Incapacitation under desistance and replacement

Crimes prevented by a sentence of S periods served from career age t0 when the offending rate is a non-increasing path \(\lambda(t)\) (desistance) and a share replacement of the prevented crimes is committed by others. The sum \(\sum_{s<S}\lambda(t_0+s)\) is at most \(S\lambda(t_0)\) (Research.P17Replacement.prevented_le_const) and at least \(S\lambda(t_0+S)\) (prevented_ge_const); the same sentence served later prevents fewer crimes (later_sentence_prevents_less); replacement scales the result by \(1 - r\) (replaced_le, replaced_antitone, replaced_full), so the constant-rate estimate from the entry-age rate is an upper bound under both (prevented_net_le).

Usage

morie_incapacitation_career(lambda_path, t0, S, replacement = 0)

Arguments

lambda_path

Non-increasing offending rates by career age, the first element being age 0; length at least t0 + S + 1.

t0

Career age (0-based) at which the sentence starts.

S

Sentence length in periods, a positive integer.

replacement

Replacement share \(r \in [0, 1]\).

Value

A list with prevented, prevented_net (\((1-r)\) times it), upper (\(S\lambda(t_0)\)), lower (\(S\lambda(t_0+S)\)), later (the sum for a start one period later, NA when the path is too short), replacement, factor (\(1-r\)) and theorems.

Examples

lam <- c(12, 10, 8, 6, 5, 4, 3, 2, 2, 1)
r <- morie_incapacitation_career(lam, t0 = 2, S = 3, replacement = 0.25)
c(prevented = r$prevented, net = r$prevented_net, upper = r$upper, lower = r$lower, later = r$later)
#> prevented       net     upper     lower     later 
#>     19.00     14.25     24.00     12.00     15.00