pdfpdq3 {lmomco}

Probability Density Function of the Polynomial Density-Quantile3 Distribution

Description

This function computes the probability density of the Polynomial Density-Quantile3 distribution given parameters (\alpha and \beta) computed by parpdq3. The probability density function has not explicit form. The implementation here simply uses a five-point stencil to approciate the derivative of the cumulative distribution function cdfpdq3 and hence an eps term is used and multipled to the scale parameter (\alpha) of the distribution. The distribution's canonical definition is in terms of the quantile function (quapdq3).

Usage

pdfpdq3(x, para, paracheck=TRUE, h=NA, hfactor=0.2)

Arguments

x

A real value vector.

para

The parameters from parpdq4 or vec2par.

paracheck

A logical switch as to whether the validity of the parameters should be checked. Default is paracheck=TRUE. This switch is made so that the root solution needed for cdfpdq3 shows an extreme speed increase because of the repeated calls to quapdq3.

h

The differential element of the stencil, if provided, otherwise hfactor used.

hfactor

A term multiplied to the \alpha parameter to set the h in the numerical derivative. Not optimal, but seems to work for a variety of chosen parameters for plotting the density function.

Value

Probability density (f) for x.

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. 2870–2891, doi:10.1016/j.jspi.2006.10.010.

See Also

cdfpdq3, quapdq3, lmompdq3, parpdq3

Examples

## Not run: 
  para <- list(para=c(0.6933, 1.5495, 0.5488), type="pdq3")
  X <- seq(-5, +12, by=(12 - -5) / 1000)
  plot( X, pdfpdq3(X, para), type="l", col=grey(0.8), lwd=4, ylim=c(0, 0.3))
  lines(X, c(NA, diff(pf(exp(X), df1=7, df2=1))/((12 - -5) / 1000)), lty=2)
  legend("topleft", c("log F(7,1) distribution with same L-moments",
                      "PDQ3 distribution with same L-moments as the log F(7,1)"),
                    lwd=c(1, 4), lty=c(2, 1), col=c(1, grey(0.8)), cex=0.8)
  mtext("Mimic Hosking (2007, fig. 2 [left])")
  check.pdf(pdfpdq3, para) # 
## End(Not run)

## Not run: 
  para <- list(para=c(100, 43.32, -0.7029), type="pdq3")
  minX <- quapdq3(0.0001, para)
  maxX <- quapdq3(0.9999, para)
  X <- seq(minX, maxX, by=(maxX - minX) / 1000)
  plot( X, pdfpdq3(X, para), type="l", col=grey(0.8), lwd=4)
  check.pdf(pdfpdq3, para) # 
## End(Not run)

## Not run: 
  para <- vec2par(c(0.4729820, 3.0242067, 0.9880701), type="pdq3")
  print(lmom2par(par2lmom(para), type="pdq3"))
  # "|kappa| > 0.98, alpha (yes alpha) results could be unreliable"
  # So, we are entering into a problem for which the kappa parameter is
  # very large and instabilities in the algorithm will result, but
  # vec2par() has not mechanism for determining this type of situation.
  # Ultimately, things will manifest with a check.pdf() that fails.
  sup <- lmomco::supdist(para)$support
  xx <- seq(sup[1], sup[2], by=diff(range(sup)) / 2000)
  plot(xx, pdfpdq3(xx, para), type="l", col=grey(0.8))
  plot(xx, pdfpdq3(xx, para), type="l", col=grey(0.8), xlim=c(-1,10))
  # See hints of instability in the density shape in the second plot
  check.pdf(pdfpdq3, para) # non-finite function value 
## End(Not run)