Maps the user-specified intrinsic time v_opt (the point at which a
boundary is tightest) to the rho tuning parameter, via the Lambert W
formula of Howard et al. (2021), Proposition 3 (paper-exact).
Usage
rho_from_vopt(v_opt = 10, alpha = 0.025)
Arguments
- v_opt
Numeric > 0. Intrinsic time at which the boundary is tightest.
Recommended default from CR23: 10.
- alpha
Numeric in (0,1). Significance level (one-sided).
For a two-sided boundary at level alpha, pass alpha/2 here.
Value
Numeric > 0. The rho tuning parameter.
Details
$$\rho = \frac{v_{opt}}{-W_{-1}(-\alpha^2 / e) - 1}$$
The lower branch \(W_{-1}\) is defined for x in [-1/e, 0) and returns
values <= -1. For alpha in (0, 1), -alpha^2/e is always in
(-1/e, 0), so the branch is well-defined.
Examples
rho_from_vopt(v_opt = 10, alpha = 0.025)
#> [1] 1.025332