# qua.ostat: Compute the Quantiles of the Distribution of an Order... In lmomco: L-Moments, Censored L-Moments, Trimmed L-Moments, L-Comoments, and Many Distributions

## Description

This function computes a specified quantile by nonexceedance probability F for the jth-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,Θ) represent the quantile function of the given distribution and Θ represents a vector of distribution parameters. The quantile function of the distribution of the jth-order statistic is

x(\mathrm{B}^{(-1)}(F,j,n-j+1),Θ) \mbox{.}

## Usage

 1 qua.ostat(f,j,n,para=NULL) 

## Arguments

 f The nonexceedance probability F for the quantile. j The jth-order statistic x_{1:n} ≤ x_{2:n} ≤ … ≤ x_{j:n} ≤ x_{n:n}. n The sample size. para A distribution parameter list from a function such as lmom2par or vec2par.

## Value

The quantile of the distribution of the jth-order statistic is returned.

W.H. Asquith

## References

Gilchrist, W.G., 2000, Statistical modelling with quantile functions: Chapman and Hall/CRC, Boca Raton, Fla.

lmom2par, vec2par
  1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 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) ## Not run: # 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,"\n")) ## End(Not run)