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 a 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 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)

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.

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.

The increment for the seq() between abeg and aend.

Value

An R list is returned.
tau2A vector of the $\tau^{(t_1,t_2)}_2$ values.
tau3A vector of the $\tau^{(t_1,t_2)}_3$ values.
tau4A vector of the $\tau^{(t_1,t_2)}_4$ values.
tau5A vector of the $\tau^{(t_1,t_2)}_5$ values.
tau6A vector of the $\tau^{(t_1,t_2)}_6$ values.

Examples

Run this code

tlmrpe3(leftrim=2, rightrim=4, xi=0, beta=2)
tlmrpe3(leftrim=2, rightrim=4, xi=100, beta=20)

# 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.
# 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.

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