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Taylor Approximation and the Delta Method

Taylor Approximation and the Delta MethodAlex Papanicolaou April 28, 20091 Taylor Motivating Example: Estimating the oddsSuppose we observeX1,..,Xnindependent Bernoulli(p) random variables. Typically, we areinterested inpbut there is also interest in the parameterp1 p, which is known as theodds. Forexample, if the outcomes of a medical treatment occur withp= 2/3, then the odds of getting betteris 2 : 1. Furthermore, if there is another treatment with success probabilityr, we might also beinterested in theodds ratiop1 p/r1 r, which gives the relative odds of one treatment over we wished to estimatep, we would typically estimate this quantity with the observed successprobability p= iXi/n.

1The statement of the Delta Method allows for great generality of sequences Y n satisfying the CLT. This is because there are multiple forms of the CLT. Typically, the sample mean is used in these types of approximations and from elementary probability, we know the sample mean is one such sequence of random variables that satis es the CLT. 4

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  Methods, Sequence, Delta, Taylor, Approximation, The delta, Taylor approximation and the delta method

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