Foundations of Mathematics I Set Theory (onlyadraft

Description

Foundations of Mathematics I Set Theory (only a draft) Ali Nesin Mathematics Department Istanbul Bilgi University Kus¸tepe S¸is¸li Istanbul Turkey February 12, 2004

• kus¸tepe s¸is¸li istanbul turkey
• 16.4 addition
• 17.2 multiplication
• geometric automorphism groups
• 3.4 operations
• definition
• functions
• natural numbers
• sets

Sujets

• Academic term
• Operation
• Numbers
• Automorphism
• Multiplication
• Function
• Définition
• Addition
• Istanbul
• Group

Informations

• Publié par: edwui
• Nombre de lectures: 22
• Langue: English

Extrait

Taylor Approximation and the Delta Method

1TaylorApproximation

∗ Alex Papanicolaou

April 28, 2009

1.1 MotivatingExample: Estimatingthe odds

Suppose we observeX 1 , . . . , X n independent Bernoulli(pTypically, we are) random variables. p interested inpbut there is also interest in the parameter, which is known as theodds. For 1−p example, if the outcomes of a medical treatment occur withp= 2/3, then the odds of getting better is 2: 1. Furthermore,if there is another treatment with success probabilityr, we might also be p r interested in theodds ratio/, which gives the relative odds of one treatment over another. 1−p1−r If we wished to estimatep, we would typically estimate this quantity with the observed success P pˆ probabilitypˆ =X i /n. Toestimate the odds, it then seems perfectly natural to useas an i 1−pˆ p estimate for. Butwhereas we know the variance of our estimatorpˆ isp(1−p) (check this 1−p pˆ be computing var(phow can we approximate its samplingˆ)), what is the variance of? Or, 1−pˆ distribution? The Delta Method gives a technique for doing this and is based on using a Taylor series approxi- mation.

1.2 TheTaylor Series

r (r) d Definition:If a functiong(x)has derivatives of orderr, that isg(x) = r g(x)exists, then for dx any constanta, theTaylor polynomial of orderraboutais r (k) X g(a) k T r (x() =x−a). k! k=0 While the Taylor polynomial was introduced as far back as beginning calculus, the major theorem from Taylor is that theremainderfrom the approximation, namelyg(x)−T r (x), tends to 0 faster than the highest-order term inT r (x). r (r) d Theorem:Ifg(a) = r g(x)| x=a exists, then dx g(x)−T r (x) lim =0. r (x−a) x→a ∗ The material here is almost word for word from pp.240-245 ofStatistical Inferenceby George Casella and Roger L. Berger and credit is really to them. 1