# Returns the mean and (asymptotic) variance of arc density of Proportional Edge Proximity Catch Digraph (PE-PCD) for 1D data - middle interval case 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 (if(!exists(".Rdpack.currefs")) .Rdpack.currefs \<-new.env();Rdpack::insert_citeOnly(keys="ceyhan:metrika-2012;textual",package="pcds",cached_env=.Rdpack.currefs) ). ```r 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 $M_c=a+c(b-a)$. ## Returns `muPE1D` returns the mean and `asy.varPE1D` returns the asymptotic variance of the arc density of PE-PCD for $U(a,b)$ data ## Examples ```r #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") ``` ## References if(!exists(".Rdpack.currefs")) .Rdpack.currefs \<-new.env();Rdpack::insert_all_ref(.Rdpack.currefs) ## See Also `muCS1D` and `asy.varCS1D` ## Author(s) Elvan Ceyhan

funsMuVarPE1D function