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

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,\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 jth-order statistic is

x\bigl(\mathrm{B}^{(-1)}(F,j,n-j+1),\Theta\bigr) \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} \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 lmom2par or vec2par.

Value

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

Author(s)

W.H. Asquith

References

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

See Also

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_len(50000)) {
  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 May 29, 2024, 10:06 a.m.