lmomco (version 1.7.3)

qua.ostat: Compute the Quantiles of the Distribution of an Order Statistic

Description

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

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

Usage

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

Arguments

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.

Value

• The quantile of the distribution of the $j$th-order statistic is returned.

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)

# As of version 1.6.2, it is felt that in spirit of CRAN CPU
# reduction that the intensive operations of this example should
# be kept a bay.

# 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"))