View source: R/deviationindex.R

deviationindex | R Documentation |

This function returns the deviation index and the signed deviation index for a rule with respect to another rule.

deviationindex(E, d, R, S)

`E` |
The endowment. |

`d` |
The vector of claims. |

`R` |
A rule : AA, APRO, CE, CEA, CEL, DT, MO, PIN, PRO, RA, Talmud. |

`S` |
A rule: AA, APRO, CE, CEA, CEL, DT, MO, PIN, PRO, RA, Talmud. |

Let *E> 0* be the endowment to be divided and *d* the vector of claims
with *d≥ 0* and such that *D=∑ di ≥ E*, the sum of claims *D* exceeds the endowment.

Rearrange the claims from small to large, *
0 ≤ d1 ≤...≤ dn*.
The signed deviation index of the rule *S* with respect to the rule *R* for the problem *(E,d)*, denoted by *I(R(E,d),S(E,d))*, is
the ratio of the area that lies between the identity line and the cumulative curve over the total area under the identity line.

Let *R0=0* and *S=0*. For each *k=1,…,n* define
*Xk=(R0+…+Rk)/E* and
*Yk=(S0+…+Sk)/E*. Then

*I(R(E,d),S(E,d))=1-∑ (Xk-X(k-1))(Yk+Y(k-1)) where the sum goes from k=0 to n.*

In general *-1 ≤ I(R(E,d),S(E,d)) ≤ 1*.

The deviation index of the rule *S* with respect to the rule *R* for the problem *(E,d)*, denoted by *I+(R(E,d),S(E,d))*, is
the ratio of the area between the line of the cumulative sum of the distribution proposed by the rule *R* and the cumulative curve over the area under the line *x=y*.

In general *0 ≤ I+(R(E,d),S(E,d)) ≤ 1*.

The proportionality deviation index is the deviation index when *R = PRO*. The proportionality deviation index of the proportional rule is zero for all claims problems.
The signed proportionality deviation index is the signed deviation index with *R = PRO*.

*proportionalityindex* function of version 0.1.0 returned the the signed proportionality index.

The deviation index and the signed deviation index of a rule for a claims problem.

Ceriani, L. and Verme, P. (2012). The origins of the Gini index: extracts from Variabilitá e Mutabilitá (1912) by Corrado Gini. The Journal of Economic Inequality, 10(3), 421-443.

Mirás Calvo, M.Á., Núñez Lugilde, I., Quinteiro Sandomingo, C., and Sánchez Rodríguez, E. (2022). Deviation from proportionality and Lorenz-domination for claims problems. Rev Econ Design. doi: 10.1007/s10058-022-00300-y

indexgpath, cumawardscurve, lorenzcurve, giniindex, lorenzdominance, allrules.

E=10 d=c(2,4,7,8) R=CEA S=AA deviationindex(E,d,R,S) #The deviation index of rule R with respect of the rule R is 0. deviationindex(E,d,PRO,PRO)

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