funsMuVarPE1D {pcds}

Returns the mean and (asymptotic) variance of arc density of Proportional Edge Proximity Catch Digraph (PE-PCD) for 1D data - middle interval case

Description

The functions muPE1D and asy.varPE1D and their auxiliary functions.

muPE1D returns the mean of the (arc) density of PE-PCD and asy.varPE1D returns the (asymptotic) variance of the arc density of PE-PCD for a given centrality parameter c \in (0,1) and an expansion parameter r \ge 1 and for 1D uniform data in a finite interval (a,b), i.e., data from U(a,b) distribution.

muPE1D uses auxiliary (internal) function mu1PE1D which yields mean (i.e., expected value) of the arc density of PE-PCD for a given c \in (0,1/2) and r \ge 1.

asy.varPE1D uses auxiliary (internal) functions fvar1 which yields asymptotic variance of the arc density of PE-PCD for c \in (1/4,1/2) and r \ge 1; and fvar2 which yields asymptotic variance of the arc density of PE-PCD for c \in (0,1/4) and r \ge 1.

See also (Ceyhan (2012)).

Usage

mu1PE1D(r, c)

muPE1D(r, c)

fvar1(r, c)

fvar2(r, c)

asy.varPE1D(r, c)

Arguments

r

A positive real number which serves as the expansion parameter in PE proximity region; must be \ge 1.

c

A positive real number in (0,1) parameterizing the center inside int=(a,b). For the interval, (a,b), the parameterized center is M_c=a+c(b-a).

Value

muPE1D returns the mean and asy.varPE1D returns the asymptotic variance of the arc density of PE-PCD for U(a,b) data

Author(s)

Elvan Ceyhan

References

Ceyhan E (2012). “The Distribution of the Relative Arc Density of a Family of Interval Catch Digraph Based on Uniform Data.” Metrika, 75(6), 761-793.

See Also

muCS1D and asy.varCS1D

Examples

#Examples for muPE1D
muPE1D(1.2,.4)
muPE1D(1.2,.6)

rseq<-seq(1.01,5,by=.1)
cseq<-seq(0.01,.99,by=.1)

lrseq<-length(rseq)
lcseq<-length(cseq)

mu.grid<-matrix(0,nrow=lrseq,ncol=lcseq)
for (i in 1:lrseq)
  for (j in 1:lcseq)
  {
    mu.grid[i,j]<-muPE1D(rseq[i],cseq[j])
  }

persp(rseq,cseq,mu.grid, xlab="r", ylab="c", zlab="mu(r,c)", theta = -30, phi = 30,
expand = 0.5, col = "lightblue", ltheta = 120, shade = 0.05, ticktype = "detailed")



#Examples for asy.varPE1D
asy.varPE1D(1.2,.8)

rseq<-seq(1.01,5,by=.1)
cseq<-seq(0.01,.99,by=.1)

lrseq<-length(rseq)
lcseq<-length(cseq)

var.grid<-matrix(0,nrow=lrseq,ncol=lcseq)
for (i in 1:lrseq)
  for (j in 1:lcseq)
  {
    var.grid[i,j]<-asy.varPE1D(rseq[i],cseq[j])
  }

persp(rseq,cseq,var.grid, xlab="r", ylab="c", zlab="var(r,c)", theta = -30, phi = 30,
expand = 0.5, col = "lightblue", ltheta = 120, shade = 0.05, ticktype = "detailed")