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The dark figure of crime: bounds instead of a point

Source: R/crcap.R
dkfig.Rd

What recorded crime and a victimisation survey together say about the crime that is never recorded, as intervals that hold under stated limits on the errors (Manski 2003).

Usage

dkfig(
  v = NULL,
  r = NULL,
  amax = 0,
  bmax = 0,
  thr = NULL,
  va = NULL,
  ra = NULL,
  rb = NULL,
  lam = 1,
  k = NULL
)

Arguments

v

NULL, or survey (true) victimisation rates: numbers from 0 to 1.

r

The recorded rates: one number or one per v, from 0 to 1.

amax, bmax

The largest false-positive and false-negative rates of the survey: numbers from 0, with a sum below 1.

thr

NULL, or a threshold above every v.

va, ra, rb

NULL, or the survey's offences of the covered type, the recorded ones of that type (both above 0, ra at most va) and the recorded offences of the uncovered type (from 0).

lam

The least ratio of the uncovered type's recording rate to the covered type's: above 0, at most 1.

k

NULL, or the number of offences in each incident: whole numbers from 1.

Value

An object of class dkfig, a list: bnd (with v: v, r, lo, hi, dlo, dhi the dark figure's bounds, ratio the upper bound over the recorded rate, and with thr brk), cmp (with va: rate A's recording rate, rateup, the total's bounds totlo, tothi, the dark figure's dklo, dkhi and width) and hier (with k: incid, offen, ratio, extra, lower, upper); the parts not asked for are NULL.

Details

  • With a survey rate v and a recorded rate r, and survey false positives at most amax and false negatives at most bmax, the true rate lies in [max(r, (v - amax) / (1 - amax)), min(1, v / (1 - bmax))] and the dark figure (true minus recorded) in that interval shifted by r. Given thr, brk is the largest bmax under which the conclusion "the true rate is below thr" survives: 1 - v / thr.

  • When the survey covers only some offences (type A, va of them, ra recorded) and the recorded count of the rest is rb, recorded at no less than lam times A's rate, the total and the dark figure have the bounds in cmp.

  • With k, the number of offences in each recorded incident, the hierarchy rule's arithmetic: one incident in the counts stands for between 1 and max(k) offences.

See proof("dkfig").

References

Manski, C. F. (2003). Partial Identification of Probability Distributions. Springer. doi:10.1007/b97478

Examples

dkfig(v = c(0.12, 0.08), r = 0.03, amax = 0.01, bmax = 0.2, thr = 0.15)
#> dkfig: bounds on the dark figure
#>      v    r         lo   hi        dlo  dhi    ratio       brk
#> 1 0.12 0.03 0.11111111 0.15 0.08111111 0.12 5.000000 0.2000000
#> 2 0.08 0.03 0.07070707 0.10 0.04070707 0.07 3.333333 0.4666667
dkfig(va = 900, ra = 300, rb = 450, lam = 0.5)
#> dkfig: bounds on the dark figure
#>         rate       rateup        totlo        tothi         dklo         dkhi 
#>    0.3333333    0.5555556 1350.0000000 3600.0000000  600.0000000 2850.0000000 
#>        width 
#> 2250.0000000 
dkfig(k = c(1, 1, 2, 1, 3))
#> dkfig: bounds on the dark figure
#> incid offen ratio extra lower upper 
#>   5.0   8.0   1.6   0.6   5.0  15.0