quapdq4: Quantile Function of the Polynomial Density-Quantile4...

In lmomco: L-Moments, Censored L-Moments, Trimmed L-Moments, L-Comoments, and Many Distributions

Quantile Function of the Polynomial Density-Quantile4 Distribution

Description

This function computes the quantiles of the Polynomial Density-Quantile4 distribution (PDQ4) given parameters (\xi, \alpha, and \kappa) computed by parpdq4. The quantile function for 0 < \kappa < 1 is

x(F) = \xi + \alpha \biggl[\log\biggl(\frac{F}{1-F}\biggr) - 2\kappa\;\mathrm{atanh}(\kappa[2F-1])\biggr] \mbox{\ and}

for -\infty < \kappa < 0 is

x(F) = \xi + \alpha \biggl[\log\biggl(\frac{F}{1-F}\biggr) + 2\kappa\;\mathrm{atan}(\kappa[2F-1])\biggr] \mbox{,}

where x(F) is the quantile for nonexceedance probability F, \xi is a location parameter, \alpha is a scale parameter, and \kappa is a shape parameter. The range of the distribution is -\infty < x < \infty.

Usage

quapdq4(f, para, paracheck=TRUE)

Arguments

f	Nonexceedance probability (0 \le F \le 1).
para	The parameters from parpdq4 or vec2par.
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 \lambda_1, \lambda_2, and \tau_4 values of the L-moments. (See also the Polynomial Density-Quantile3 function for constraint on \lambda_1, \lambda_2, and \tau_3 values of the L-moments, quapdq3.)

The PDQ4 is a symmetrical distribution (\tau_3 = 0 everywhere) that has maximum entropy conditional on having specified values for the L-moments of \lambda_1, \lambda_2, and \lambda_4 = \tau_4\lambda_2 with \lambda_3 = \tau_3 = 0. 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 -1/4 < \tau_4 < 1 with the sign change from negative to positive of \kappa occurring at \tau_4 = 1/6. Finally, PDQ4 generalizes the logistic distribution, which is the special case \kappa \rightarrow 0, and contains distributions both lighter-tailed (\kappa < 0) and heavier-tailed (\kappa > 0) than the logistic.

Value

Quantile value for nonexceedance probability F.

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, \Sexpr[results=rd]{tools:::Rd_expr_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)

lmomco documentation built on Aug. 30, 2023, 5:10 p.m.