pvrank (version 1.1)

VGGFR: Vianelli density

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.

Value

• The value returned is a list contaning:
• st.dev
  standard deviation
• kurt
  kurtosis
• oam
  ordinate at the mode

Details

The VGGFR density is given by $$f(r;\lambda_1,\lambda_2)=\lambda_1(1-|r|^{\lambda_2})^{\lambda_1}/[2B(1/\lambda_1,\lambda_2+1)]$$ where $\lambda_1,\lambda_2>0$ and $B()$ is the beta function.

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])
			}}