# 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

 qua.ostat R Documentation

## Compute the Quantiles of the Distribution of an Order Statistic

### 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

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

### Examples

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)


lmomco documentation built on Aug. 27, 2022, 1:06 a.m.