The dark figure from a survey that covers only some offences

A victimisation survey covers an offence set A; the police record covers A and the rest, B. With \(V = V_A + V_B\) true offences and \(R = R_A + R_B\) recorded ones, the survey identifies \(V_A\) and so the recording rate on A, \(r_A = R_A / V_A\) (Research.P1.Comparable.rate_on_A); \(V_B\) is never observed. The overall rate is at most \(R / (V_A + R_B)\) (total_rate_le) and has no positive lower bound without an assumption on B (total_rate_unbounded_below). Under \(r_B \ge \lambda r_A\) (the non-comparable offences are recorded at least a fraction \(\lambda\) as well as the comparable ones) the total count lies in \([V_A + R_B, V_A + R_B / (\lambda r_A)]\) (total_interval) and the dark figure in \([V_A - R_A, V_A - R_A + R_B (1 / (\lambda r_A) - 1)]\) (dark_interval), an interval of width \(R_B (1 - \lambda r_A) / (\lambda r_A)\) (dark_figure_width).

Usage

morie_dark_figure_comparable(VA, RA, RB, lambda = 1)

Arguments

VA

Survey estimate of the true offences in the comparable set.

RA

Recorded offences in the comparable set.

RB

Recorded offences outside it.

lambda

The assumed floor on the non-comparable recording rate relative to the comparable one, in (0, 1]; 1 says B is recorded at least as well as A.

Value

A list with rate_A, rate_upper (the total rate's upper bound), total and dark_figure (each a two-element interval), width and theorems.

Examples

morie_dark_figure_comparable(VA = 1000, RA = 400, RB = 300, lambda = 0.5)
#> $rate_A
#> [1] 0.4
#> 
#> $rate_upper
#> [1] 0.5384615
#> 
#> $total
#> lower upper 
#>  1300  2500 
#> 
#> $dark_figure
#> lower upper 
#>   600  1800 
#> 
#> $width
#> [1] 1200
#> 
#> $theorems
#> [1] "Research.P1.Comparable.rate_on_A"        
#> [2] "Research.P1.Comparable.total_rate_le"    
#> [3] "Research.P1.Comparable.total_interval"   
#> [4] "Research.P1.Comparable.dark_interval"    
#> [5] "Research.P1.Comparable.dark_figure_width"
#>