qua.ostat {lmomco} | R Documentation |
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(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 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 vec2par or lmom2par . |
The quantile of the distribution of the jth-order statistic is returned.
W.H. Asquith
Gilchrist, W.G., 2000, Statistical modelling with quantile functions: Chapman and Hall/CRC, Boca Raton, Fla.
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,"\n"))