Gamma-exponential mixture boundary

Gamma-exponential mixture boundary

Usage

ge_boundary(v, alpha, rho, c, s_lo = -10, s_hi = 500)

Arguments

v

Numeric vector >= 0. Intrinsic time values.

alpha

Numeric in (0,1). ONE-SIDED significance level.

rho

Numeric > 0. From rho_from_vopt().

c

Numeric > 0. Sub-exponential scale.

s_lo

Numeric. Lower search bound for uniroot. Default: -10.

s_hi

Numeric. Upper search bound for uniroot. Default: 500.

Value

Numeric vector of boundary values (cumulative-sum scale).

Details

Computes \(u_{GE}(v; \alpha, \rho, c) = \sup\{s : m(s,v) < 1/\alpha\}\) by solving m(s, v) = 1/alpha numerically for s via uniroot(), separately for each v_i.

Root-finding fallback: the search starts in [s_lo, s_hi]; if m(s_hi, v_i) has not yet crossed the target, s_hi is doubled once and retried. If it still fails, a warning is issued and s_hi is returned as a conservative fallback value. Increase s_hi directly if this warning appears often (e.g. at large v or small alpha); increase abs(s_lo) if no root is found at small v.

Computed elementwise; can be slow for long vectors — consider caching boundary values when the same (alpha, rho, c) are reused.

Examples

rho <- rho_from_vopt(v_opt = 10, alpha = 0.025)
ge_boundary(v = 1:3, alpha = 0.025, rho = rho, c = 2)
#> [1] 10.12051 11.43416 12.49748