6
$\begingroup$

Suppose we are living in a frequentist world and want to compute confidence intervals on some quantity that is a complicated function of the parameters $q_1 = f(\Theta)$ (i.e., there's no closed-form solution that would let us re-express this in terms of one of the parameters $\theta_j = g(\Theta_{-j}, q_1)$.

  • The classical way to do this, AFAIK, would be to compute likelihood profile confidence intervals by computing $\max {\cal L}(\Theta)|q_1 = \hat {q_1}$ for a series of $\hat {q_1}$ around $f(\Theta_{\text{MLE}})$, then find the critical values of $\hat q_1$ such that the difference in the log-likelihood is half of the critical value of $\chi^2_1$ (Wilks' theorem, blah blah blah). Problem: we need an efficient, robust algorithm for equality-constrained nonlinear optimization (these do exist, e.g. using Lagrange multipliers, but are much less available/tested than those for unconstrained or box-constrained optimization).
  • We could do parametric bootstrapping (sorry, no link). Problem: slow.
  • We could assume that Wald statistics are OK and use the delta method to approximate the variances of the derived quantity (assuming that we can compute the Jacobian of $f(\Theta)$ by finite differences or automatic differentiation or ...). Problem: combines the assumption of "multivariate Normal sampling distribution"/"quadratic log-likelihood surface" with "$f$ has constant curvature".

A shortcut that I've seen used, and used myself, is to assume (approximate) multivariate Normality of the sampling distribution of the parameters; draw a MVN sample (based on the observed information matrix); compute $f(\Theta)$ for each sample; and find the quantiles of the computed values.

Lande et al (2003) call these "population prediction intervals". Bolker (2008) says that a problem with this approach is that:

It blurs the line between frequentist and Bayesian approaches. Several papers (including some of mine, e.g. Vonesh and Bolker (2005)) have used this approach, but I have yet to see a solidly grounded justification for propagating the sampling distributions of the parameters in this way.

Is anyone aware of such a justification, or is this really just cheesy pseudo-Bayesianism?

(Alternatively, are there elegant ways of computing well-justified confidence intervals on complex (not-easily-invertible) functions of parameters?)


Lande, R., S. Engen, and B.-E. Sæther. 2003. Stochastic Population Dynamics in Ecology and Conservation. Oxford University Press, Oxford, UK.

$\endgroup$

1 Answer 1

3
+50
$\begingroup$

In the econometrics literature this is referred to as the method of Krinsky and Robb (1986, 1990, & 1991).

I think the argument goes as follows:
Assume we have consistently estimated the expectation and the covariance matrix of an asymptotically normal estimator $\hat{\Theta}$ of $\Theta$. Then, by the law of large numbers, we can estimate the expectation, variance, and quantiles of the distribution of $f(\hat{\Theta})$ by drawing a random sample $\left\{\tilde{\Theta}^{(1)},\ldots,\tilde{\Theta}^{(M)}\right\}$ of size $M$ (where $M$ is large) from the asymptotic normal distribution of $\hat{\Theta}$ and using the sample mean/variance/quantiles of $\left\{f\left(\tilde{\Theta}^{(1)}\right),\ldots,f\left(\tilde{\Theta}^{(M)}\right)\right\}$ as consistent estimates for their population counterparts.

From the simulation studies I've read about this method I remember that it was, overall, not superior to the delta method.

$\endgroup$
2
  • $\begingroup$ Can you add references to some of this simulation studies? $\endgroup$ Commented Oct 2, 2022 at 13:21
  • $\begingroup$ @kjetilbhalvorsen Besides Krinsky and Robb (1990, 1991) I recall Hole (2006) off the top of my head. In any case, which one performs better depends on the function of interest $f$ and the data at hand. $\endgroup$
    – statmerkur
    Commented Oct 3, 2022 at 10:19

Your Answer

By clicking “Post Your Answer”, you agree to our terms of service and acknowledge you have read our privacy policy.

Not the answer you're looking for? Browse other questions tagged or ask your own question.