Computes Extinction Probability for a Density-Independent Model

Compute $G(w,z)$ and its complement $Q(w,z)=1-G(w,z)$ for a density-independent (drifted Wiener) model, on both linear and log scales.

Usage

ext_prob_di(w, z)

log_ext_prob_di(w, z)

log_ext_comp_di(w, z)

ext_prob_format_di(w, z, digits = 5L)

Arguments

w: Numeric; transformed parameter $w=(\mu t+x_d)/(\sigma\sqrt{t})$.
z: Numeric; transformed parameter $z=(-\mu t+x_d)/(\sigma\sqrt{t})$.
digits: Integer; significant digits for formatting (only for ext_prob_format_di).

Value

For ext_prob_di: numeric scalar $G(w,z)$.
For log_ext_prob_di: numeric scalar $\log G(w,z)$.
For log_ext_comp_di: numeric scalar $\log Q(w,z)$.
For ext_prob_format_di: character string formatted for display.

Details

For any $t>0$ with $w+z>0$, $$ \Pr[T \leq t] = G(w,z)=\Phi(-w)+ \exp\!\left(\tfrac{z^2-w^2}{2}\right)\Phi(-z), \qquad Q(w,z)=1-G(w,z). $$ Here $\Phi$ and $\phi$ denote the standard normal CDF and PDF.

Stability strategy. (i) For large $z$, rewrite the product $\exp((z^2-w^2)/2)\,\Phi(-z)$ via the Mills ratio and replace it by an 8-term asymptotic series when $z \ge 19$. (ii) On the log scale, use log-sum-exp and a stable log-difference (log1mexp) built from log1p/expm1 to retain tail info.

Domain. Scalar inputs are assumed and require $w+z>0$.

Functions.

ext_prob_di(w,z): returns $G(w,z)$ (linear scale).
log_ext_prob_di(w,z): returns $\log G(w,z)$.
log_ext_comp_di(w,z): returns $\log Q(w,z)$.
ext_prob_format_di(w,z,digits): formats a point estimate using repr_mode() and format_by_mode().

Author

Hiroshi Hakoyama, hiroshi.hakoyama@gmail.com