GGFR: Vianelli density

In pvrank: Rank Correlations

Description

Plots Vianelly (generalized Gaussian) density with finite range in [-1,1].

Usage

VGGFR(L1, L2, add=FALSE, lwd=2, lty=5, col="blue", ylim=c(0,1), Main="", np=201)

Arguments

L1	positive shape parameter.
L2	positive shape parameter. Impacts more on the tails.
add	when add=TRUE the plot is superimposed to an existing graph.
lwd	weight of the line.
lty	the type of the line.
col	color of the curve.
ylim	numeric vectors of length 2, giving the y coordinates ranges.
Main	a string describing the graph.
np	number of points to be plotted.

Details

The VGGFR density is given by

f(r;λ_1,λ_2)=λ_1(1-|r|^{λ_2})^{λ_1}/[2B(1/λ_1,λ_2+1)]

where λ_1,λ_2>0 and B() is the beta function.

Value

The value returned is a list contaning:

st.dev	standard deviation
kurt	kurtosis
oam	ordinate at the mode

Note

If you want to use the output from VGGFR save the result and then select from the list the value(s) needed.

Author(s)

Agostino Tarsitano and Ilaria Lucrezia Amerise

References

Tarsitano, A. and Amerise, I. L. (2016). "Modelling of the null distribution of rank correlations". Submitted.

Vianelli, S. (1968). "Sulle curve normali di ordine $r$ per intervalli finiti delle variabili statistiche". Annali della Facolt\‘a di Economia e Commercio dell’Universit\'a di Palermo, 2.

Vianelli, S. (1983). "The family of normal and lognormal distributions of order r". Metron, 41, 3-10.

Examples

	
# Density curve of a VGGFR model
VGGFR(2, 12, add=FALSE, lwd=2, lty=5, col="darkgreen", ylim=c(0,2), Main="", np=201)
#####
#
a<-ranktes(0.5, 28, "r4", "vg",FALSE, "two", FALSE)
b<-VGGFR(a$Lambda, add = FALSE, lwd = 2, lty = 5, col = "blue", ylim=c(0,2.5),np = 201)
#####
#
# Lambert's semicircular distribution of errors (1760,1765).
# Given a probability distribution, the value with the higher probability density is 
# deemed to be more probable than the value with the lower probability density.
#
VGGFR(2,0.5,col="red",ylim=c(0,0.75),Main="Lambert's distribution of errors")
#
#####
# Pearson type II used as an approximation to the null distribution of the Fisher-Yates
# rank correlation. Fieller, E. C. and Pearson, E. S. (1961). Tests for rank correlation 
# coefficients: II. Biometrika, 48, 29-40.
n<-10
VGGFR(2, (n-4)/2, add=FALSE, lwd=2, lty=5, col="magenta2", ylim=c(0,1.1), Main="", np=201)
abline(h=0);abline(v=0,lty=2,lwd=2,col="pink2")
#####
# 
# Save and use the results 
res<-VGGFR(1.5,5.5,add = FALSE, lwd = 2, lty = 1, col = "blue", ylim=c(0,2.5),np = 201)
res$kurt-res$oam/res$st.dev
#####
# 
# A family of symmetrical beta densities
VGGFR(2,1,col="black",ylim=c(0,1.4),Main="Symmetrical beta densities")
La<-seq(1,6,0.5)
for (L1 in La){VGGFR(2,L1,add=TRUE, lwd = 1, lty = 1, col=gray(L1/6))}
#####
# 
# A family of GGFR curves
VGGFR(1,2, lwd = 1, lty = 1,col="black",ylim=c(0,5))
La<-seq(1,6,0.5);Lg<-seq(0,1,1/12)
for (L1 in La){
		c2<-gray(Lg, alpha= 2/6)
		for (L2 in seq(1,12,1)){
			VGGFR(L1,L2,add=TRUE, lwd = 1, lty = 1, col=c2[L2])
			}}	

pvrank documentation built on May 17, 2018, 9:03 a.m.