SarabiaE {biogeom}

Sarabia Equation

Description

SarabiaE is used to calculate y values at given x values using the Sarabia equation. The equation describes the y coordinates of the Lorenz curve.

Usage

SarabiaE(P, x)

Arguments

P

the parameters of the Sarabia equation.

x

the given x values ranging between 0 and 1.

Details

y = \left(1-\lambda+\eta\right)x+\lambda x^{a_1 + 1}-\eta \left[1-\left(1-x\right)^{a_2 + 1}\right].

Here, x and y represent the independent and dependent variables, respectively; and \lambda, \eta, a_1 and a_2 are constants to be estimated, where a_1 \ge 0, a_2 + 1 \ge 0, \eta\,a_2 + \lambda \le 1, \lambda \ge 0, and \eta\,a_2 \ge 0. There are four elements in P, representing the values of \lambda, \eta, a_1 and a_2, respectively.

Value

The y values predicted by the Sarabia equation.

Note

The numerical range of x should range between 0 and 1.

Author(s)

Peijian Shi [e-mail], Johan Gielis [e-mail], Brady K. Quinn [e-mail].

References

Sarabia, J.-M. (1997) A hierarchy of Lorenz curves based on the generalized Tukey's lambda distribution. Econometric Reviews 16, 305-320. doi:10.1080/07474939708800389

Sitthiyot, T., Holasut, K. (2023) A universal model for the Lorenz curve with novel applications for datasets containing zeros and/or exhibiting extreme inequality. Scientific Reports 13, 4729. doi:10.1038/s41598-023-31827-x

See Also

fitLorenz, MPerformanceE, SCSE, SHE

Examples

X1  <- seq(0, 1, len=2000)
Pa1 <- c(0.295, 101.485, 0.705, 0.003762)
Y1  <- SarabiaE(P=Pa1, x=X1)

dev.new()
plot( X1, Y1, cex.lab=1.5, cex.axis=1.5, type="l", asp=1, xaxs="i", 
      yaxs="i", xlim=c(0, 1), ylim=c(0, 1), 
      xlab="Cumulative proportion of the number of infructescences", 
      ylab="Cumulative proportion of the infructescence length" )

graphics.off()