroductio to Aalysis of Algorithms Fall 3 Asymptotic Growth of Fuctios We itroduce several types of asymptotic otatio which are used to compare the performace ad efficiecy of algorithms As we ll

### An example of non-quenched convergence in the conditional central limit theorem for partial sums of a linear process

A example of o-queched covergece i the coditioal cetral limit theorem for partial sums of a liear process Dalibor Volý ad Michael Woodroofe Abstract A causal liear processes X,X 0,X is costructed for which

### ORDERS OF GROWTH KEITH CONRAD

ORDERS OF GROWTH KEITH CONRAD Itroductio Gaiig a ituitive feel for the relative growth of fuctios is importat if you really wat to uderstad their behavior It also helps you better grasp topics i calculus

### NPTEL STRUCTURAL RELIABILITY

NPTEL Course O STRUCTURAL RELIABILITY Module # 0 Lecture 1 Course Format: Web Istructor: Dr. Aruasis Chakraborty Departmet of Civil Egieerig Idia Istitute of Techology Guwahati 1. Lecture 01: Basic Statistics

### Chapter 7 - Sampling Distributions. 1 Introduction. What is statistics? It consist of three major areas:

Chapter 7 - Samplig Distributios 1 Itroductio What is statistics? It cosist of three major areas: Data Collectio: samplig plas ad experimetal desigs Descriptive Statistics: umerical ad graphical summaries

### Case Study. Normal and t Distributions. Density Plot. Normal Distributions

Case Study Normal ad t Distributios Bret Halo ad Bret Larget Departmet of Statistics Uiversity of Wiscosi Madiso October 11 13, 2011 Case Study Body temperature varies withi idividuals over time (it ca

### In nite Sequences. Dr. Philippe B. Laval Kennesaw State University. October 9, 2008

I ite Sequeces Dr. Philippe B. Laval Keesaw State Uiversity October 9, 2008 Abstract This had out is a itroductio to i ite sequeces. mai de itios ad presets some elemetary results. It gives the I ite Sequeces

### Hypothesis Tests Applied to Means

The Samplig Distributio of the Mea Hypothesis Tests Applied to Meas Recall that the samplig distributio of the mea is the distributio of sample meas that would be obtaied from a particular populatio (with

### Characterizing End-to-End Packet Delay and Loss in the Internet

Characterizig Ed-to-Ed Packet Delay ad Loss i the Iteret Jea-Chrysostome Bolot Xiyu Sog Preseted by Swaroop Sigh Layout Itroductio Data Collectio Data Aalysis Strategy Aalysis of packet delay Aalysis of

### Maximum Likelihood Estimators.

Lecture 2 Maximum Likelihood Estimators. Matlab example. As a motivatio, let us look at oe Matlab example. Let us geerate a radom sample of size 00 from beta distributio Beta(5, 2). We will lear the defiitio

### Confidence Intervals for One Mean with Tolerance Probability

Chapter 421 Cofidece Itervals for Oe Mea with Tolerace Probability Itroductio This procedure calculates the sample size ecessary to achieve a specified distace from the mea to the cofidece limit(s) with

### A probabilistic proof of a binomial identity

A probabilistic proof of a biomial idetity Joatho Peterso Abstract We give a elemetary probabilistic proof of a biomial idetity. The proof is obtaied by computig the probability of a certai evet i two

### Our aim is to show that under reasonable assumptions a given 2π-periodic function f can be represented as convergent series

8 Fourier Series Our aim is to show that uder reasoable assumptios a give -periodic fuctio f ca be represeted as coverget series f(x) = a + (a cos x + b si x). (8.) By defiitio, the covergece of the series

### 1 Computing the Standard Deviation of Sample Means

Computig the Stadard Deviatio of Sample Meas Quality cotrol charts are based o sample meas ot o idividual values withi a sample. A sample is a group of items, which are cosidered all together for our aalysis.

### Hypothesis testing. Null and alternative hypotheses

Hypothesis testig Aother importat use of samplig distributios is to test hypotheses about populatio parameters, e.g. mea, proportio, regressio coefficiets, etc. For example, it is possible to stipulate

### 1 Correlation and Regression Analysis

1 Correlatio ad Regressio Aalysis I this sectio we will be ivestigatig the relatioship betwee two cotiuous variable, such as height ad weight, the cocetratio of a ijected drug ad heart rate, or the cosumptio

### 5: Introduction to Estimation

5: Itroductio to Estimatio Cotets Acroyms ad symbols... 1 Statistical iferece... Estimatig µ with cofidece... 3 Samplig distributio of the mea... 3 Cofidece Iterval for μ whe σ is kow before had... 4 Sample

### Modified Line Search Method for Global Optimization

Modified Lie Search Method for Global Optimizatio Cria Grosa ad Ajith Abraham Ceter of Excellece for Quatifiable Quality of Service Norwegia Uiversity of Sciece ad Techology Trodheim, Norway {cria, ajith}@q2s.tu.o

### 0.7 0.6 0.2 0 0 96 96.5 97 97.5 98 98.5 99 99.5 100 100.5 96.5 97 97.5 98 98.5 99 99.5 100 100.5

Sectio 13 Kolmogorov-Smirov test. Suppose that we have a i.i.d. sample X 1,..., X with some ukow distributio P ad we would like to test the hypothesis that P is equal to a particular distributio P 0, i.e.

### Economics 140A Confidence Intervals and Hypothesis Testing

Ecoomics 140A Cofidece Itervals ad Hypothesis Testig Obtaiig a estimate of a parameter is ot the al purpose of statistical iferece because it is highly ulikely that the populatio value of a parameter is

### SAMPLE QUESTIONS FOR FINAL EXAM. (1) (2) (3) (4) Find the following using the definition of the Riemann integral: (2x + 1)dx

SAMPLE QUESTIONS FOR FINAL EXAM REAL ANALYSIS I FALL 006 3 4 Fid the followig usig the defiitio of the Riema itegral: a 0 x + dx 3 Cosider the partitio P x 0 3, x 3 +, x 3 +,......, x 3 3 + 3 of the iterval

### if A S, then X \ A S, and if (A n ) n is a sequence of sets in S, then n A n S,

Lecture 5: Borel Sets Topologically, the Borel sets i a topological space are the σ-algebra geerated by the ope sets. Oe ca build up the Borel sets from the ope sets by iteratig the operatios of complemetatio

### MATH 361 Homework 9. Royden Royden Royden

MATH 61 Homework 9 Royde..9 First, we show that for ay subset E of the real umbers, E c + y = E + y) c traslatig the complemet is equivalet to the complemet of the traslated set). Without loss of geerality,

### Module 4: Mathematical Induction

Module 4: Mathematical Iductio Theme 1: Priciple of Mathematical Iductio Mathematical iductio is used to prove statemets about atural umbers. As studets may remember, we ca write such a statemet as a predicate

### Sequences and Series

CHAPTER 9 Sequeces ad Series 9.. Covergece: Defiitio ad Examples Sequeces The purpose of this chapter is to itroduce a particular way of geeratig algorithms for fidig the values of fuctios defied by their

### Measurable Functions

Measurable Fuctios Dug Le 1 1 Defiitio It is ecessary to determie the class of fuctios that will be cosidered for the Lebesgue itegratio. We wat to guaratee that the sets which arise whe workig with these

### Non-life insurance mathematics. Nils F. Haavardsson, University of Oslo and DNB Skadeforsikring

No-life isurace mathematics Nils F. Haavardsso, Uiversity of Oslo ad DNB Skadeforsikrig Mai issues so far Why does isurace work? How is risk premium defied ad why is it importat? How ca claim frequecy

### Section IV.5: Recurrence Relations from Algorithms

Sectio IV.5: Recurrece Relatios from Algorithms Give a recursive algorithm with iput size, we wish to fid a Θ (best big O) estimate for its ru time T() either by obtaiig a explicit formula for T() or by

### Definition. Definition. 7-2 Estimating a Population Proportion. Definition. Definition

7- stimatig a Populatio Proportio I this sectio we preset methods for usig a sample proportio to estimate the value of a populatio proportio. The sample proportio is the best poit estimate of the populatio

### FIBONACCI NUMBERS: AN APPLICATION OF LINEAR ALGEBRA. 1. Powers of a matrix

FIBONACCI NUMBERS: AN APPLICATION OF LINEAR ALGEBRA. Powers of a matrix We begi with a propositio which illustrates the usefuless of the diagoalizatio. Recall that a square matrix A is diogaalizable if

### 8.5 Alternating infinite series

65 8.5 Alteratig ifiite series I the previous two sectios we cosidered oly series with positive terms. I this sectio we cosider series with both positive ad egative terms which alterate: positive, egative,

### Chapter 6: Variance, the law of large numbers and the Monte-Carlo method

Chapter 6: Variace, the law of large umbers ad the Mote-Carlo method Expected value, variace, ad Chebyshev iequality. If X is a radom variable recall that the expected value of X, E[X] is the average value

### Section 7-3 Estimating a Population. Requirements

Sectio 7-3 Estimatig a Populatio Mea: σ Kow Key Cocept This sectio presets methods for usig sample data to fid a poit estimate ad cofidece iterval estimate of a populatio mea. A key requiremet i this sectio

### Overview of some probability distributions.

Lecture Overview of some probability distributios. I this lecture we will review several commo distributios that will be used ofte throughtout the class. Each distributio is usually described by its probability

### LECTURE 13: Cross-validation

LECTURE 3: Cross-validatio Resampli methods Cross Validatio Bootstrap Bias ad variace estimatio with the Bootstrap Three-way data partitioi Itroductio to Patter Aalysis Ricardo Gutierrez-Osua Texas A&M

### Class Meeting # 16: The Fourier Transform on R n

MATH 18.152 COUSE NOTES - CLASS MEETING # 16 18.152 Itroductio to PDEs, Fall 2011 Professor: Jared Speck Class Meetig # 16: The Fourier Trasform o 1. Itroductio to the Fourier Trasform Earlier i the course,

### Irreducible polynomials with consecutive zero coefficients

Irreducible polyomials with cosecutive zero coefficiets Theodoulos Garefalakis Departmet of Mathematics, Uiversity of Crete, 71409 Heraklio, Greece Abstract Let q be a prime power. We cosider the problem

### THE REGRESSION MODEL IN MATRIX FORM. For simple linear regression, meaning one predictor, the model is. for i = 1, 2, 3,, n

We will cosider the liear regressio model i matrix form. For simple liear regressio, meaig oe predictor, the model is i = + x i + ε i for i =,,,, This model icludes the assumptio that the ε i s are a sample

### Stat 104 Lecture 16. Statistics 104 Lecture 16 (IPS 6.1) Confidence intervals - the general concept

Statistics 104 Lecture 16 (IPS 6.1) Outlie for today Cofidece itervals Cofidece itervals for a mea, µ (kow σ) Cofidece itervals for a proportio, p Margi of error ad sample size Review of mai topics for

### PROCEEDINGS OF THE YEREVAN STATE UNIVERSITY AN ALTERNATIVE MODEL FOR BONUS-MALUS SYSTEM

PROCEEDINGS OF THE YEREVAN STATE UNIVERSITY Physical ad Mathematical Scieces 2015, 1, p. 15 19 M a t h e m a t i c s AN ALTERNATIVE MODEL FOR BONUS-MALUS SYSTEM A. G. GULYAN Chair of Actuarial Mathematics

### Lecture 13. Lecturer: Jonathan Kelner Scribe: Jonathan Pines (2009)

18.409 A Algorithmist s Toolkit October 27, 2009 Lecture 13 Lecturer: Joatha Keler Scribe: Joatha Pies (2009) 1 Outlie Last time, we proved the Bru-Mikowski iequality for boxes. Today we ll go over the

### Joint Probability Distributions and Random Samples

STAT5 Sprig 204 Lecture Notes Chapter 5 February, 204 Joit Probability Distributios ad Radom Samples 5. Joitly Distributed Radom Variables Chapter Overview Joitly distributed rv Joit mass fuctio, margial

### PSYCHOLOGICAL STATISTICS

UNIVERSITY OF CALICUT SCHOOL OF DISTANCE EDUCATION B Sc. Cousellig Psychology (0 Adm.) IV SEMESTER COMPLEMENTARY COURSE PSYCHOLOGICAL STATISTICS QUESTION BANK. Iferetial statistics is the brach of statistics

### SECTION 1.5 : SUMMATION NOTATION + WORK WITH SEQUENCES

SECTION 1.5 : SUMMATION NOTATION + WORK WITH SEQUENCES Read Sectio 1.5 (pages 5 9) Overview I Sectio 1.5 we lear to work with summatio otatio ad formulas. We will also itroduce a brief overview of sequeces,

### The second difference is the sequence of differences of the first difference sequence, 2

Differece Equatios I differetial equatios, you look for a fuctio that satisfies ad equatio ivolvig derivatives. I differece equatios, istead of a fuctio of a cotiuous variable (such as time), we look for

Advaced Probability Theory Math5411 HKUST Kai Che (Istructor) Chapter 1. Law of Large Numbers 1.1. σ-algebra, measure, probability space ad radom variables. This sectio lays the ecessary rigorous foudatio

### Plug-in martingales for testing exchangeability on-line

Plug-i martigales for testig exchageability o-lie Valetia Fedorova, Alex Gammerma, Ilia Nouretdiov, ad Vladimir Vovk Computer Learig Research Cetre Royal Holloway, Uiversity of Lodo, UK {valetia,ilia,alex,vovk}@cs.rhul.ac.uk

### Key Ideas Section 8-1: Overview hypothesis testing Hypothesis Hypothesis Test Section 8-2: Basics of Hypothesis Testing Null Hypothesis

Chapter 8 Key Ideas Hypothesis (Null ad Alterative), Hypothesis Test, Test Statistic, P-value Type I Error, Type II Error, Sigificace Level, Power Sectio 8-1: Overview Cofidece Itervals (Chapter 7) are

### A Gentle Introduction to Algorithms: Part II

A Getle Itroductio to Algorithms: Part II Cotets of Part I:. Merge: (to merge two sorted lists ito a sigle sorted list.) 2. Bubble Sort 3. Merge Sort: 4. The Big-O, Big-Θ, Big-Ω otatios: asymptotic bouds

### Annuities Under Random Rates of Interest II By Abraham Zaks. Technion I.I.T. Haifa ISRAEL and Haifa University Haifa ISRAEL.

Auities Uder Radom Rates of Iterest II By Abraham Zas Techio I.I.T. Haifa ISRAEL ad Haifa Uiversity Haifa ISRAEL Departmet of Mathematics, Techio - Israel Istitute of Techology, 3000, Haifa, Israel I memory

### Z-TEST / Z-STATISTIC: used to test hypotheses about. µ when the population standard deviation is unknown

Z-TEST / Z-STATISTIC: used to test hypotheses about µ whe the populatio stadard deviatio is kow ad populatio distributio is ormal or sample size is large T-TEST / T-STATISTIC: used to test hypotheses about

### Section 11.3: The Integral Test

Sectio.3: The Itegral Test Most of the series we have looked at have either diverged or have coverged ad we have bee able to fid what they coverge to. I geeral however, the problem is much more difficult

### ARTICLE IN PRESS. Statistics & Probability Letters ( ) A Kolmogorov-type test for monotonicity of regression. Cecile Durot

STAPRO 66 pp: - col.fig.: il ED: MG PROD. TYPE: COM PAGN: Usha.N -- SCAN: il Statistics & Probability Letters 2 2 2 2 Abstract A Kolmogorov-type test for mootoicity of regressio Cecile Durot Laboratoire

### Approximating Area under a curve with rectangles. To find the area under a curve we approximate the area using rectangles and then use limits to find

1.8 Approximatig Area uder a curve with rectagles 1.6 To fid the area uder a curve we approximate the area usig rectagles ad the use limits to fid 1.4 the area. Example 1 Suppose we wat to estimate 1.

### Normal Distribution.

Normal Distributio www.icrf.l Normal distributio I probability theory, the ormal or Gaussia distributio, is a cotiuous probability distributio that is ofte used as a first approimatio to describe realvalued

### Department of Computer Science, University of Otago

Departmet of Computer Sciece, Uiversity of Otago Techical Report OUCS-2006-09 Permutatios Cotaiig May Patters Authors: M.H. Albert Departmet of Computer Sciece, Uiversity of Otago Micah Colema, Rya Fly

### Notes on Hypothesis Testing

Probability & Statistics Grishpa Notes o Hypothesis Testig A radom sample X = X 1,..., X is observed, with joit pmf/pdf f θ x 1,..., x. The values x = x 1,..., x of X lie i some sample space X. The parameter

### Confidence Intervals for One Mean

Chapter 420 Cofidece Itervals for Oe Mea Itroductio This routie calculates the sample size ecessary to achieve a specified distace from the mea to the cofidece limit(s) at a stated cofidece level for a

### Probabilistic Engineering Mechanics. Do Rosenblatt and Nataf isoprobabilistic transformations really differ?

Probabilistic Egieerig Mechaics 4 (009) 577 584 Cotets lists available at ScieceDirect Probabilistic Egieerig Mechaics joural homepage: wwwelseviercom/locate/probegmech Do Roseblatt ad Nataf isoprobabilistic

### Confidence Intervals for the Mean of Non-normal Data Class 23, 18.05, Spring 2014 Jeremy Orloff and Jonathan Bloom

Cofidece Itervals for the Mea of No-ormal Data Class 23, 8.05, Sprig 204 Jeremy Orloff ad Joatha Bloom Learig Goals. Be able to derive the formula for coservative ormal cofidece itervals for the proportio

### LIMIT DISTRIBUTION FOR THE WEIGHTED RANK CORRELATION COEFFICIENT, r W

REVSTAT Statistical Joural Volume 4, Number 3, November 2006, 189 200 LIMIT DISTRIBUTION FOR THE WEIGHTED RANK CORRELATION COEFFICIENT, r W Authors: Joaquim F. Pito da Costa Dep. de Matemática Aplicada,

### Chapter 14 Nonparametric Statistics

Chapter 14 Noparametric Statistics A.K.A. distributio-free statistics! Does ot deped o the populatio fittig ay particular type of distributio (e.g, ormal). Sice these methods make fewer assumptios, they

THE ABRACADABRA PROBLEM FRANCESCO CARAVENNA Abstract. We preset a detailed solutio of Exercise E0.6 i [Wil9]: i a radom sequece of letters, draw idepedetly ad uiformly from the Eglish alphabet, the expected

### THE COMPLETENESS OF CONVERGENT SEQUENCES SPACE OF FUZZY NUMBERS. Hee Chan Choi

Kagweo-Kyugki Math. Jour. 4 (1996), No. 2, pp. 117 124 THE COMPLETENESS OF CONVERGENT SEQUENCES SPACE OF FUZZY NUMBERS Hee Cha Choi Abstract. I this paper we defie a ew fuzzy metric θ of fuzzy umber sequeces,

### 1. C. The formula for the confidence interval for a population mean is: x t, which was

s 1. C. The formula for the cofidece iterval for a populatio mea is: x t, which was based o the sample Mea. So, x is guarateed to be i the iterval you form.. D. Use the rule : p-value

### 3. Continuous Random Variables

Statistics ad probability: 3-1 3. Cotiuous Radom Variables A cotiuous radom variable is a radom variable which ca take values measured o a cotiuous scale e.g. weights, stregths, times or legths. For ay

### Riemann Sums y = f (x)

Riema Sums Recall that we have previously discussed the area problem I its simplest form we ca state it this way: The Area Problem Let f be a cotiuous, o-egative fuctio o the closed iterval [a, b] Fid

### University of California, Los Angeles Department of Statistics. Distributions related to the normal distribution

Uiversity of Califoria, Los Ageles Departmet of Statistics Statistics 100B Istructor: Nicolas Christou Three importat distributios: Distributios related to the ormal distributio Chi-square (χ ) distributio.

### Lesson 12. Sequences and Series

Retur to List of Lessos Lesso. Sequeces ad Series A ifiite sequece { a, a, a,... a,...} ca be thought of as a list of umbers writte i defiite order ad certai patter. It is usually deoted by { a } =, or

### π d i (b i z) (n 1)π )... sin(θ + )

SOME TRIGONOMETRIC IDENTITIES RELATED TO EXACT COVERS Joh Beebee Uiversity of Alaska, Achorage Jauary 18, 1990 Sherma K Stei proves that if si π = k si π b where i the b i are itegers, the are positive

### The Euler Totient, the Möbius and the Divisor Functions

The Euler Totiet, the Möbius ad the Divisor Fuctios Rosica Dieva July 29, 2005 Mout Holyoke College South Hadley, MA 01075 1 Ackowledgemets This work was supported by the Mout Holyoke College fellowship

### A Study for the (μ,s) n Relation for Tent Map

Applied Mathematical Scieces, Vol. 8, 04, o. 60, 3009-305 HIKARI Ltd, www.m-hikari.com http://dx.doi.org/0.988/ams.04.4437 A Study for the (μ,s) Relatio for Tet Map Saba Noori Majeed Departmet of Mathematics

### A CHARACTERIZATION OF MINIMAL ZERO-SEQUENCES OF INDEX ONE IN FINITE CYCLIC GROUPS

INTEGERS: ELECTRONIC JOURNAL OF COMBINATORIAL NUMBER THEORY 5(1) (2005), #A27 A CHARACTERIZATION OF MINIMAL ZERO-SEQUENCES OF INDEX ONE IN FINITE CYCLIC GROUPS Scott T. Chapma 1 Triity Uiversity, Departmet

### Center, Spread, and Shape in Inference: Claims, Caveats, and Insights

Ceter, Spread, ad Shape i Iferece: Claims, Caveats, ad Isights Dr. Nacy Pfeig (Uiversity of Pittsburgh) AMATYC November 2008 Prelimiary Activities 1. I would like to produce a iterval estimate for the

### THE HEIGHT OF q-binary SEARCH TREES

THE HEIGHT OF q-binary SEARCH TREES MICHAEL DRMOTA AND HELMUT PRODINGER Abstract. q biary search trees are obtaied from words, equipped with the geometric distributio istead of permutatios. The average

### Methods of Evaluating Estimators

Math 541: Statistical Theory II Istructor: Sogfeg Zheg Methods of Evaluatig Estimators Let X 1, X 2,, X be i.i.d. radom variables, i.e., a radom sample from f(x θ), where θ is ukow. A estimator of θ is

### Gregory Carey, 1998 Linear Transformations & Composites - 1. Linear Transformations and Linear Composites

Gregory Carey, 1998 Liear Trasformatios & Composites - 1 Liear Trasformatios ad Liear Composites I Liear Trasformatios of Variables Meas ad Stadard Deviatios of Liear Trasformatios A liear trasformatio

### Inference on Proportion. Chapter 8 Tests of Statistical Hypotheses. Sampling Distribution of Sample Proportion. Confidence Interval

Chapter 8 Tests of Statistical Hypotheses 8. Tests about Proportios HT - Iferece o Proportio Parameter: Populatio Proportio p (or π) (Percetage of people has o health isurace) x Statistic: Sample Proportio

### 1 Introduction to reducing variance in Monte Carlo simulations

Copyright c 007 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 uow mea µ = E(X) of a distributio by

### Estimating the Mean and Variance of a Normal Distribution

Estimatig the Mea ad Variace of a Normal Distributio Learig Objectives After completig this module, the studet will be able to eplai the value of repeatig eperimets eplai the role of the law of large umbers

### 4.1 Sigma Notation and Riemann Sums

0 the itegral. Sigma Notatio ad Riema Sums Oe strategy for calculatig the area of a regio is to cut the regio ito simple shapes, calculate the area of each simple shape, ad the add these smaller areas

### Infinite Sequences and Series

CHAPTER 4 Ifiite Sequeces ad Series 4.1. Sequeces A sequece is a ifiite ordered list of umbers, for example the sequece of odd positive itegers: 1, 3, 5, 7, 9, 11, 13, 15, 17, 19, 21, 23, 25, 27, 29...

### Determining the sample size

Determiig the sample size Oe of the most commo questios ay statisticia gets asked is How large a sample size do I eed? Researchers are ofte surprised to fid out that the aswer depeds o a umber of factors

### Lecture 10: Hypothesis testing and confidence intervals

Eco 514: Probability ad Statistics Lecture 10: Hypothesis testig ad cofidece itervals Types of reasoig Deductive reasoig: Start with statemets that are assumed to be true ad use rules of logic to esure

### Matrix Transforms of A-statistically Convergent Sequences with Speed

Filomat 27:8 2013, 1385 1392 DOI 10.2298/FIL1308385 Published by Faculty of Scieces ad Mathematics, Uiversity of Niš, Serbia vailable at: http://www.pmf.i.ac.rs/filomat Matrix Trasforms of -statistically

### Vladimir N. Burkov, Dmitri A. Novikov MODELS AND METHODS OF MULTIPROJECTS MANAGEMENT

Keywords: project maagemet, resource allocatio, etwork plaig Vladimir N Burkov, Dmitri A Novikov MODELS AND METHODS OF MULTIPROJECTS MANAGEMENT The paper deals with the problems of resource allocatio betwee

### Lecture 7: Borel Sets and Lebesgue Measure

EE50: Probability Foudatios for Electrical Egieers July-November 205 Lecture 7: Borel Sets ad Lebesgue Measure Lecturer: Dr. Krisha Jagaatha Scribes: Ravi Kolla, Aseem Sharma, Vishakh Hegde I this lecture,

### arxiv:1506.03481v1 [stat.me] 10 Jun 2015

BEHAVIOUR OF ABC FOR BIG DATA By Wetao Li ad Paul Fearhead Lacaster Uiversity arxiv:1506.03481v1 [stat.me] 10 Ju 2015 May statistical applicatios ivolve models that it is difficult to evaluate the likelihood,

### A Faster Clause-Shortening Algorithm for SAT with No Restriction on Clause Length

Joural o Satisfiability, Boolea Modelig ad Computatio 1 2005) 49-60 A Faster Clause-Shorteig Algorithm for SAT with No Restrictio o Clause Legth Evgey Datsi Alexader Wolpert Departmet of Computer Sciece

### Universal coding for classes of sources

Coexios module: m46228 Uiversal codig for classes of sources Dever Greee This work is produced by The Coexios Project ad licesed uder the Creative Commos Attributio Licese We have discussed several parametric

### Alternatives To Pearson s and Spearman s Correlation Coefficients

Alteratives To Pearso s ad Spearma s Correlatio Coefficiets Floreti Smaradache Chair of Math & Scieces Departmet Uiversity of New Mexico Gallup, NM 8730, USA Abstract. This article presets several alteratives