tlmrpe3: Compute Select TL-moment ratios of the Pearson Type III

Description

This function computes select TL-moment ratios of the Pearson Type III distribution for defaults of \(\xi = 0\) and \(\beta = 1\). This function can be useful for plotting the trajectory of the distribution on TL-moment ratio diagrams of \(\tau^{(t_1,t_2)}_2\), \(\tau^{(t_1,t_2)}_3\), \(\tau^{(t_1,t_2)}_4\), \(\tau^{(t_1,t_2)}_5\), and \(\tau^{(t_1,t_2)}_6\). In reality, \(\tau^{(t_1,t_2)}_2\) is dependent on the values for \(\xi\) and \(\alpha\). If the message

Error in integrate(XofF, 0, 1) : the integral is probably divergent

occurs then careful adjustment of the shape parameter \(\beta\) parameter range is very likely required. Remember that TL-moments with nonzero trimming permit computation of TL-moments into parameter ranges beyond those recognized for the usual (untrimmed) L-moments. The function uses numerical integration of the quantile function of the distribution through the theoTLmoms function.

Usage

tlmrpe3(trim=NULL, leftrim=NULL, rightrim=NULL,
        xi=0, beta=1, abeg=-.99, aend=0.99, by=.1)

Value

An R

list is returned.

tau2: A vector of the \(\tau^{(t_1,t_2)}_2\) values.
tau3: A vector of the \(\tau^{(t_1,t_2)}_3\) values.
tau4: A vector of the \(\tau^{(t_1,t_2)}_4\) values.
tau5: A vector of the \(\tau^{(t_1,t_2)}_5\) values.
tau6: A vector of the \(\tau^{(t_1,t_2)}_6\) values.

Arguments

trim: Level of symmetrical trimming to use in the computations. Although NULL in the argument list, the default is 0---the usual L-moment ratios are returned.
leftrim: Level of trimming of the left-tail of the sample.
rightrim: Level of trimming of the right-tail of the sample.
xi: Location parameter of the distribution.
beta: Scale parameter of the distribution.
abeg: The beginning \(\alpha\) value of the distribution.
aend: The ending \(\alpha\) value of the distribution.
by: The increment for the seq() between abeg and aend.

Author

W.H. Asquith

Examples

Run this code

if (FALSE) {
tlmrpe3(leftrim=2, rightrim=4, xi=0, beta=2)
tlmrpe3(leftrim=2, rightrim=4, xi=100, beta=20) # another slow example
  # Plot and L-moment ratio diagram of Tau3 and Tau4
  # with exclusive focus on the PE3 distribution.
  plotlmrdia(lmrdia(), autolegend=TRUE, xleg=-.1, yleg=.6,
             xlim=c(-.8, .7), ylim=c(-.1, .8),
             nolimits=TRUE, nogev=TRUE, nogpa=TRUE, noglo=TRUE,
             nogno=TRUE, nocau=TRUE, noexp=TRUE, nonor=TRUE,
             nogum=TRUE, noray=TRUE, nouni=TRUE)

  # Compute the TL-moment ratios for trimming of one
  # value on the left and four on the right. Notice the
  # expansion of the alpha parameter space from
  # -1 < a < -1 to something larger based on manual
  # adjustments until blue curve encompassed the plot.
  J <- tlmrpe3(abeg=-15, aend=6, leftrim=1, rightrim=4)
  lines(J$tau3, J$tau4, lwd=2, col=2) # RED CURVE

  # Compute the TL-moment ratios for trimming of four
  # values on the left and one on the right.
  J <- tlmrpe3(abeg=-6, aend=10, leftrim=4, rightrim=1)
  lines(J$tau3, J$tau4, lwd=2, col=4) # BLUE CURVE

  # The abeg and aend can be manually changed to see how
  # the resultant curve expands or contracts on the
  # extent of the L-moment ratio diagram.
}
if (FALSE) {
  # Following up, let us plot the two quantile functions
  LM  <- vec2par(c(0,1,0.99), type='pe3', paracheck=FALSE)
  TLM <- vec2par(c(0,1,3.00), type='pe3', paracheck=FALSE)
  F <- nonexceeds()
  plot(qnorm(F),  quape3(F, LM), type="l")
  lines(qnorm(F), quape3(F, TLM, paracheck=FALSE), col=2)
  # Notice how the TLM parameterization runs off towards
  # infinity much much earlier than the conventional
  # near limits of the PE3.
}