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(This article was first published on <strong><a href="https://xianblog.wordpress.com/2017/02/07/an-accurate-variance-approximation/"> R – Xi'an's Og</a></strong>, and kindly contributed to <a rel="nofollow" href="https://www.r-bloggers.com/">R-bloggers)</a> </div>
<p><a target="_blank" rel="nofollow" href="http://stats.stackexchange.com/a/259014/7224"><img alt="" src="https://i2.wp.com/i.stack.imgur.com/6igw3.jpg?resize=456%2C456&ssl=1" data-recalc-dims="1"></a><strong>I</strong>n answering <a target="_blank" rel="nofollow" href="http://stats.stackexchange.com/a/259014/7224">a simple question</a> 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.</p>
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