Confidence Intervals for Extinction Probability (w–z, DI model)
wz_ci_di.RdComputes confidence intervals (CIs) for extinction probability in a density-independent (drifted Wiener) model using the w–z method, and provides a formatter for display-ready CI strings.
Usage
confidence_interval_wz_di(mu, xd, s, th, tq, qq, alpha, prob_fun = ext_prob_di)
ci_wz_format_di(mu, xd, s, th, tq, qq, alpha, digits = 5L)Arguments
- mu
Numeric. Estimated growth rate \(\hat{\mu}\).
- xd
Numeric. Log-distance to threshold \(x_d=\log(n_q/n_e)\).
- s
Numeric. Estimated environmental variance \(\hat{\sigma}^2\).
- th
Numeric. Time horizon \(t^{\ast}\) for evaluation.
- tq
Numeric. Observation span \(t_q\) (first to last time).
Integer. Number of intervals \(q\) (sample size minus 1).
- alpha
Numeric. Significance level \(\alpha\)\,=\,1\,-\,CL.
- prob_fun
Function. One of
ext_prob_di,log_ext_prob_di,log_ext_comp_di.- digits
Integer. Significant digits for formatting (used only by
ci_wz_format_di).
Value
For confidence_interval_wz_di: numeric vector c(lower, upper)
on the chosen scale (natural log if log_*).
For ci_wz_format_di: named character vector c(lower, upper)
with values preformatted for display.
Details
The w–z method derives CIs by inverting noncentral-\(t\) distributions for the transformed statistics \(w\) and \(z\), then combining them to form bounds on \(G(w,z)\).
Exact confidence intervals for \(w\) and \(z\) are obtained by numerically solving the noncentral-\(t\) quantile equations corresponding to the observed statistics.
The CI for \(G(w,z)\) is then approximated by $$ \bigl( G(\overline{w},\,\underline{z}),\; G(\underline{w},\,\overline{z}) \bigr), $$ where \(\overline{w}\), \(\underline{w}\), \(\overline{z}\), and \(\underline{z}\) are the upper and lower confidence limits for \(w\) and \(z\), respectively. Across the full parameter space, this approach achieves near-nominal coverage. When $z$ is large and positive, $G$ depends only on $w$, so exact CIs are available.
The argument prob_fun selects which probability is evaluated:
ext_prob_di,log_ext_prob_di: CIs for \(G(w,z)\) and \(\log G(w,z)\); returned as (lower, upper).log_ext_comp_di: CIs for \(\log Q(w,z)= \log (1-G(w,z))\); returned as (lower, upper).
Author
Hiroshi Hakoyama, hiroshi.hakoyama@gmail.com