# justification for 'population prediction intervals'?

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.

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.
• @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. Oct 3, 2022 at 10:19