biogeom (version 1.4.3)

SarabiaE: 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)

Value

The \(y\) values predicted by the Sarabia equation.

Arguments

P

the parameters of the Sarabia equation.

x

the given \(x\) values ranging between 0 and 1.

Author

Peijian Shi , Johan Gielis , Brady K. Quinn .

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.

References

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