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Foundations of Mathematics I Set Theory (onlyadraft

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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



Publié par
Nombre de lectures 25
Langue English
Taylor Approximation and the Delta Method
Alex Papanicolaou
April 28, 2009
1.1 MotivatingExample: Estimatingthe odds
Suppose we observeX1, . . . , Xnindependent Bernoulli(pTypically, we are) random variables. p interested inpbut there is also interest in the parameter, which is known as theodds. For 1p 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. 1p1r If we wished to estimatep, we would typically estimate this quantity with the observed success P pˆ probabilitypˆ =Xi/n. Toestimate the odds, it then seems perfectly natural to useas an i1pˆ p estimate for. Butwhereas we know the variance of our estimatorpˆ isp(1p) (check this 1p pˆ be computing var(phow can we approximate its samplingˆ)), what is the variance of? Or, 1pˆ 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) =rg(x)exists, then for dx any constanta, theTaylor polynomial of orderraboutais r (k) X g(a) k Tr(x() =xa). 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)Tr(x), tends to 0 faster than the highest-order term inTr(x). r (r)d Theorem:Ifg(a) =rg(x)|x=aexists, then dx g(x)Tr(x) lim =0. r (xa) xa 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