Largest admissible shape parameter n — pm_suggest

Implements the step-2 calibration rule. The worst-case scenario-I risk increases with n, while the information loss decreases with n at every rho < 1 and is unchanged at rho = 1. Both objectives are therefore monotone and opposite in n: there is no interior optimum, and the best choice is the upper edge of the admissible set, n_max = sup{n : max_rho mu_I <= tau}. Because the risk is monotone, that edge is found by root-finding rather than by a grid search.

Usage

pm_suggest_n(
  params = NULL,
  sigma_nu = NULL,
  beta = NULL,
  tau = NULL,
  sigma_eps = NULL,
  margin = 0,
  n_range = c(1, 30),
  integer = FALSE,
  level = 0.95,
  rho = NULL
)

Arguments

params: Optional pm_params; supplies sigma_eps, beta and tau.
sigma_nu: Numeric vector of candidate values. Default seq(0.05, 0.5, 0.05).
beta, tau: Scenario-I threshold and ceiling. Default to the dominance policy of params.
sigma_eps: Fixed differencing noise. Defaults to params$mechanism$sigma_eps.
margin: Safety margin in [0,1): the effective ceiling is tau * (1 - margin). n_max is a boundary solution with zero slack, so a small margin guards against later revisions of the policy. Default 0.
n_range: Search interval for n. Default c(1, 30).
integer: If TRUE, round n_max down to an integer – the only conservative rounding, the risk being increasing in n.
level: Confidence level for the reported CI loss (default 0.95).
rho: Grid used to locate the worst case. Default seq(0.001, 1, 0.001).

Value

A data.frame with one row per sigma_nu: n_max, risk_at_n_max, and the information loss at rho = 1 and at the dominance threshold 1 - beta. n_max is NA when no n meets the ceiling (raise sigma_nu), and the upper end of n_range when the whole range qualifies.

Details

The rule holds whatever the dominance profile of the table: since utility improves at every rho, no weighting by the actual distribution of rho could favour a smaller n.

Note this determines n given sigma_nu. Every point of the returned frontier meets the ceiling exactly, so choosing among them is a pure utility arbitrage – larger sigma_nu costs more at rho = 1 but spares cells of intermediate dominance. The reported losses are there to settle it.

Examples

para <- pm_commit_diff(pm_params())
#> Differencing step committed: beta = 0.05, tau = 0.95  ->  sigma_eps = 0.0255
#>   loss floor at rho -> 0:  E|Z| = 2.04%,  upper 95% CI = 5.00%
pm_suggest_n(para, sigma_nu = c(0.3, 0.4, 0.5))
#>   sigma_nu  sigma_eps beta tau target    n_max risk_at_n_max  EZ_rho1
#> 1      0.3 0.02551067  0.2 0.9    0.9 15.76402           0.9 24.02292
#> 2      0.4 0.02551067  0.2 0.9    0.9 18.42437           0.9 31.98022
#> 3      0.5 0.02551067  0.2 0.9    0.9 20.37695           0.9 39.94612
#>   EZ_rho_dom  CI_rho1 CI_rho_dom
#> 1   2.155793 59.01113   5.295600
#> 2   2.101572 78.55784   5.162408
#> 3   2.078914 98.12567   5.106749