This function computes a specified quantile by nonexceedance probability $F$ for the $j$th-order statistic of a sample of size $n$ for a given distribution. Let the quantile function (inverse distribution) of the Beta distribution be$$\mathrm{B}^{-1}(F,j,n-j+1) \mbox{,}$$
and let $x(F,\Theta)$ represent the quantile function of the given distribution and $\Theta$ represents a vector of distribution parameters. The quantile function of the distribution of the $j$th-order statistic is
$$x(\mathrm{B}^{-1}(F,j,n-j+1),\Theta) \mbox{.}$$
qua.ostat(f,j,n,para=NULL)
- f
{The nonexceedance probability $F$ for the quantile.}
- j
{The $j$th-order statistic $x_{1:n} \le x_{2:n} \le \ldots \le x_{j:n} \le x_{n:n}.$}
- n
{The sample size.}
- para
{A distribution parameter list from a function such as vec2par or lmom2par.}
The quantile of the distribution of the $j$th-order statistic is returned.
Gilchrist, W.G., 2000, Statistical modelling with quantile functions: Chapman and Hall/CRC, Boca Raton, Fla.
[object Object]
lmom2par, vec2par
gpa <- vec2par(c(100,500,0.5),type='gpa')
n <- 20 # the sample size
j <- 15 # the 15th order statistic
F <- 0.99 # the 99th percentile
theoOstat <- qua.ostat(F,j,n,gpa)
# Let us test this value against a brute force estimate.
Jth <- vector(mode = "numeric")
for(i in seq(1,10000)) {
Q <- sort(rlmomco(n,gpa))
Jth[i] <- Q[j]
}
bruteOstat <- quantile(Jth,F) # estimate by built-in function
theoOstat <- signif(theoOstat,digits=5)
bruteOstat <- signif(bruteOstat,digits=5)
cat(c("Theoretical=",theoOstat," Simulated=",bruteOstat,""))
distribution