quapdq4 {lmomco} | R Documentation |
Quantile Function of the Polynomial Density-Quantile4 Distribution
Description
This function computes the quantiles of the Polynomial Density-Quantile4
distribution (PDQ4) given parameters (,
, and
) computed by
parpdq4
. The quantile function
for is
for is
where is the quantile for nonexceedance probability
,
is a location parameter,
is a scale parameter,
and
is a shape parameter. The range of the distribution is
.
Usage
quapdq4(f, para, paracheck=TRUE)
Arguments
f |
Nonexceedance probability ( |
para |
|
paracheck |
A logical controlling whether the parameters are checked for validity. Overriding of this check might be extremely important and needed for use of the quantile function in the context of TL-moments with nonzero trimming. |
Details
The PDQ4 was proposed by Hosking (2007) with the core justification of maximizing entropy and that “maximizing entropy subject to a set of constraints can be regarded as deriving a distribution that is consistent with the information specified in the constraints while making minimal assumptions about the form of the distribution other than those embodied in the constraints.” The PDQ4 is that family constrained to the ,
, and
values of the L-moments. (See also the Polynomial Density-Quantile3 function for constraint on
,
, and
values of the L-moments,
quapdq3
.)
The PDQ4 is a symmetrical distribution ( everywhere) that has maximum entropy conditional on having specified values for the L-moments of
,
, and
with
. The tails of the PDQ4 are exponentially decreasing and the distribution could be useful in distributional analysis with data showing similar tail characteristics. The attainable L-kurtosis range is
with the sign change from negative to positive of
occurring at
. Finally, PDQ4 generalizes the logistic distribution, which is the special case
, and contains distributions both lighter-tailed (
) and heavier-tailed (
) than the logistic.
Value
Quantile value for nonexceedance probability .
Author(s)
W.H. Asquith
References
Hosking, J.R.M., 2007, Distributions with maximum entropy subject to constraints on their L-moments or expected order statistics: Journal of Statistical Planning and Inference, v. 137, no. 9, pp. 2,870–2891, doi:10.1016/j.jspi.2006.10.010.
See Also
cdfpdq4
, pdfpdq4
, lmompdq4
, parpdq4
Examples
lmr <- lmoms(c(123, 34, 4, 654, 37, 78))
quapdq4(0.5, parpdq4(lmr)) # [1] 155
## Not run:
FF <- seq(0.0001, 0.9999, by=0.001)
para <- list(para=c(0, 0.4332, -0.7029), type="pdq4")
plot( qnorm(FF, sd=1), quapdq4(FF, para), type="l", col=grey(0.8), lwd=4,
xlab="Standard normal variate", ylab="Quantiles, Q(f)")
lines(qnorm(FF, sd=1), qnorm(FF, sd=1), lty=2)
legend("topleft", c("Standard normal distribution",
"PDQ4 distribution with same L-moments as the standard normal"),
lwd=c(1, 4), lty=c(2, 1), col=c(1, grey(0.8)), cex=0.8)
mtext("Mimic Hosking (2007, fig. 3 [right])") #
## End(Not run)
## Not run:
# A quick recipe to look at the shapes of quantile functions.
FF <- seq(0.001, 0.999, by=0.001)
plot(qnorm(FF), qnorm(FF), type="n", ylim=c(-7, 7),
xlab="Standard normal variate", ylab="PDQ4 variate")
abline(h=0, lty=2, lwd=0.9); abline(v=0, lty=2, lwd=0.9)
lscale <- 1 / sqrt(pi)
tau4s <- seq(-1/4, 0.7, by=.05)
tau4s[1] <- tau4s[1] + 0.001
for(i in 1:length(tau4s)) {
lmr <- vec2lmom(c(0, lscale, 0, tau4s[i]))
if(! are.lmom.valid(lmr)) next
pdq4 <- parpdq4(lmr, snapt4uplimit=FALSE)
lines(qnorm(FF), qlmomco(FF, pdq4), col=rgb(abs(tau4s[i]), 0, 1))
}
abline(0,1, col="darkgreen", lwd=3)
txt <- "Standard normal distribution (Tau4=0.122602)"
txt <- c(txt, paste0("PDQ4 distribution for varying Tau4 values",
" (color varies for accenting)"))
legend("topleft", txt, col=c("darkgreen", rgb(0.2, 0, 1)),
cex=0.9, bty="n", lwd=c(3,1)) #
## End(Not run)