# Order statistics in R

#Example: max{X} from Unif(0,theta)

unifmax <- function(theta,n) {
   y <- (0:10000)*theta/10000
   Gyn <- (y/theta)^n
   plot(y,Gyn,type="l",main=expression(paste(
        "Cdf of ",Y[n]," from Uniform(0,",theta,")")),
        sub = paste("theta = ",theta,"; n = ",n ),ylab = "G(y)"  )
}

unifmaxdens <- function(theta,n) {
   y <- (0:10000)*theta/10000
   gyn <- n*(y/theta)^(n-1)/theta
   plot(y,gyn,type="l",main=expression(paste(
        "pdf of ",Y[n]," from Uniform(0,",theta,")")),
        sub = paste("theta = ",theta,"; n = ",n ),ylab = "g(y)"  )
}

p2xmin <- function(n) {
  # Plots density of minimum order statistic if f(x) = 2x, 0 < x < 1
  y <- (0:10000)/10000
   gy1 <- n*(1-y^2)^(n-1)*2*y
   plot(y,gy1,type="l",main=expression(paste(
        "pdf of ",Y[1]," from f(x) = 2x")),
        sub = paste("n = ",n ),ylab = "g(y)"  )
}

p2xmax <- function(n) {
  # Plots density of maximum order statistic if f(x) = 2x, 0 < x < 1
  y <- (0:10000)/10000
   gyn <- 2*n*y^(2*n-1)
   plot(y,gyn,type="l",main=expression(paste(
        "pdf of ",Y[n]," from f(x) = 2x")),
        sub = paste("n = ",n ),ylab = "g(y)"  )
}


unifmax(2,1)

unifmax(2,5)

unifmax(2,100)

unifmax(2,1000)

# Example: X_i iid Unif(0,theta)
# Let Z_n = n(theta-Y_n)

unifz <- function(theta,n) {
   z <- (0:10000)*theta*n/10000
   Hz <- ( 1 - (1 - z/(n*theta) )^n )
   plot(z,Hz,type="l",main=expression(paste(
        "Cdf of ",Z[n],"; Uniform(0,",theta,")")),
        sub = paste("theta = ",theta,"; n = ",n ),ylab = "H(z)",xlim = c(0,12)  )
}

unifz(2,1)

unifz(2,5)

unifz(2,100)

unifz(2,1000)


# Plot Exp(2) Cdf

   z <- (0:10000)*13/10000
   Hz <- ( 1 - exp(-z/2))
   plot(z,Hz,type="l",main="Cdf of Exp(2)",xlim = c(0,12)  )

# Problem 2, page 259

prob2cdf <- function(n) {
   y <- (-10000:10000)*20/21000
   Gy <- (1 + exp(-y-log(n)))^(-n)
   plot(y,Gy,type="l",sub=paste("n= ",n))
}

prob2cdf(1)
prob2cdf(10)
prob2cdf(100)

  # Limiting distribution

   y <- (-10000:10000)*20/21000
   Gy <- exp(-exp(-y))
   plot(y,Gy,type="l",sub="limiting distribution")