(This article was originally published at R – Xi'an's Og, and syndicated at StatsBlogs.)
In answering a simple question on X validated about producing Monte Carlo estimates of the variance of estimators of exp(-θ) in a Poisson model, I wanted to illustrate the accuracy of these estimates against the theoretical values. While one case was easy, since the estimator was a Binomial B(n,exp(-θ)) variate [in yellow on the graph], the other one being the exponential of the negative of the Poisson sample average did not enjoy a closed-form variance and I instead used a first order (δ-method) approximation for this variance which ended up working surprisingly well [in brown] given that the experiment is based on an n=20 sample size.
Thanks to the comments of George Henry, I stand corrected: the variance of the exponential version is easily manageable with two lines of summation! As
![\text{var}(\exp\{-\bar{X}_n\})=\exp\left\{-n\theta[1-\exp\{-2/n\}]\right\}](https://s0.wp.com/latex.php?latex=%5Ctext%7Bvar%7D%28%5Cexp%5C%7B-%5Cbar%7BX%7D_n%5C%7D%29%3D%5Cexp%5Cleft%5C%7B-n%5Ctheta%5B1-%5Cexp%5C%7B-2%2Fn%5C%7D%5D%5Cright%5C%7D&bg=000000&%23038;fg=B0B0B0&%23038;s=0 "\text{var}(\exp\{-\bar{X}_n\})=\exp\left\{-n\theta[1-\exp\{-2/n\}]\right\}")
![-\exp\left\{-2n\theta[1-\exp\{-1/n\}]\right\}](https://s0.wp.com/latex.php?latex=-%5Cexp%5Cleft%5C%7B-2n%5Ctheta%5B1-%5Cexp%5C%7B-1%2Fn%5C%7D%5D%5Cright%5C%7D&bg=000000&%23038;fg=B0B0B0&%23038;s=0 "-\exp\left\{-2n\theta[1-\exp\{-1/n\}]\right\}")
which allows for a comparison with its second order Taylor approximation:
Filed under: Books, Kids, pictures, R, Statistics Tagged: binomial distribution, cross validated, Monte Carlo Statistical Methods, Poisson distribution, R, simulation, variance estimation
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