Let's say I estimating some parameter vector $\theta$ with some estimator $\hat \theta$. I have that asymptotically,

$\hat \theta \text{ }\dot \sim \text{ } N(\theta, \Sigma)$

Now let's say I want to compute a confidence interval for $f(\theta)$, where $f$ is some differentiable function. Given a consistent estimator of $\Sigma$ defined as $\hat \Sigma$, I can create an asymptotic confidence interval using the delta method, i.e.

$\hat {f(\theta)} \text{ } \dot \sim \text{ } N\left(f(\theta),\text{ } f'(\theta)^T \Sigma \text{ } f'(\theta) \right)$

For a variety of reasons, this may be suboptimal in my case. Mostly the issue that $f$ is very non-linear about $\theta$, relative to area covered by $N(\theta, \Sigma)$.

In my head, a much better solution to this is to actually sample from

$N(\hat \theta, \hat \Sigma)$

and then plug directly into $f$. We can then form confidence intervals in exact same manner we normally form credible intervals from MCMC samples.

Is this a well known practice? Is there reference or a name for this procedure? I realize it's not possible that I'm the first person to think of this, but I don't know where to start looking about references for this.

  • $\begingroup$ Yes, this is resampling which is broad topic that includes (1) the jackknife (2) subsampling (3) the bootstrap and (4) permutation methods. These methods are analogous to the delta method. $\endgroup$ Commented Mar 31, 2017 at 1:09
  • 1
    $\begingroup$ I would not call this a bootstrap, since you're not resampling your data at all. It is just regular integration by sampling (Monte-Carlo integration). Note that you can also compute the distribution accurately using change-of-variable techniques. I'm not sure if this is valid in an asymptotic setting though. $\endgroup$ Commented Mar 31, 2017 at 1:57
  • $\begingroup$ If I used resampling or Monte-Carlo integration to describe the procedure above, I think the typically reader would be very confused about what I was doing. Were this a Bayesian method, I could simply say "drew samples from the posterior of $f(\theta)$". But since it's not, I'm not sure what the best way to describe it would be. $\endgroup$
    – Cliff AB
    Commented Mar 31, 2017 at 3:47

1 Answer 1


The procedure you describe can be used to compute the standard errors of $f(\hat{\theta})$. Let us assume that knowledge of $\theta$ is available and that we are able to sample from: $$ \tilde\theta_1,...,\tilde\theta_M \sim N(\theta, \Sigma). $$ Then an estimate for the bias of $f(\hat\theta)$ is: $$ \frac{1}{M}\sum_{m=1}^{M} f(\tilde\theta_m) - f(\theta), $$ and an estimate of the variance is given by: $$ \hat\Sigma_f = \frac{1}{M-1}\sum_{m=1}^{M} \left(f(\tilde\theta_m) - \bar f \right) \left(f(\tilde\theta_m) - \bar f \right)^{T}. $$

If we choose to believe in the asymptotic normality of $f(\hat\theta)$ then symmetric confidence intervals can be constructed based on $\hat\Sigma_f$ in the usual way.

If a normality assumption is undesirable, it is possible to make a weaker assumption that: $$ f(\hat\theta) - f(\theta) \approx^D f(\tilde\theta) - f(\hat\theta) $$ Assume w.l.o.g. that $f$ is one dimensional and denote by $D_q$ the $q$th quantile of the distribution of $f(\tilde\theta) - f(\hat\theta)$. Under the assumption above, a valid confidence intervals at a level $\alpha$ for $f(\hat\theta)$ is given by: $$ \left( f(\hat\theta) - D_{1 - \alpha/2}, f(\hat\theta) - D_{\alpha / 2} \right) $$

About the first step where we assumed knowledge of $\theta$, in practice we will sample: $$ \hat\theta_1,...,\hat\theta_M \sim N(\hat\theta, \Sigma). $$ However, it is important to keep in mind that if $\hat\theta$ is far from $\theta$ or not normal, then the resulting confidence intervals may be inaccurate.

  • $\begingroup$ Thank you for you answer. I am aware that these procedure should provide valid confidence intervals under the assumption of normality of $\hat \theta$, (and the second procedure should work better than the first if $f$ is very locally non-linear). What's not clear to me is what to call this procedure, i.e. if I would like to reference this in a paper, I think it would waste space to write it all out, but don't know the technical term nor the relevant literature about this. $\endgroup$
    – Cliff AB
    Commented Mar 31, 2017 at 3:57
  • $\begingroup$ The closest thing I can think of is parametric bootstrap. $\endgroup$ Commented Mar 31, 2017 at 6:51

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