## Saturday, 30 March 2013

### Simulation Based Confidence Intervals for Functions with Complicated Derivatives

Micha Mandel has a new paper in the American Statistician. This concerns the simulation based Delta method approach to obtaining asymptotic confidence intervals which has been used quite extensively for obtaining confidence intervals of transition probabilities in multi-state models. This essentially involves assuming $\inline \hat\theta \sim N(\theta, I(\theta)^{-1})$ and constructing confidence intervals for $\inline g(\hat\theta)$ by simulating $\inline \hat\theta^{*}_1 , \ldots, \hat\theta^{*}_B$ and then considering the empirical distribution of $\inline g(\hat\theta^{*}_1) , \ldots, g(\hat\theta^{*}_B)$

A formal justification for the approach is laid out and some simulations of its performance compared to the standard Delta method in some important cases is given. It is stated that the utility of the simulation method is not in situations when the Delta method fails, but in situations where the calculating the derivatives needed for the Delta method is difficult. In particular, it still requires the functional g to be differentiable. This seems to down play the simulation method slightly. One advantage of the simulation method is that it is not necessary to linearize g around the MLE. To take a pathological example consider data that are $\inline N(\mu, 1)$. Suppose we are interested in estimating $\inline \mu^3$. When $\inline \mu = 0$ the coverage of the Delta method confidence interval will be anti-conservative because $\inline \mu^3$ has a point of inflection at $\inline \mu = 0$. The simulation based will work fine in this situation (as obviously would just constructing a confidence interval for $\inline \mu$ and just cubing the upper and lower limits!).