Theorem 8 Let φ be the most powerful size α test of H
|
|
- Μελίνα Γιαννόπουλος
- 5 χρόνια πριν
- Προβολές:
Transcript
1 Testing composite hypotheses Θ = Θ 0 Θ c 0 H 0 : θ Θ 0 H 1 : θ Θ c 0 Definition 16 A test φ is a uniformly most powerful (UMP) level α test for H 0 vs. H 1 if φ has level α and for any other level α test φ E θ [φ(x)] E θ [φ (X)] θ Θ c 0. Theorem 8 Let φ be the most powerful size α test of H 0 : θ = θ 0 vs. H 1 : θ = θ 1, where θ 0 Θ 0 and θ 1 Θ c 0. (φ is assumed to be of the form in a) of Neyman-Pearson lemma.) Suppose that 1) φ does not depend on θ 1, and 2) E θ [φ(x)] α θ Θ 0. Then φ is the UMP level α test of H 0 vs. H
2 Proof 2) that φ is level α. Let φ be any other level α test of H 0 vs. H 1. Since φ does not depend on θ 1, it follows from the Neyman- Pearson lemma that E θ [φ(x)] E θ [φ (X)] θ Θ c 0 and any test φ such that E θ0 [φ (X)] α. But φ is of this form since it is level α for testing H 0 vs. H 1. The result follows immediately. Note In most cases where Theorem 8 is applied, θ 0 is chosen to be on the boundary between Θ 0 and Θ c 0. Example 33 Let X 1,..., X n be i.i.d. N(θ, 1) observations. Want to test H 0 : θ θ 0 vs. H 1 : θ > θ 0. Let θ 1 > θ 0. In Example 30 we found the most powerful size α test of H 0 : θ = θ 0 vs. H 1 : θ = θ 1 is φ(x) = { 1, x θ0 + z α / n 0, x < θ 0 + z α / n. 154
3 This test does not depend on θ 1. Does it have level α when used to test H 0 vs. H 1? Let θ < θ 0. P θ ( X θ 0 + z α n ) = P θ ( X θ 1/ n ) n(θ 0 θ) + z α = P (Z n(θ 0 θ) + z α ) < α since θ 0 > θ. This explains why θ 0 was used in H 0. In many cases the most powerful level α test of H 0 : θ = θ 0 vs. H 1 : θ = θ 1 (θ 1 > θ 0 ) will depend on θ 1. In such cases a UMP test for H 0 : θ θ 0 vs. H 1 : θ > θ 0 does not exist. However, there is a large class of distributions for which UMP tests do exist. 155
4 Let X 1,..., X n be i.i.d. observations with pdf or pmf p(x θ), θ Θ R. Define f(x θ) = p(x 1 θ) p(x n θ). Definition 17 The family of distributions {f(x θ) : θ Θ} is said to have the monotone likelihood ratio property in the statistic T (X) if for θ 1 < θ 2 the ratio f(x θ 2 )/f(x θ 1 ) is a nondecreasing function of T (x) on {x : f(x θ 1 ) > 0 or f(x θ 2 ) > 0}. Example 34 Let X 1,..., X n be i.i.d. U(0, θ) observations, where θ > 0. The joint pdf is f(x θ) = θ n I (0,x(n) ) (x (1) )I (0,θ) (x (n) ). 156
5 Let 0 < θ 1 < θ 2, and x be such that 0 < x (n) < θ 2. (Otherwise both densities are 0.) f(x θ 2 ) f(x θ 1 ) = { (θ1 /θ 2 ) n, 0 < x (n) < θ 1, θ 1 x (n) < θ 2. Hence, the family of U(0, θ) densities has MLR in T (X) = X (n). Theorem 9 family The one-parameter exponential f(x θ) = exp [c(θ)t (x) + d(θ) + S(x)] I A (x) has MLR in T when c is nondecreasing in θ. Proof Let θ 1 < θ 2 and x A. (Why do we assume x A?) 157
6 f(x θ 2 ) f(x θ 1 ) = exp {T (x)[c(θ 2) c(θ 1 )]+ d(θ 2 ) d(θ 1 )}. Since c(θ 1 ) c(θ 2 ), this is a nondecreasing function of T (x). Normal, Poisson, binomial, and exponential distributions all have the MLR property. Theorem 10 Let X 1,..., X n be i.i.d. observations with joint pdf f(x θ), θ Θ, and let the family {f(x θ) : θ Θ} have the MLR property in T. Let θ 0 Θ. Then for testing the composite hypotheses H 0 : θ θ 0 vs. H 1 : θ > θ 0 at level α, there exists a test which is UMP among all level α tests. The UMP test is φ(x) = 1, if T (x) > k γ, if T (x) = k 0, if T (x) < k 158
7 where k and γ are determined by α = E θ0 [φ(x)] = P θ0 [T (X) > k] + γp θ0 [T (X) = k]. Proof Roussas, p. 279; Lehmann, pp ; Casella and Berger, p Note The UMP test for H 0 : θ θ 0 vs. H 1 : θ < θ 0 in the situation of Theorem 10 is given by reversing the inequalities in the definition of φ. Example 35 Let X 1,..., X n be i.i.d. U(0, θ). From Example 34 {f(x θ) : θ > 0} has MLR in X (n). By Theorem 10 the UMP test of is H 0 : θ θ 0 vs. H 1 : θ > θ 0 φ(x) = { 1, if x(n) c 0, if x (n) < c, where E θ0 [φ(x)] = α = P θ0 (X (n) c). 159
8 P θ0 (X (n) c) = 1 P θ0 (X 1 < c,..., X n < c) = 1 [P (X 1 < c)] n = 1 ( c θ 0 ) n The last quantity is α iff c = θ 0 (1 α) 1/n. For θ > θ 0 (1 α) 1/n, the power curve is β(θ) = P θ (X (n) c) = 1 [ θ0 θ (1 α)1/n ] n = 1 ( θ0 θ ) n (1 α). For θ θ 0 (1 α) 1/n, β(θ) = 0. Note: As θ, β(θ)
9 Example 36 Let X 1,..., X n be i.i.d. N(0, θ). Θ = {θ : θ > 0}. Find UMP test of H 0 : θ θ 0 vs. H 1 : θ < θ 0. Defining η = 1/θ and η 0 = 1/θ 0, these hypotheses are equivalent to H 0 : η η 0 vs. H 1 : η > η 0. Check for the MLR property. f(x η) = exp η 2 n i=1 x 2 i n 2 log(2πη 1 ) Since η/2 is an increasing function of η, Theorems 9 and 10 tell us that the UMP level α test of H 0 vs. H 1 has the form φ(x) = { 1, if ni=1 x 2 i c 0, if n i=1 x 2 i < c. 161
10 The test on the last page can also be written { 1, if ni=1 x 2 φ(x) = i c 0, if n i=1 x 2 i > c. The constant c is such that P η0 ( n i=1 X 2 i c ) = α. When η = η 0, we know that η 0 X i N(0, 1), and so η 0 X 2 i χ 2 1 and η 0 ni=1 X 2 i χ 2 n. It follows that c = χ 2 n,α/η 0, where χ 2 n,p is the 100pth percentile of the χ 2 n distribution. So, we have found the most powerful level α test of H 0 vs. H 1, and hence of H 0 vs. H 1. β(θ) = P θ [ χ 2 n ( θ0 θ ) ] χ 2 n,α Limiting cases: θ = 0 and θ =. 162
11 Example 37: UMP tests do not always exist Let the probability model be as in Example 36. Want to test H 0 : θ = θ 0 vs. H 1 : θ θ 0. Θ 0 = {θ 0 } Θ c 0 = (0, ) {θ 0} c MP test of H 0 : θ = θ 0 vs. H 1 : θ = θ 1 for θ 1 > θ 0 is { 1, if ni=1 x 2 φ 1 (x) = i c 0, if n i=1 x 2 i < c. MP test of H 0 : θ = θ 0 vs. H 1 : θ = θ 2 for θ 2 < θ 0 is { 1, if ni=1 x 2 φ 2 (x) = i c 1 0, if n i=1 x 2 i > c 1. Assume φ is UMP for testing H 0 vs. H 1. Then it is most powerful for H 0 : θ = θ 0 vs. H 1 : θ = θ 1 and hence agrees with φ 1 by N-P lemma. Also, it must agree with φ 2 by the same logic. But φ 1 φ 2, which yields a contradiction. Hence, there is no UMP test. 163
12 When a UMP test doesn t exist, one can look at a smaller class of tests and try to find the most powerful test within the smaller class. Examples of such tests: (i) Class of unbiased tests A test is said to be unbiased if β(θ) size of test for all θ Θ c 0. (ii) Class of invariant tests See pp of Lehmann, Testing Statistical Hypotheses. 164
13 Likelihood ratio tests Neyman-Pearson says that the most powerful level α test of H 0 : θ = θ 0 vs. H 1 : θ = θ 1 rejects H 0 for large values of f(x θ 1 ) f(x θ 0 ) = L(θ 1 x) L(θ 0 x). The test statistic is a ratio of likelihoods. Extend this notion to testing composite hypotheses. Let X 1,..., X n be observations with joint pdf f(x θ), θ Θ = Θ 0 Θ c 0, and consider sup θ Θ c 0 L(θ x) sup θ Θ0 L(θ x). The numerator is the likelihood function at the most plausible value of θ in Θ c 0. Likewise, the denominator is L at the most plausible value of θ in Θ
14 It seems reasonable to test H 0 : θ Θ 0 vs. H 1 : θ Θ c 0 by rejecting H 0 if and only if the likelihood ratio on the previous page is sufficiently large. In practice, the test statistic most often used is λ(x), where λ(x) = sup θ Θ L(θ x) sup θ Θ0 L(θ x). Define and L(Θ 0 ) = sup θ Θ 0 L(θ x) L(Θ c 0 ) = sup θ Θ c L(θ x). 166
15 λ(x) = max ( L(Θ 0 ), L(Θ c 0 )) L(Θ 0 ) ( ) L(Θ c = max 0 ) L(Θ 0 ), 1. The likelihood ratio test is { 1, λ(x) > c φ L (x) = 0, λ(x) < c, where c is chosen so that E θ [φ L (X)] α for all θ Θ 0. Note that λ(x) 1 for all x. Recall that under general conditions MLEs are consistent. If H 0 is true, the MLE ˆθ will be close to H 0. If ˆθ Θ 0, then λ(x) = 1, and we won t reject H 0. When ˆθ Θ c 0 and far enough from Θ 0, then we will reject H
16 To calculate λ(x) we need ˆθ, the MLE of θ, and the restricted MLE ˆθ R, where L(ˆθ R x) = sup θ Θ 0 L(θ x). Usually we try to find a statistic T (X) such that λ(x) is a monotone function of T (x). The likelihood ratio test can then be formulated in terms of T (x). Likelihood ratio tests are especially useful in two situations: (i) Two-sided tests (ii) Tests in the presence of nuisance parameters 168
17 Example 38: Two-sided test X 1,..., X n is a random sample from N(0, θ). H 0 : θ = θ 0 vs. H 1 : θ θ 0 The unrestricted MLE of θ is ˆθ = n i=1 Xi 2/n, and, trivially, the restricted MLE of θ is θ 0. [ L(ˆθ x) L(θ 0 x) = (2πˆθ) n/2 exp ni=1 x 2 i /(2ˆθ) ] (2πθ 0 ) n/2 exp [ n i=1 x 2 i /(2θ 0) ] Now, = ( θ0ˆθ ) n/2 exp [ n 2 + nˆθ 2θ 0 log(λ(x)) = n 2 log(θ 0/ˆθ) n 2 (1 ˆθ/θ 0 ) ] = n 2 [ log(ˆθ/θ 0 ) + (1 ˆθ/θ 0 ) ]. 169
18 Define g(y) = [log y + (1 y)]. We have g (y) = 1 y + 1, and hence g (y) < 0 for y < 1 and g (y) > 0 for y > 1. So, log λ(x) > c iff ˆθ/θ 0 < c 1 or ˆθ/θ 0 > c 2, where c 1 and c 2 are such that g(c 1 ) = g(c 2 ). We also know that n(ˆθ/θ 0 ) χ 2 n when θ = θ 0. The precise likelihood ratio test is thus φ L (x) = { 1, ˆθ θ 0 c 1 or ˆθ θ 0 c 2 0, θ 0 c 1 < ˆθ < θ 0 c 2, where g(c 1 ) = g(c 2 ) and P (χ 2 n nc 1) + P (χ 2 n nc 2) = α. 170
Other Test Constructions: Likelihood Ratio & Bayes Tests
Other Test Constructions: Likelihood Ratio & Bayes Tests Side-Note: So far we have seen a few approaches for creating tests such as Neyman-Pearson Lemma ( most powerful tests of H 0 : θ = θ 0 vs H 1 :
Διαβάστε περισσότεραST5224: Advanced Statistical Theory II
ST5224: Advanced Statistical Theory II 2014/2015: Semester II Tutorial 7 1. Let X be a sample from a population P and consider testing hypotheses H 0 : P = P 0 versus H 1 : P = P 1, where P j is a known
Διαβάστε περισσότεραStatistical Inference I Locally most powerful tests
Statistical Inference I Locally most powerful tests Shirsendu Mukherjee Department of Statistics, Asutosh College, Kolkata, India. shirsendu st@yahoo.co.in So far we have treated the testing of one-sided
Διαβάστε περισσότεραSTAT200C: Hypothesis Testing
STAT200C: Hypothesis Testing Zhaoxia Yu Spring 2017 Some Definitions A hypothesis is a statement about a population parameter. The two complementary hypotheses in a hypothesis testing are the null hypothesis
Διαβάστε περισσότεραEcon 2110: Fall 2008 Suggested Solutions to Problem Set 8 questions or comments to Dan Fetter 1
Eon : Fall 8 Suggested Solutions to Problem Set 8 Email questions or omments to Dan Fetter Problem. Let X be a salar with density f(x, θ) (θx + θ) [ x ] with θ. (a) Find the most powerful level α test
Διαβάστε περισσότερα2 Composition. Invertible Mappings
Arkansas Tech University MATH 4033: Elementary Modern Algebra Dr. Marcel B. Finan Composition. Invertible Mappings In this section we discuss two procedures for creating new mappings from old ones, namely,
Διαβάστε περισσότεραExercise 2: The form of the generalized likelihood ratio
Stats 2 Winter 28 Homework 9: Solutions Due Friday, March 6 Exercise 2: The form of the generalized likelihood ratio We want to test H : θ Θ against H : θ Θ, and compare the two following rules of rejection:
Διαβάστε περισσότερα557: MATHEMATICAL STATISTICS II RESULTS FROM CLASSICAL HYPOTHESIS TESTING
Most Powerful Tests 557: MATHEMATICAL STATISTICS II RESULTS FROM CLASSICAL HYPOTHESIS TESTING To construct and assess the quality of a statistical test, we consider the power function β(θ). Consider a
Διαβάστε περισσότεραSolution Series 9. i=1 x i and i=1 x i.
Lecturer: Prof. Dr. Mete SONER Coordinator: Yilin WANG Solution Series 9 Q1. Let α, β >, the p.d.f. of a beta distribution with parameters α and β is { Γ(α+β) Γ(α)Γ(β) f(x α, β) xα 1 (1 x) β 1 for < x
Διαβάστε περισσότερα6. MAXIMUM LIKELIHOOD ESTIMATION
6 MAXIMUM LIKELIHOOD ESIMAION [1] Maximum Likelihood Estimator (1) Cases in which θ (unknown parameter) is scalar Notational Clarification: From now on, we denote the true value of θ as θ o hen, view θ
Διαβάστε περισσότερα557: MATHEMATICAL STATISTICS II HYPOTHESIS TESTING
557: MATHEMATICAL STATISTICS II HYPOTHESIS TESTING A statistical hypothesis test is a decision rule that takes as an input observed sample data and returns an action relating to two mutually exclusive
Διαβάστε περισσότεραOrdinal Arithmetic: Addition, Multiplication, Exponentiation and Limit
Ordinal Arithmetic: Addition, Multiplication, Exponentiation and Limit Ting Zhang Stanford May 11, 2001 Stanford, 5/11/2001 1 Outline Ordinal Classification Ordinal Addition Ordinal Multiplication Ordinal
Διαβάστε περισσότερα4.6 Autoregressive Moving Average Model ARMA(1,1)
84 CHAPTER 4. STATIONARY TS MODELS 4.6 Autoregressive Moving Average Model ARMA(,) This section is an introduction to a wide class of models ARMA(p,q) which we will consider in more detail later in this
Διαβάστε περισσότεραExample Sheet 3 Solutions
Example Sheet 3 Solutions. i Regular Sturm-Liouville. ii Singular Sturm-Liouville mixed boundary conditions. iii Not Sturm-Liouville ODE is not in Sturm-Liouville form. iv Regular Sturm-Liouville note
Διαβάστε περισσότεραEvery set of first-order formulas is equivalent to an independent set
Every set of first-order formulas is equivalent to an independent set May 6, 2008 Abstract A set of first-order formulas, whatever the cardinality of the set of symbols, is equivalent to an independent
Διαβάστε περισσότεραLast Lecture. Biostatistics Statistical Inference Lecture 19 Likelihood Ratio Test. Example of Hypothesis Testing.
Last Lecture Biostatistics 602 - Statistical Iferece Lecture 19 Likelihood Ratio Test Hyu Mi Kag March 26th, 2013 Describe the followig cocepts i your ow words Hypothesis Null Hypothesis Alterative Hypothesis
Διαβάστε περισσότεραC.S. 430 Assignment 6, Sample Solutions
C.S. 430 Assignment 6, Sample Solutions Paul Liu November 15, 2007 Note that these are sample solutions only; in many cases there were many acceptable answers. 1 Reynolds Problem 10.1 1.1 Normal-order
Διαβάστε περισσότερα3.4 SUM AND DIFFERENCE FORMULAS. NOTE: cos(α+β) cos α + cos β cos(α-β) cos α -cos β
3.4 SUM AND DIFFERENCE FORMULAS Page Theorem cos(αβ cos α cos β -sin α cos(α-β cos α cos β sin α NOTE: cos(αβ cos α cos β cos(α-β cos α -cos β Proof of cos(α-β cos α cos β sin α Let s use a unit circle
Διαβάστε περισσότεραSOLUTIONS TO MATH38181 EXTREME VALUES AND FINANCIAL RISK EXAM
SOLUTIONS TO MATH38181 EXTREME VALUES AND FINANCIAL RISK EXAM Solutions to Question 1 a) The cumulative distribution function of T conditional on N n is Pr T t N n) Pr max X 1,..., X N ) t N n) Pr max
Διαβάστε περισσότεραProblem Set 3: Solutions
CMPSCI 69GG Applied Information Theory Fall 006 Problem Set 3: Solutions. [Cover and Thomas 7.] a Define the following notation, C I p xx; Y max X; Y C I p xx; Ỹ max I X; Ỹ We would like to show that C
Διαβάστε περισσότεραCHAPTER 25 SOLVING EQUATIONS BY ITERATIVE METHODS
CHAPTER 5 SOLVING EQUATIONS BY ITERATIVE METHODS EXERCISE 104 Page 8 1. Find the positive root of the equation x + 3x 5 = 0, correct to 3 significant figures, using the method of bisection. Let f(x) =
Διαβάστε περισσότεραΜηχανική Μάθηση Hypothesis Testing
ΕΛΛΗΝΙΚΗ ΔΗΜΟΚΡΑΤΙΑ ΠΑΝΕΠΙΣΤΗΜΙΟ ΚΡΗΤΗΣ Μηχανική Μάθηση Hypothesis Testing Γιώργος Μπορμπουδάκης Τμήμα Επιστήμης Υπολογιστών Procedure 1. Form the null (H 0 ) and alternative (H 1 ) hypothesis 2. Consider
Διαβάστε περισσότεραEE512: Error Control Coding
EE512: Error Control Coding Solution for Assignment on Finite Fields February 16, 2007 1. (a) Addition and Multiplication tables for GF (5) and GF (7) are shown in Tables 1 and 2. + 0 1 2 3 4 0 0 1 2 3
Διαβάστε περισσότεραLecture 34 Bootstrap confidence intervals
Lecture 34 Bootstrap confidence intervals Confidence Intervals θ: an unknown parameter of interest We want to find limits θ and θ such that Gt = P nˆθ θ t If G 1 1 α is known, then P θ θ = P θ θ = 1 α
Διαβάστε περισσότεραInverse trigonometric functions & General Solution of Trigonometric Equations. ------------------ ----------------------------- -----------------
Inverse trigonometric functions & General Solution of Trigonometric Equations. 1. Sin ( ) = a) b) c) d) Ans b. Solution : Method 1. Ans a: 17 > 1 a) is rejected. w.k.t Sin ( sin ) = d is rejected. If sin
Διαβάστε περισσότεραSection 8.3 Trigonometric Equations
99 Section 8. Trigonometric Equations Objective 1: Solve Equations Involving One Trigonometric Function. In this section and the next, we will exple how to solving equations involving trigonometric functions.
Διαβάστε περισσότεραFinite Field Problems: Solutions
Finite Field Problems: Solutions 1. Let f = x 2 +1 Z 11 [x] and let F = Z 11 [x]/(f), a field. Let Solution: F =11 2 = 121, so F = 121 1 = 120. The possible orders are the divisors of 120. Solution: The
Διαβάστε περισσότεραAn Introduction to Signal Detection and Estimation - Second Edition Chapter II: Selected Solutions
An Introduction to Signal Detection Estimation - Second Edition Chapter II: Selected Solutions H V Poor Princeton University March 16, 5 Exercise : The likelihood ratio is given by L(y) (y +1), y 1 a With
Διαβάστε περισσότεραSolutions to Exercise Sheet 5
Solutions to Eercise Sheet 5 jacques@ucsd.edu. Let X and Y be random variables with joint pdf f(, y) = 3y( + y) where and y. Determine each of the following probabilities. Solutions. a. P (X ). b. P (X
Διαβάστε περισσότεραConcrete Mathematics Exercises from 30 September 2016
Concrete Mathematics Exercises from 30 September 2016 Silvio Capobianco Exercise 1.7 Let H(n) = J(n + 1) J(n). Equation (1.8) tells us that H(2n) = 2, and H(2n+1) = J(2n+2) J(2n+1) = (2J(n+1) 1) (2J(n)+1)
Διαβάστε περισσότεραSOLUTIONS TO MATH38181 EXTREME VALUES AND FINANCIAL RISK EXAM
SOLUTIONS TO MATH38181 EXTREME VALUES AND FINANCIAL RISK EXAM Solutions to Question 1 a) The cumulative distribution function of T conditional on N n is Pr (T t N n) Pr (max (X 1,..., X N ) t N n) Pr (max
Διαβάστε περισσότεραLecture 2. Soundness and completeness of propositional logic
Lecture 2 Soundness and completeness of propositional logic February 9, 2004 1 Overview Review of natural deduction. Soundness and completeness. Semantics of propositional formulas. Soundness proof. Completeness
Διαβάστε περισσότεραFourier Series. MATH 211, Calculus II. J. Robert Buchanan. Spring Department of Mathematics
Fourier Series MATH 211, Calculus II J. Robert Buchanan Department of Mathematics Spring 2018 Introduction Not all functions can be represented by Taylor series. f (k) (c) A Taylor series f (x) = (x c)
Διαβάστε περισσότεραThe challenges of non-stable predicates
The challenges of non-stable predicates Consider a non-stable predicate Φ encoding, say, a safety property. We want to determine whether Φ holds for our program. The challenges of non-stable predicates
Διαβάστε περισσότεραHomework for 1/27 Due 2/5
Name: ID: Homework for /7 Due /5. [ 8-3] I Example D of Sectio 8.4, the pdf of the populatio distributio is + αx x f(x α) =, α, otherwise ad the method of momets estimate was foud to be ˆα = 3X (where
Διαβάστε περισσότεραThe Simply Typed Lambda Calculus
Type Inference Instead of writing type annotations, can we use an algorithm to infer what the type annotations should be? That depends on the type system. For simple type systems the answer is yes, and
Διαβάστε περισσότεραLecture 12: Pseudo likelihood approach
Lecture 12: Pseudo likelihood approach Pseudo MLE Let X 1,...,X n be a random sample from a pdf in a family indexed by two parameters θ and π with likelihood l(θ,π). The method of pseudo MLE may be viewed
Διαβάστε περισσότεραPhys460.nb Solution for the t-dependent Schrodinger s equation How did we find the solution? (not required)
Phys460.nb 81 ψ n (t) is still the (same) eigenstate of H But for tdependent H. The answer is NO. 5.5.5. Solution for the tdependent Schrodinger s equation If we assume that at time t 0, the electron starts
Διαβάστε περισσότεραNowhere-zero flows Let be a digraph, Abelian group. A Γ-circulation in is a mapping : such that, where, and : tail in X, head in
Nowhere-zero flows Let be a digraph, Abelian group. A Γ-circulation in is a mapping : such that, where, and : tail in X, head in : tail in X, head in A nowhere-zero Γ-flow is a Γ-circulation such that
Διαβάστε περισσότερα5.4 The Poisson Distribution.
The worst thing you can do about a situation is nothing. Sr. O Shea Jackson 5.4 The Poisson Distribution. Description of the Poisson Distribution Discrete probability distribution. The random variable
Διαβάστε περισσότεραFractional Colorings and Zykov Products of graphs
Fractional Colorings and Zykov Products of graphs Who? Nichole Schimanski When? July 27, 2011 Graphs A graph, G, consists of a vertex set, V (G), and an edge set, E(G). V (G) is any finite set E(G) is
Διαβάστε περισσότεραUniform Convergence of Fourier Series Michael Taylor
Uniform Convergence of Fourier Series Michael Taylor Given f L 1 T 1 ), we consider the partial sums of the Fourier series of f: N 1) S N fθ) = ˆfk)e ikθ. k= N A calculation gives the Dirichlet formula
Διαβάστε περισσότεραTridiagonal matrices. Gérard MEURANT. October, 2008
Tridiagonal matrices Gérard MEURANT October, 2008 1 Similarity 2 Cholesy factorizations 3 Eigenvalues 4 Inverse Similarity Let α 1 ω 1 β 1 α 2 ω 2 T =......... β 2 α 1 ω 1 β 1 α and β i ω i, i = 1,...,
Διαβάστε περισσότεραORDINAL ARITHMETIC JULIAN J. SCHLÖDER
ORDINAL ARITHMETIC JULIAN J. SCHLÖDER Abstract. We define ordinal arithmetic and show laws of Left- Monotonicity, Associativity, Distributivity, some minor related properties and the Cantor Normal Form.
Διαβάστε περισσότεραMatrices and Determinants
Matrices and Determinants SUBJECTIVE PROBLEMS: Q 1. For what value of k do the following system of equations possess a non-trivial (i.e., not all zero) solution over the set of rationals Q? x + ky + 3z
Διαβάστε περισσότερα6.3 Forecasting ARMA processes
122 CHAPTER 6. ARMA MODELS 6.3 Forecasting ARMA processes The purpose of forecasting is to predict future values of a TS based on the data collected to the present. In this section we will discuss a linear
Διαβάστε περισσότεραAreas and Lengths in Polar Coordinates
Kiryl Tsishchanka Areas and Lengths in Polar Coordinates In this section we develop the formula for the area of a region whose boundary is given by a polar equation. We need to use the formula for the
Διαβάστε περισσότεραSection 9.2 Polar Equations and Graphs
180 Section 9. Polar Equations and Graphs In this section, we will be graphing polar equations on a polar grid. In the first few examples, we will write the polar equation in rectangular form to help identify
Διαβάστε περισσότερα( y) Partial Differential Equations
Partial Dierential Equations Linear P.D.Es. contains no owers roducts o the deendent variables / an o its derivatives can occasionall be solved. Consider eamle ( ) a (sometimes written as a ) we can integrate
Διαβάστε περισσότεραMath221: HW# 1 solutions
Math: HW# solutions Andy Royston October, 5 7.5.7, 3 rd Ed. We have a n = b n = a = fxdx = xdx =, x cos nxdx = x sin nx n sin nxdx n = cos nx n = n n, x sin nxdx = x cos nx n + cos nxdx n cos n = + sin
Διαβάστε περισσότεραAreas and Lengths in Polar Coordinates
Kiryl Tsishchanka Areas and Lengths in Polar Coordinates In this section we develop the formula for the area of a region whose boundary is given by a polar equation. We need to use the formula for the
Διαβάστε περισσότεραBayesian statistics. DS GA 1002 Probability and Statistics for Data Science.
Bayesian statistics DS GA 1002 Probability and Statistics for Data Science http://www.cims.nyu.edu/~cfgranda/pages/dsga1002_fall17 Carlos Fernandez-Granda Frequentist vs Bayesian statistics In frequentist
Διαβάστε περισσότεραA Note on Intuitionistic Fuzzy. Equivalence Relation
International Mathematical Forum, 5, 2010, no. 67, 3301-3307 A Note on Intuitionistic Fuzzy Equivalence Relation D. K. Basnet Dept. of Mathematics, Assam University Silchar-788011, Assam, India dkbasnet@rediffmail.com
Διαβάστε περισσότεραHomework 3 Solutions
Homework 3 Solutions Igor Yanovsky (Math 151A TA) Problem 1: Compute the absolute error and relative error in approximations of p by p. (Use calculator!) a) p π, p 22/7; b) p π, p 3.141. Solution: For
Διαβάστε περισσότεραMore Notes on Testing. Large Sample Properties of the Likelihood Ratio Statistic. Let X i be iid with density f(x, θ). We are interested in testing
Ulrich Müller Economics 2110 Fall 2008 More Notes on Testing Large Sample Properties of the Likelihood Ratio Statistic Let X i be iid with density f(x, θ). We are interested in testing H 0 : θ = θ 0 against
Διαβάστε περισσότεραAquinas College. Edexcel Mathematical formulae and statistics tables DO NOT WRITE ON THIS BOOKLET
Aquinas College Edexcel Mathematical formulae and statistics tables DO NOT WRITE ON THIS BOOKLET Pearson Edexcel Level 3 Advanced Subsidiary and Advanced GCE in Mathematics and Further Mathematics Mathematical
Διαβάστε περισσότεραLecture 21: Properties and robustness of LSE
Lecture 21: Properties and robustness of LSE BLUE: Robustness of LSE against normality We now study properties of l τ β and σ 2 under assumption A2, i.e., without the normality assumption on ε. From Theorem
Διαβάστε περισσότεραMath 446 Homework 3 Solutions. (1). (i): Reverse triangle inequality for metrics: Let (X, d) be a metric space and let x, y, z X.
Math 446 Homework 3 Solutions. (1). (i): Reverse triangle inequalit for metrics: Let (X, d) be a metric space and let x,, z X. Prove that d(x, z) d(, z) d(x, ). (ii): Reverse triangle inequalit for norms:
Διαβάστε περισσότεραHOMEWORK 4 = G. In order to plot the stress versus the stretch we define a normalized stretch:
HOMEWORK 4 Problem a For the fast loading case, we want to derive the relationship between P zz and λ z. We know that the nominal stress is expressed as: P zz = ψ λ z where λ z = λ λ z. Therefore, applying
Διαβάστε περισσότεραMath 6 SL Probability Distributions Practice Test Mark Scheme
Math 6 SL Probability Distributions Practice Test Mark Scheme. (a) Note: Award A for vertical line to right of mean, A for shading to right of their vertical line. AA N (b) evidence of recognizing symmetry
Διαβάστε περισσότεραk A = [k, k]( )[a 1, a 2 ] = [ka 1,ka 2 ] 4For the division of two intervals of confidence in R +
Chapter 3. Fuzzy Arithmetic 3- Fuzzy arithmetic: ~Addition(+) and subtraction (-): Let A = [a and B = [b, b in R If x [a and y [b, b than x+y [a +b +b Symbolically,we write A(+)B = [a (+)[b, b = [a +b
Διαβάστε περισσότεραb. Use the parametrization from (a) to compute the area of S a as S a ds. Be sure to substitute for ds!
MTH U341 urface Integrals, tokes theorem, the divergence theorem To be turned in Wed., Dec. 1. 1. Let be the sphere of radius a, x 2 + y 2 + z 2 a 2. a. Use spherical coordinates (with ρ a) to parametrize.
Διαβάστε περισσότεραF19MC2 Solutions 9 Complex Analysis
F9MC Solutions 9 Complex Analysis. (i) Let f(z) = eaz +z. Then f is ifferentiable except at z = ±i an so by Cauchy s Resiue Theorem e az z = πi[res(f,i)+res(f, i)]. +z C(,) Since + has zeros of orer at
Διαβάστε περισσότεραChapter 6: Systems of Linear Differential. be continuous functions on the interval
Chapter 6: Systems of Linear Differential Equations Let a (t), a 2 (t),..., a nn (t), b (t), b 2 (t),..., b n (t) be continuous functions on the interval I. The system of n first-order differential equations
Διαβάστε περισσότεραΕΙΣΑΓΩΓΗ ΣΤΗ ΣΤΑΤΙΣΤΙΚΗ ΑΝΑΛΥΣΗ
ΕΙΣΑΓΩΓΗ ΣΤΗ ΣΤΑΤΙΣΤΙΚΗ ΑΝΑΛΥΣΗ ΕΛΕΝΑ ΦΛΟΚΑ Επίκουρος Καθηγήτρια Τµήµα Φυσικής, Τοµέας Φυσικής Περιβάλλοντος- Μετεωρολογίας ΓΕΝΙΚΟΙ ΟΡΙΣΜΟΙ Πληθυσµός Σύνολο ατόµων ή αντικειµένων στα οποία αναφέρονται
Διαβάστε περισσότεραSection 7.6 Double and Half Angle Formulas
09 Section 7. Double and Half Angle Fmulas To derive the double-angles fmulas, we will use the sum of two angles fmulas that we developed in the last section. We will let α θ and β θ: cos(θ) cos(θ + θ)
Διαβάστε περισσότερα2. Let H 1 and H 2 be Hilbert spaces and let T : H 1 H 2 be a bounded linear operator. Prove that [T (H 1 )] = N (T ). (6p)
Uppsala Universitet Matematiska Institutionen Andreas Strömbergsson Prov i matematik Funktionalanalys Kurs: F3B, F4Sy, NVP 2005-03-08 Skrivtid: 9 14 Tillåtna hjälpmedel: Manuella skrivdon, Kreyszigs bok
Διαβάστε περισσότερα6.1. Dirac Equation. Hamiltonian. Dirac Eq.
6.1. Dirac Equation Ref: M.Kaku, Quantum Field Theory, Oxford Univ Press (1993) η μν = η μν = diag(1, -1, -1, -1) p 0 = p 0 p = p i = -p i p μ p μ = p 0 p 0 + p i p i = E c 2 - p 2 = (m c) 2 H = c p 2
Διαβάστε περισσότεραReminders: linear functions
Reminders: linear functions Let U and V be vector spaces over the same field F. Definition A function f : U V is linear if for every u 1, u 2 U, f (u 1 + u 2 ) = f (u 1 ) + f (u 2 ), and for every u U
Διαβάστε περισσότεραJesse Maassen and Mark Lundstrom Purdue University November 25, 2013
Notes on Average Scattering imes and Hall Factors Jesse Maassen and Mar Lundstrom Purdue University November 5, 13 I. Introduction 1 II. Solution of the BE 1 III. Exercises: Woring out average scattering
Διαβάστε περισσότεραEstimation for ARMA Processes with Stable Noise. Matt Calder & Richard A. Davis Colorado State University
Estimation for ARMA Processes with Stable Noise Matt Calder & Richard A. Davis Colorado State University rdavis@stat.colostate.edu 1 ARMA processes with stable noise Review of M-estimation Examples of
Διαβάστε περισσότεραBounding Nonsplitting Enumeration Degrees
Bounding Nonsplitting Enumeration Degrees Thomas F. Kent Andrea Sorbi Università degli Studi di Siena Italia July 18, 2007 Goal: Introduce a form of Σ 0 2-permitting for the enumeration degrees. Till now,
Διαβάστε περισσότεραPractice Exam 2. Conceptual Questions. 1. State a Basic identity and then verify it. (a) Identity: Solution: One identity is csc(θ) = 1
Conceptual Questions. State a Basic identity and then verify it. a) Identity: Solution: One identity is cscθ) = sinθ) Practice Exam b) Verification: Solution: Given the point of intersection x, y) of the
Διαβάστε περισσότεραLecture 7: Overdispersion in Poisson regression
Lecture 7: Overdispersion in Poisson regression Claudia Czado TU München c (Claudia Czado, TU Munich) ZFS/IMS Göttingen 2004 0 Overview Introduction Modeling overdispersion through mixing Score test for
Διαβάστε περισσότεραSCITECH Volume 13, Issue 2 RESEARCH ORGANISATION Published online: March 29, 2018
Journal of rogressive Research in Mathematics(JRM) ISSN: 2395-028 SCITECH Volume 3, Issue 2 RESEARCH ORGANISATION ublished online: March 29, 208 Journal of rogressive Research in Mathematics www.scitecresearch.com/journals
Διαβάστε περισσότεραAn Inventory of Continuous Distributions
Appendi A An Inventory of Continuous Distributions A.1 Introduction The incomplete gamma function is given by Also, define Γ(α; ) = 1 with = G(α; ) = Z 0 Z 0 Z t α 1 e t dt, α > 0, >0 t α 1 e t dt, α >
Διαβάστε περισσότεραApproximation of distance between locations on earth given by latitude and longitude
Approximation of distance between locations on earth given by latitude and longitude Jan Behrens 2012-12-31 In this paper we shall provide a method to approximate distances between two points on earth
Διαβάστε περισσότερα5. Choice under Uncertainty
5. Choice under Uncertainty Daisuke Oyama Microeconomics I May 23, 2018 Formulations von Neumann-Morgenstern (1944/1947) X: Set of prizes Π: Set of probability distributions on X : Preference relation
Διαβάστε περισσότεραCongruence Classes of Invertible Matrices of Order 3 over F 2
International Journal of Algebra, Vol. 8, 24, no. 5, 239-246 HIKARI Ltd, www.m-hikari.com http://dx.doi.org/.2988/ija.24.422 Congruence Classes of Invertible Matrices of Order 3 over F 2 Ligong An and
Διαβάστε περισσότεραThe Probabilistic Method - Probabilistic Techniques. Lecture 7: The Janson Inequality
The Probabilistic Method - Probabilistic Techniques Lecture 7: The Janson Inequality Sotiris Nikoletseas Associate Professor Computer Engineering and Informatics Department 2014-2015 Sotiris Nikoletseas,
Διαβάστε περισσότεραStatistics 104: Quantitative Methods for Economics Formula and Theorem Review
Harvard College Statistics 104: Quantitative Methods for Economics Formula and Theorem Review Tommy MacWilliam, 13 tmacwilliam@college.harvard.edu March 10, 2011 Contents 1 Introduction to Data 5 1.1 Sample
Διαβάστε περισσότεραP AND P. P : actual probability. P : risk neutral probability. Realtionship: mutual absolute continuity P P. For example:
(B t, S (t) t P AND P,..., S (p) t ): securities P : actual probability P : risk neutral probability Realtionship: mutual absolute continuity P P For example: P : ds t = µ t S t dt + σ t S t dw t P : ds
Διαβάστε περισσότεραDepth versus Rigidity in the Design of International Trade Agreements. Leslie Johns
Depth versus Rigidity in the Design of International Trade Agreements Leslie Johns Supplemental Appendix September 3, 202 Alternative Punishment Mechanisms The one-period utility functions of the home
Διαβάστε περισσότεραLecture 2: Dirac notation and a review of linear algebra Read Sakurai chapter 1, Baym chatper 3
Lecture 2: Dirac notation and a review of linear algebra Read Sakurai chapter 1, Baym chatper 3 1 State vector space and the dual space Space of wavefunctions The space of wavefunctions is the set of all
Διαβάστε περισσότεραD Alembert s Solution to the Wave Equation
D Alembert s Solution to the Wave Equation MATH 467 Partial Differential Equations J. Robert Buchanan Department of Mathematics Fall 2018 Objectives In this lesson we will learn: a change of variable technique
Διαβάστε περισσότεραANSWERSHEET (TOPIC = DIFFERENTIAL CALCULUS) COLLECTION #2. h 0 h h 0 h h 0 ( ) g k = g 0 + g 1 + g g 2009 =?
Teko Classes IITJEE/AIEEE Maths by SUHAAG SIR, Bhopal, Ph (0755) 3 00 000 www.tekoclasses.com ANSWERSHEET (TOPIC DIFFERENTIAL CALCULUS) COLLECTION # Question Type A.Single Correct Type Q. (A) Sol least
Διαβάστε περισσότεραA Conception of Inductive Logic: Example
A Conception of Inductive Logic: Example Patrick Maher Department of Philosophy, University of Illinois at Urbana-Champaign This paper presents a simple example of inductive logic done according to the
Διαβάστε περισσότεραHomework 8 Model Solution Section
MATH 004 Homework Solution Homework 8 Model Solution Section 14.5 14.6. 14.5. Use the Chain Rule to find dz where z cosx + 4y), x 5t 4, y 1 t. dz dx + dy y sinx + 4y)0t + 4) sinx + 4y) 1t ) 0t + 4t ) sinx
Διαβάστε περισσότεραChapter 3: Ordinal Numbers
Chapter 3: Ordinal Numbers There are two kinds of number.. Ordinal numbers (0th), st, 2nd, 3rd, 4th, 5th,..., ω, ω +,... ω2, ω2+,... ω 2... answers to the question What position is... in a sequence? What
Διαβάστε περισσότεραLocal Approximation with Kernels
Local Approximation with Kernels Thomas Hangelbroek University of Hawaii at Manoa 5th International Conference Approximation Theory, 26 work supported by: NSF DMS-43726 A cubic spline example Consider
Διαβάστε περισσότεραSrednicki Chapter 55
Srednicki Chapter 55 QFT Problems & Solutions A. George August 3, 03 Srednicki 55.. Use equations 55.3-55.0 and A i, A j ] = Π i, Π j ] = 0 (at equal times) to verify equations 55.-55.3. This is our third
Διαβάστε περισσότεραAPPENDICES APPENDIX A. STATISTICAL TABLES AND CHARTS 651 APPENDIX B. BIBLIOGRAPHY 677 APPENDIX C. ANSWERS TO SELECTED EXERCISES 679
APPENDICES APPENDIX A. STATISTICAL TABLES AND CHARTS 1 Table I Summary of Common Probability Distributions 2 Table II Cumulative Standard Normal Distribution Table III Percentage Points, 2 of the Chi-Squared
Διαβάστε περισσότεραderivation of the Laplacian from rectangular to spherical coordinates
derivation of the Laplacian from rectangular to spherical coordinates swapnizzle 03-03- :5:43 We begin by recognizing the familiar conversion from rectangular to spherical coordinates (note that φ is used
Διαβάστε περισσότεραECE598: Information-theoretic methods in high-dimensional statistics Spring 2016
ECE598: Information-theoretic methods in high-dimensional statistics Spring 06 Lecture 7: Information bound Lecturer: Yihong Wu Scribe: Shiyu Liang, Feb 6, 06 [Ed. Mar 9] Recall the Chi-squared divergence
Διαβάστε περισσότεραExercises 10. Find a fundamental matrix of the given system of equations. Also find the fundamental matrix Φ(t) satisfying Φ(0) = I. 1.
Exercises 0 More exercises are available in Elementary Differential Equations. If you have a problem to solve any of them, feel free to come to office hour. Problem Find a fundamental matrix of the given
Διαβάστε περισσότεραLecture 26: Circular domains
Introductory lecture notes on Partial Differential Equations - c Anthony Peirce. Not to be copied, used, or revised without eplicit written permission from the copyright owner. 1 Lecture 6: Circular domains
Διαβάστε περισσότεραENGR 691/692 Section 66 (Fall 06): Machine Learning Assigned: August 30 Homework 1: Bayesian Decision Theory (solutions) Due: September 13
ENGR 69/69 Section 66 (Fall 06): Machine Learning Assigned: August 30 Homework : Bayesian Decision Theory (solutions) Due: Septemer 3 Prolem : ( pts) Let the conditional densities for a two-category one-dimensional
Διαβάστε περισσότεραDESIGN OF MACHINERY SOLUTION MANUAL h in h 4 0.
DESIGN OF MACHINERY SOLUTION MANUAL -7-1! PROBLEM -7 Statement: Design a double-dwell cam to move a follower from to 25 6, dwell for 12, fall 25 and dwell for the remader The total cycle must take 4 sec
Διαβάστε περισσότεραMATH423 String Theory Solutions 4. = 0 τ = f(s). (1) dτ ds = dxµ dτ f (s) (2) dτ 2 [f (s)] 2 + dxµ. dτ f (s) (3)
1. MATH43 String Theory Solutions 4 x = 0 τ = fs). 1) = = f s) ) x = x [f s)] + f s) 3) equation of motion is x = 0 if an only if f s) = 0 i.e. fs) = As + B with A, B constants. i.e. allowe reparametrisations
Διαβάστε περισσότεραΕΛΛΗΝΙΚΗ ΔΗΜΟΚΡΑΤΙΑ ΠΑΝΕΠΙΣΤΗΜΙΟ ΚΡΗΤΗΣ. Ψηφιακή Οικονομία. Διάλεξη 10η: Basics of Game Theory part 2 Mαρίνα Μπιτσάκη Τμήμα Επιστήμης Υπολογιστών
ΕΛΛΗΝΙΚΗ ΔΗΜΟΚΡΑΤΙΑ ΠΑΝΕΠΙΣΤΗΜΙΟ ΚΡΗΤΗΣ Ψηφιακή Οικονομία Διάλεξη 0η: Basics of Game Theory part 2 Mαρίνα Μπιτσάκη Τμήμα Επιστήμης Υπολογιστών Best Response Curves Used to solve for equilibria in games
Διαβάστε περισσότερα