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Taylor Approximation and the Delta Method - Rice …

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.

2.2 Delta Method: A Generalized CLT Theorem: Let Y n be a sequence of random variables that satis es p n(Y n ) !N(0;˙2) in distribution. For a given function and a speci c value of , suppose that g0( ) exists and is not 0. Then, p n(g(Y

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