Computes the theoretical quantile ratio index (QRI) for measuring inequality for a given parametric distribution.
Arguments
- qfunction
A quantile function (e.g.,
qnorm,qlnorm,qgamma).- lower
Lower bound of integration. Default is 0.
- upper
Upper bound of integration. Default is 1.
- subdivisions
Maximum number of subintervals for integration. Default is 1000L.
- ...
Additional parameters to pass to
qfunction(e.g., distribution parameters).
Value
A numeric value representing the theoretical QRI for the specified parametric distribution. Values range from 0 (perfect equality) to 1 (maximum inequality).
Details
The QRI was proposed by (Prendergast and Staudte 2018) for measuring economic inequality. Consider a random variable \(Y\) with positive support, which admits a continuous CDF \(F\) and quantile function \(Q(p) = F^{-1}(p)\), for any \(p \in (0, 1)\). It is calculated as: $$QRI = 1 - \int_0^1 R(p) dp$$ where \(R(p) = Q(p/2) / Q(1 - p/2)\) is the ratio of symmetric quantiles, with \(R(0) = 0\) and \(R(1) = 1\).
This function computes the (superpopulation) QRI for
theoretical parametric distributions, as opposed to qri which estimates
the QRI from sample data.
References
Prendergast LA, Staudte RG (2018). “A simple and effective inequality measure.” The American Statistician, 72, 328–343.
See also
qri for the sample-based QRI estimator, plot_inequality_curve for its representation
Other inequality indicators based on quantiles:
inequantiles(),
plot_inequality_curve(),
qri(),
ratio_quantiles(),
share_ratio()
Examples
# Log-normal distribution
superpop_qri(qlnorm, meanlog = 9, sdlog = 0.3)
#> [1] 0.3433188
superpop_qri(qlnorm, meanlog = 9, sdlog = 1.4)
#> [1] 0.7424406
# Weibull distribution
superpop_qri(qweibull, shape = 1.7, scale = 30000)
#> [1] 0.5669101
superpop_qri(qweibull, shape = 3, scale = 30000)
#> [1] 0.4140496