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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.

a binomial success probability. Using the notation described in the previous section, we take g(p) = p 1 p so that g 0(p) = 1 (1 2p) (this is a univariate this case, so k= 1 and thus there is only one derivative) and Var p^ 1 p^ ˇg0(p)2Var(^p) = 1 (1 p)2 2 p(1 p) n = p n(1 p)3; giving us an approximation for the variance of our estimator. k

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  Methods, Probability, Delta, Taylor, Univariate, Approximation, Taylor approximation and the delta method

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