Sharp bounds on a true victimisation rate and its dark figure

A recorded rate r (police records per capita) never exceeds the true rate v, and a survey rate v_obs is a noisy proxy $v(1-\beta) + (1-v)\alpha$ with under-reporting $\beta$ (stigma, non-recall) and over-reporting $\alpha$ (misclassified incidents). With only the boxes $\alpha \in [0, a]$, $\beta \in [0, b]$, $a + b < 1$ credible, the true rate is identified exactly up to $$\max\!\left(r, \frac{v_{obs}-a}{1-a}\right) \le v \le \frac{v_{obs}}{1-b},$$ and the dark figure $v - r$ up to that interval shifted by r (Research.P1.true_rate_bounds, Research.P1.dark_figure_bounds). Both ends of the survey part are attained, so nothing narrower follows from the boxes alone.

Usage

morie_dark_figure_bounds(v_obs, r, alpha_max, beta_max)

Arguments

v_obs: Survey victimisation rate(s) in [0, 1].
r: Recorded rate(s) in [0, 1], same length as v_obs or length 1.
alpha_max: Largest credible over-reporting rate.
beta_max: Largest credible under-reporting rate; alpha_max + beta_max must be below 1.

Value

A data frame with v_obs, r, v_lower, v_upper, dark_lower, dark_upper and ratio_upper (the largest credible ratio of true to recorded rate, NA when r = 0).

Examples

# survey says 6 percent were victimised, police recorded 2 percent
morie_dark_figure_bounds(v_obs = 0.06, r = 0.02, alpha_max = 0.01, beta_max = 0.30)
#>   v_obs    r    v_lower    v_upper dark_lower dark_upper ratio_upper
#> 1  0.06 0.02 0.05050505 0.08571429 0.03050505 0.06571429    4.285714