Order statistics plots

There are two plots can be obtained from extreme order statistics; see, Elamir (2016) International Journal of Statistics and Applications, pp.314-324. The first plot is minimum order statistics and the second plot is maximum order statistics. These plots focus on the information at the extreme tails of the distributions while few values can describe the middle of the distribution.

Min. and Max. order statistics

Min. order statistics plot depends on \[ {E(X_{1:n}), n=1,2,...} \] while Max. order statistics plot depends on \[ {E(X_{n:n}), n=1,2,...} \]

These parameters can be estimated from the data using equations in Elamir (2016).

These graphs can describe data well in terms of

  • skeweness (symmeric or asymmetric)
  • kurtosis
  • esimate location and scale of the distribution
  • comparison with known distribution
  • compare two samples to see if they are from same distribution.

This post only on how to graph min. and max. plot

Examples

Example 1: Normal distribution

# function for estimated Min order statistics
minord <- function(x,t){ 
    n = length(x)  
    i = 1:n
    x = sort(x)
   c1 = 1/choose(n,t)
   t1 = choose(n-i,t-1)*x
        c1*sum(t1)}

# function for estimated Max order statistic
maxord <- function(x,t){
   n  = length(x) 
   i  = 1:n 
   x  = sort(x)
   c1 = 1/choose(n,t)
   t1 = choose(i-1,t-1)*x
        c1*sum(t1)}
 
# estimate normal data
 n = 100; k = 1:n 
 y = rnorm(n,50,7) ### simulated normal data
 wdy = 0; woy = 0

 for (i in 1:n){
  wdy[i] = minord(y,i) # estimate min order stat.
  woy[i] = maxord(y,i) # estimate Max order stat.
}

plot(wdy, ylim = c(min(wdy),max(woy)),ylab = "Min-Max", 
     main = "Min-Max plot for simulated normal data")
points(woy)
abline(h=mean(y))

Example 2: Exponential distribution

 n = 100; k = 1:n  # no. of observations
 y = rexp(n, 0.7)  # simulated exp. data
 wdy = 0; woy = 0

 for (i in 1:n){
  wdy[i] = minord(y,i) # estimate min order stat.
  woy[i] = maxord(y,i) # estimate Max order stat.
}

plot(wdy, ylim = c(min(wdy),max(woy)),ylab = "Min-Max", 
     main = "Min-Max plot for simulated exponential data")
points(woy)
abline(h=mean(y))

Note how the shapes of the two graphs are different.