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
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- kus¸tepe s¸is¸li istanbul turkey
- 16.4 addition
- 17.2 multiplication
- geometric automorphism groups
- 3.4 operations
- definition
- functions
- natural numbers
- sets
Taylor Approximation and the Delta Method
1TaylorApproximation
∗ 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 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 =Xi/n. Toestimate the odds, it then seems perfectly natural to useas an i1−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) =rg(x)exists, then for dx any constanta, theTaylor polynomial of orderraboutais r (k) X g(a) k Tr(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)−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 (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
