Package 'lmomco'

Description

The lmomco package is a comparatively comprehensive implementation of L-moments in addition to probability-weighted moments, and parameter estimation for numerous familiar and not-so-familiar distributions. L-moments and their cousins are based on certain linear combinations of order statistic expectations. Being based on linear mathematics and thus especially robust compared to conventional moments, they are particular suitable for analysis of rare events of non-Normal data. L-moments are consistent and often have smaller sampling variances than maximum likelihood in small to moderate sample sizes. L-moments are especially useful in the context of quantile functions. The method of L-moments (lmr2par) is augmented here with access to the methods of maximum likelihood (mle2par) and maximum product of spacings (mps2par) as alternatives for parameter estimation bound into the distributions of the lmomco package.

About 370 user-level functions are implemented in lmomco that range from low-level utilities forming an application programming interface (API) to high-level sophisticated data analysis and visualization operators. The “See Also” section lists recommended function entry points for new users. The nomenclature (d, p, r, q)-lmomco is directly analogous to that for distributions built-in to R. To conclude, the R packages lmom (Hosking), lmomRFA (Hosking), Lmoments (Karvanen) might also be of great interest.

How does lmomco basically work? The design of lmomco is to fit distributions to the L-moments of sample data. Distributions are specified by a type argument for very many functions. The package stores both L-moments (see vec2lmom) and parameters (see vec2par) in simple R list structures—very elementary. The following code shows a comparison of parameter estimation for a random sample (rlmomco) of a GEV distribution using L-moments (lmoms coupled with lmom2par or simply lmr2par), maximum likelihood (MLE, mle2par), and maximum product of spacings (MPS, mps2par). (A note of warning, the MLE and MPS algorithms might not converge with the initial parameters—for purposes of “learning” about this package just rerun the code below again for another random sample.)

  parent.lmoments <- vec2lmom(c(3.08, 0.568, -0.163)); ty <- "gev"
  Q <- rlmomco(63, lmom2par(parent.lmoments, type=ty)) # random sample
  init <- lmoms(Q); init$ratios[3] <- 0 # failure rates for mps and mle are
  # substantially lowered if starting from the middle of the distribution's
  # shape to form the initial parameters for init.para
  lmr  <- lmr2par(Q, type=ty)                # method of L-moments
  mle  <- mle2par(Q, type=ty, init.lmr=init) # method of MLE
  mps  <- mps2par(Q, type=ty, init.lmr=init) # method of MPS
  lmr1 <- lmr$para; mle1 <- mle$para; mps1 <- mps$para

The lmr1, mle1, and mps1 variables each contain distribution parameter estimates, but before they are inspected, how about quick comparison to another R package (eva)?

  lmr2 <- eva::gevrFit(Q, method="pwm")$par.ests # PWMs == L-moments
  mle2 <- eva::gevrFit(Q, method="mle")$par.ests # method of MLE
  mps2 <- eva::gevrFit(Q, method="mps")$par.ests # method of MPS
  # Package eva uses a different sign convention on the GEV shape parameter
  mle2[3] <- -mle2[3]; mps2[3] <- -mps2[3]; lmr2[3] <- -lmr2[3];

Now let us inspect the contents of the six estimates of the three GEV parameters by three different methods:

  message("LMR(lmomco): ", paste(round(lmr1, digits=5), collapse="  "))
  message("LMR(   eva): ", paste(round(lmr2, digits=5), collapse="  "))
  message("MLE(lmomco): ", paste(round(mle1, digits=5), collapse="  "))
  message("MLE(   eva): ", paste(round(mle2, digits=5), collapse="  "))
  message("MPS(lmomco): ", paste(round(mps1, digits=5), collapse="  "))
  message("MPS(   eva): ", paste(round(mps2, digits=5), collapse="  "))

The results show compatible estimates between the two packages. Lastly, let us plot what these distributions look like using the lmomco functions: add.lmomco.axis, nonexceeds, pp, and qlmomco.

  par(las=2, mgp=c(3, 0.5, 0)); FF <- nonexceeds(); qFF <- qnorm(FF)
  PP <- pp(Q); qPP <- qnorm(PP); Q <- sort(Q)
  plot(  qFF, qlmomco(FF, lmr), xaxt="n", xlab="", tcl=0.5,
                                ylab="QUANTILE", type="l")
  lines( qFF, qlmomco(FF, mle), col="blue")
  lines( qFF, qlmomco(FF, mps), col="red" )
  points(qPP, Q, lwd=0.6, cex=0.8, col=grey(0.3)); par(las=1)
  add.lmomco.axis(las=2, tcl=0.5, side.type="NPP")

Author(s)

William Asquith [email protected]

References

Asquith, W.H., 2007, L-moments and TL-moments of the generalized lambda distribution: Computational Statistics and Data Analysis, v. 51, no. 9, pp. 4484–4496, doi:10.1016/j.csda.2006.07.016.

Asquith, W.H., 2011, Distributional analysis with L-moment statistics using the R environment for statistical computing: Createspace Independent Publishing Platform, ISBN 978–146350841–8, https://www.amazon.com/dp/1463508417/.

Asquith, W.H., 2014, Parameter estimation for the 4-parameter asymmetric exponential power distribution by the method of L-moments using R: Computational Statistics and Data Analysis, v. 71, pp. 955–970, doi:10.1016/j.csda.2012.12.013.

Dey, D.K., Roy, Dooti, Yan, Jun, 2016, Univariate extreme value analysis, chapter 1, in Dey, D.K., and Yan, Jun, eds., Extreme value modeling and risk analysis—Methods and applications: Boca Raton, FL, CRC Press, pp. 1–22.

Elamir, E.A.H., and Seheult, A.H., 2003, Trimmed L-moments: Computational statistics and data analysis, vol. 43, pp. 299-314, doi:10.1016/S0167-9473(02)00250-5.

Hosking, J.R.M., 1990, L-moments—Analysis and estimation of distributions using linear combinations of order statistics: Journal of the Royal Statistical Society, Series B, v. 52, pp. 105–124, doi:10.1111/j.2517-6161.1990.tb01775.x.

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.

Hosking, J.R.M., and Wallis, J.R., 1997, Regional frequency analysis—An approach based on L-moments: Cambridge University Press, https://www.amazon.com/dp/0521019400/.

Nair, N.U., Sankaran, P.G., and Balakrishnan, N., 2013, Quantile-based reliability analysis: Springer, New York, https://www.amazon.com/dp/0817683607/.

Serfling, R., and Xiao, P., 2007, A contribution to multivariate L-moments—L-comoment matrices: Journal of Multivariate Analysis, v. 98, pp. 1765–1781, doi:10.1016/j.jmva.2007.01.008.

See Also

lmoms, dlmomco, plmomco, rlmomco, qlmomco, lmom2par, plotlmrdia, lcomoms2

Description

This is a hidden data object contained in the R/sysdata.rda file of the lmomco package. The system files inst/doc/SysDataBuilder01.R and SysDataBuilder02.R of the package are responsible for the construction of these data with the exception of the Eta-Mu and Kappa-Mu distribution content.

Format

An R environment with entries:

AEPkh2lmrTable

A data.frame of asymmetric exponential power (4-parameter) relations between its two shape parameters, numerical, and theoretical L-skew and L-kurtosis. The table stems from inst/doc/SysDataBuilder01.R. (See also paraep4)

EMU_lmompara_byeta

A data.frame of pre-computed table of relations between the parameters and L-moments of the Eta-Mu distribution. (See also lmomemu, paremu)

KMU_lmompara_bykappa

A data.frame of pre-computed table of relations between the parameters and L-moments of the Kappa-Mu distribution. (See also lmomkmu, parkmu)

RiceTable

A data.frame with coefficient of L-variation, signal to noise ratio, a parameter G, and L-skew and L-kurtosis of the Rice distribution. This is useful for quick parameter estimation. The table stems from inst/doc/SysDataBuilder01.R. (See also lmomrice, parrice)

RiceTable.maxLCV

Maximum coefficient of L-variation representable (or apparently so) within R. The value stems from inst/doc/SysDataBuilder01.R.

RiceTable.minLCV

Minimum coefficient of L-variation representable (or apparently so) within R. The value stems from inst/doc/SysDataBuilder01.R.

tau46list

Various relations of Tau4-Tau6 for symmetrical distributions and used to support the access layer provided by lmrdia46 for Tau4-Tau6 L-moment ratio diagrams. The tables in the list stem from inst/doc/SysDataBuilder02.R, which is designed to be run after the SysDataBuilder01.R.

Description

This function provides special support for adding probability-like axes to an existing plot. The function supports a recurrence interval (RI) axis, normal probability axis (NPP), and standard normal variate (SNV) axis. The function is built around the interface model that standard normal transformation of the values for the respective axis controlled by this function are being plotted; this means that qnorm() should be wrapped on the values of nonexceedance probability. This is an ease oversight to make (see Examples section below and note use of qnorm(pp)).

The function provides a convenient interface for labeling and titling two axes, so adjustments to default margins might be desired. The pertinent control is achieved using the par() function, which might be of the form par(mgp=c(3,0.5,0), mar=c(5,4,4,3)) say for plotting the lmomco axis both on the left and right (see z.par2cdf for an example).

Usage

add.lmomco.axis(side=1, twoside=FALSE, twoside.suppress.labels=FALSE,
                side.type=c("NPP", "RI", "SNV"),
                otherside.type=c("NA", "RI", "SNV", "NPP"),
                alt.lab=NA, alt.other.lab=NA, npp.as.aep=FALSE,
                case=c("upper", "lower"),
                NPP.control=NULL, RI.control=NULL, SNV.control=NULL, ...)

Arguments

Value

No value is returned. This function is used for its side effects.

Note

The NPP.control, RI.control, and SNV.control are R list structures that can be populated (and perhaps someday extended) to feed various settings into the respective axis types. In brief:

The NPP.control provides

The RI.control provides

The SNV.control provides

The user is responsible for appropriate construction of the control lists. Very little error trapping is made to keep the code base tight. The defaults when the function definition are likely good for many types of applications. Lastly, the manipulation of the mgp parameter in the example is to show how to handle the offset between the numbers and the ticks when the ticks are moved to pointing inward, which is opposite of the default in R.

Author(s)

W.H. Asquith

See Also

prob2T, T2prob, add.log.axis

Examples

par(mgp=c(3,0.5,0)) # going to tick to the inside, change some parameters
X <- sort(rnorm(65)); pp <- pp(X) # generate synthetic data
plot(qnorm(pp), X, xaxt="n", xlab="", ylab="QUANTILE", xlim=c(-2,3))
add.lmomco.axis(las=2, tcl=0.5, side.type="RI", otherside.type="NPP")
par(mgp=c(3,1,0)) # restore defaults

## Not run: 
opts <- options(scipen=6); par(mgp=c(3,0.5,0))
X <- sort(rexp(65, rate=.0001))*100; pp <- pp(X) # generate synthetic data
plot(qnorm(pp), X, yaxt="n", xaxt="n", xlab="", ylab="", log="y")
add.log.axis(side=2,    tcl=+0.8*abs(par()$tcl),         two.sided=TRUE)
add.log.axis(logs=c(1), tcl=-0.5*abs(par()$tcl), side=2, two.sided=TRUE)
add.log.axis(logs=c(1), tcl=+1.3*abs(par()$tcl), side=2, two.sided=TRUE)
add.log.axis(logs=1:8, side=2, make.labs=TRUE, las=1, label="QUANTILE")
add.lmomco.axis(las=2, tcl=0.5, side.type="NPP", npp.as.aep=TRUE, case="lower")
options(opts)
par(mgp=c(3,1,0)) # restore defaults
## End(Not run)

Description

This function provides special support for adding superior looking base-10 logarithmic axes relative to R's defaults, which are an embarassment. The Examples section shows an overly elaborate version made by repeated calls to this function with a drawback that each call redraws the line of the axis so deletion in editing software might be required. This function is indexed under the “lmomco functions” because of its relation to add.lmomco.axis and is not named add.lmomcolog.axis because such a name is too cumbersome.

Usage

add.log.axis(make.labs=FALSE, logs=c(2, 3, 4, 5, 6, 7, 8, 9), side=1,
             two.sided=FALSE, label=NULL, x=NULL, col.ticks=1, ...)

Arguments

Value

No value is returned, except if argument x is given, for which nice axis limits are returned. By overall design, this function is used for its side effects.

Author(s)

W.H. Asquith

See Also

add.lmomco.axis

Examples

## Not run: 
par(mgp=c(3,0.5,0)) # going to tick to the inside, change some parameters
X <- 10^sort(rnorm(65)); pp <- pp(X) # generate synthetic data
ylim <- add.log.axis(x=X) # snap to some nice integers within the cycle
plot(qnorm(pp), X, xaxt="n", yaxt="n", xlab="", ylab="", log="y",
     xlim=c(-2,3), ylim=ylim, pch=6, yaxs="i", col=4)
add.lmomco.axis(las=2, tcl=0.5, side.type="RI", otherside.type="NPP")
# Logarithmic axis: the base ticks to show logarithms
add.log.axis(side=2,      tcl=0.8*abs(par()$tcl), two.sided=TRUE)
#                   the long even-cycle tick, set to inside and outside
add.log.axis(logs=c(1),   tcl=-0.5*abs(par()$tcl), side=2, two.sided=TRUE)
add.log.axis(logs=c(1),   tcl=+1.3*abs(par()$tcl), side=2, two.sided=TRUE)
#                   now a micro tick at the 1.5 logs but only on the right
add.log.axis(logs=c(1.5), tcl=+0.5*abs(par()$tcl), side=4)
#                   and only label the micro tick at 1.5 on the right
add.log.axis(logs=c(1.5), side=4, make.labs=TRUE, las=3) # but weird rotate
#                   add the bulk tick labeling and axis label.
add.log.axis(logs=c(1, 2, 3, 4, 6), side=2, make.labs=TRUE, las=1, label="QUANTILE")
par(mgp=c(3,1,0)) # restore defaults
## End(Not run)

Description

Annual maximum precipitation data for Amarillo, Texas

Usage

data(amarilloprecip)

Format

An R data.frame with

YEAR

The calendar year of the annual maxima.

DEPTH

The depth of 7-day annual maxima rainfall in inches.

References

Asquith, W.H., 1998, Depth-duration frequency of precipitation for Texas: U.S. Geological Survey Water-Resources Investigations Report 98–4044, 107 p.

Examples

data(amarilloprecip)
summary(amarilloprecip)

Description

This function converts “A”-type probability-weighted moments (PWMs, $\beta^A_r$) to the “B”-type $\beta^B_r$. The $\beta^A_r$ are the ordinary PWMs for the $m$ left noncensored or observed values. The $\beta^B_r$ are more complex and use the $m$ observed values and the $m-n$ right-tailed censored values for which the censoring threshold is known. The “A”- and “B”-type PWMs are described in the documentation for pwmRC.

This function uses the defined relation between to two PWM types when the $\beta^A_r$ are known along with the parameters (para) of a right-tail censored distribution inclusive of the censoring fraction $\zeta=m/n$. The value $\zeta$ is the right-tail censor fraction or the probability $\mathrm{Pr}\lbrace \rbrace$ that $x$ is less than the quantile at $\zeta$ nonexceedance probability ($\mathrm{Pr}\lbrace x < X(\zeta) \rbrace$). The relation is

$\beta^B_{r-1} = r^{-1}\lbrace\zeta^r r \beta^A_{r-1} + (1-\zeta^r)X(\zeta)\rbrace \mbox{,}$

where $1 \le r \le n$ and $n$ is the number of moments, and $X(\zeta)$ is the value of the quantile function at nonexceedance probability $\zeta$. Finally, the RC in the function name is to denote Right-tail Censoring.

Usage

Apwm2BpwmRC(Apwm,para)

Arguments

Value

An R list is returned.

Author(s)

W.H. Asquith

References

Hosking, J.R.M., 1995, The use of L-moments in the analysis of censored data, in Recent Advances in Life-Testing and Reliability, edited by N. Balakrishnan, chapter 29, CRC Press, Boca Raton, Fla., pp. 546–560.

See Also

Bpwm2ApwmRC, pwmRC

Examples

# Data listed in Hosking (1995, table 29.2, p. 551)
H <- c(3,4,5,6,6,7,8,8,9,9,9,10,10,11,11,11,13,13,13,13,13,
             17,19,19,25,29,33,42,42,51.9999,52,52,52)
      # 51.9999 was really 52, a real (noncensored) data point.
z <-  pwmRC(H,52)
# The B-type PMWs are used for the parameter estimation of the
# Reverse Gumbel distribution. The parameter estimator requires
# conversion of the PWMs to L-moments by pwm2lmom().
para <- parrevgum(pwm2lmom(z$Bbetas),z$zeta) # parameter object
Bbetas <- Apwm2BpwmRC(z$Abetas,para)
Abetas <- Bpwm2ApwmRC(Bbetas$betas,para)
# Assertion that both of the vectors of B-type PWMs should be the same.
str(Abetas)   # A-type PWMs of the distribution
str(z$Abetas) # A-type PWMs of the original data

Description

The L-moments have particular constraints on magnitudes and relation to each other. This function evaluates and L-moment object whether the bounds for $\lambda_2 > 0$ (L-scale), $|\tau_3| < 1$ (L-skew), $\tau_4 < 1$ (L-kurtosis), and $|\tau_5| < 1$ are satisfied. An optional check on $\tau_4 \ge (5\tau_3^2 - 1)/4$ is made. Also for further protection, the finitenesses of the mean ($\lambda_1$) and $\lambda_2$ are also checked. These checks provide protection against say L-moments being computed on the logarithms of some data but the data themselves have values less than or equal to zero.

The TL-moments as implemented by the TL functions (TLmoms) are not applicable to the boundaries (well finiteness of course). The are.lmom.valid function should not be consulted on the TL-moments.

Usage

are.lmom.valid(lmom, checkt3t4=TRUE)

Arguments

Value

Author(s)

W.H. Asquith

References

Hosking, J.R.M., 1990, L-moments—Analysis and estimation of distributions using linear combinations of order statistics: Journal of the Royal Statistical Society, Series B, v. 52, pp. 105–124.

Hosking, J.R.M. and Wallis, J.R., 1997, Regional frequency analysis—An approach based on L-moments: Cambridge University Press.

See Also

lmom.ub, lmoms, pwm2lmom

Examples

lmr <- lmoms(rnorm(20))
if(are.lmom.valid(lmr)) print("They are.")
## Not run: 
X <- c(1.7106278,  1.7598761,  1.2111335,  0.3447490,  1.8312889,
       1.3938445, -0.5376054, -0.2341009, -0.4333601, -0.2545229)
are.lmom.valid(lmoms(X))
are.lmom.valid(pwm2lmom(pwm.pp(X, a=0.5)))

# Prior to version 2.2.6, the next line could leak through as TRUE. This was a problem.
# Nonfiniteness of the mean or L-scale should have been checked; they are for v2.2.6+
are.lmom.valid(lmoms(log10(c(1,23,235,652,0)), nmom=1)) # of other nmom

## End(Not run)

Description

This function is a dispatcher on top of the are.parCCC.valid functions, where CCC represents the distribution type: aep4, cau, emu, exp, gam, gep, gev, glo, gno, gov, gpa, gum, kap, kmu, kur, lap, ln3, nor, pe3, ray, revgum, rice, sla, smd, st3, texp, tri, wak, or wei. For lmomco functionality, are.par.valid is called only by vec2par in the process of converting a vector into a proper distribution parameter object.

Usage

are.par.valid(para, paracheck=TRUE, ...)

Arguments

Value

Author(s)

W.H. Asquith

References

Asquith, W.H., 2011, Distributional analysis with L-moment statistics using the R environment for statistical computing: Createspace Independent Publishing Platform, ISBN 978–146350841–8.

See Also

vec2par, dist.list

Examples

vec  <- c(12, 120)           # parameters of exponential distribution
para <- vec2par(vec, "exp")  # build exponential distribution parameter
                             # object
# The following two conditionals are equivalent as are.parexp.valid()
# is called within are.par.valid().
if(   are.par.valid(para)) Q <- quaexp(0.5, para)
if(are.parexp.valid(para)) Q <- quaexp(0.5, para)

Description

Is the distribution parameter object consistent with the corresponding distribution? The distribution functions (cdfaep4, pdfaep4, quaaep4, and lmomaep4) require consistent parameters to return the cumulative probability (nonexceedance), density, quantile, and L-moments of the distribution, respectively. These functions internally use the are.paraep4.valid function.

Usage

are.paraep4.valid(para, nowarn=FALSE)

Arguments

Value

Note

This function calls is.aep4 to verify consistency between the distribution parameter object and the intent of the user.

Author(s)

W.H. Asquith

References

Asquith, W.H., 2014, Parameter estimation for the 4-parameter asymmetric exponential power distribution by the method of L-moments using R: Computational Statistics and Data Analysis, v. 71, pp. 955–970.

Delicado, P., and Goria, M.N., 2008, A small sample comparison of maximum likelihood, moments and L-moments methods for the asymmetric exponential power distribution: Computational Statistics and Data Analysis, v. 52, no. 3, pp. 1661–1673.

See Also

is.aep4, paraep4

Examples

para <- vec2par(c(0,1, 0.5, 4), type="aep4")
if(are.paraep4.valid(para)) Q <- quaaep4(0.5,para)

Description

Is the distribution parameter object consistent with the corresponding distribution? The distribution functions (cdfcau, pdfcau, quacau, and lmomcau) require consistent parameters to return the cumulative probability (nonexceedance), density, quantile, and L-moments of the distribution, respectively. These functions internally use the are.parcau.valid function.

Usage

are.parcau.valid(para, nowarn=FALSE)

Arguments

Value

Note

This function calls is.cau to verify consistency between the distribution parameter object and the intent of the user.

Author(s)

W.H. Asquith

References

Elamir, E.A.H., and Seheult, A.H., 2003, Trimmed L-moments: Computational Statistics and Data Analysis, v. 43, pp. 299–314.

Gilchrist, W.G., 2000, Statistical modeling with quantile functions: Chapman and Hall/CRC, Boca Raton, FL.

See Also

is.cau, parcau

Examples

para <- vec2par(c(12,12),type='cau')
if(are.parcau.valid(para)) Q <- quacau(0.5,para)

Description

Is the distribution parameter object consistent with the corresponding distribution? The distribution functions (cdfemu, pdfemu, quaemu, lmomemu), and lmomemu require consistent parameters to return the cumulative probability (nonexceedance), density, quantile, and L-moments of the distribution, respectively. These functions internally use the are.paremu.valid function. The documentation for pdfemu provides the conditions for valid parameters.

Usage

are.paremu.valid(para, nowarn=FALSE)

Arguments

Value

Note

This function calls is.emu to verify consistency between the distribution parameter object and the intent of the user.

Author(s)

W.H. Asquith

See Also

is.emu, paremu

Examples

## Not run: 
para <- vec2par(c(0.4, .04), type="emu")
if(are.paremu.valid(para)) Q <- quaemu(0.5,para) # 
## End(Not run)

Description

Is the distribution parameter object consistent with the corresponding distribution? The distribution functions (cdfexp, pdfexp, quaexp, and lmomexp) require consistent parameters to return the cumulative probability (nonexceedance), density, quantile, and L-moments of the distribution, respectively. These functions internally use the are.parexp.valid function.

Usage

are.parexp.valid(para, nowarn=FALSE)

Arguments

Value

Note

This function calls is.exp to verify consistency between the distribution parameter object and the intent of the user.

Author(s)

W.H. Asquith

References

Hosking, J.R.M., 1996, FORTRAN routines for use with the method of L-moments: Version 3, IBM Research Report RC20525, T.J. Watson Research Center, Yorktown Heights, New York.

Hosking, J.R.M., and Wallis, J.R., 1997, Regional frequency analysis—An approach based on L-moments: Cambridge University Press.

See Also

is.exp, parexp

Examples

para <- parexp(lmoms(c(123,34,4,654,37,78)))
if(are.parexp.valid(para)) Q <- quaexp(0.5,para)

Description

Is the distribution parameter object consistent with the corresponding distribution? The distribution functions (cdfgam, pdfgam, quagam, and lmomgam) require consistent parameters to return the cumulative probability (nonexceedance), density, quantile, and L-moments of the distribution, respectively. These functions internally use the are.pargam.valid function. The parameters are restricted to the following conditions.

$\alpha > 0 \mbox{ and } \beta > 0\mbox{.}$

Alternatively, a three-parameter version is available following the parameterization of the Generalized Gamma distribution used in the gamlss.dist package and and for lmomco is documented under pdfgam. The parameters for this version are

$\mu > 0;\;\; \sigma > 0;\;\; -\infty < \nu < \infty$

for parameters number 1, 2, 3, respectively.

Usage

are.pargam.valid(para, nowarn=FALSE)

Arguments

Value

Note

This function calls is.gam to verify consistency between the distribution parameter object and the intent of the user.

Author(s)

W.H. Asquith

References

Hosking, J.R.M., and Wallis, J.R., 1997, Regional frequency analysis—An approach based on L-moments: Cambridge University Press.

See Also

is.gam, pargam

Examples

para <- pargam(lmoms(c(123,34,4,654,37,78)))
if(are.pargam.valid(para)) Q <- quagam(0.5,para)

Description

Is the distribution parameter object consistent with the corresponding distribution? The distribution functions (cdfgdd, pdfgdd, quagdd, and lmomgdd) require consistent parameters to return the cumulative probability (nonexceedance), density, quantile, and L-moments of the distribution, respectively. These functions internally use the are.pargdd.valid function.

Usage

are.pargdd.valid(para, nowarn=FALSE)

Arguments

Value

Note

This function calls is.gdd to verify consistency between the distribution parameter object and the intent of the user.

Author(s)

W.H. Asquith

References

Klar, B., 2015, A note on gamma difference distributions: Journal of Statistical Computation and Simulation v. 85, no. 18, pp. 1–8, doi:10.1080/00949655.2014.996566.

See Also

is.gdd, pargdd

Examples

#

Description

Is the distribution parameter object consistent with the corresponding distribution? The distribution functions (cdfgep, pdfgep, quagep, and lmomgep) require consistent parameters to return the cumulative probability (nonexceedance), density, quantile, and L-moments of the distribution, respectively. These functions internally use the are.pargep.valid function. The parameters must be $\beta > 0$, $\kappa > 0$, and $h > 0$.

Usage

are.pargep.valid(para, nowarn=FALSE)

Arguments

Value

Note

This function calls is.gep to verify consistency between the distribution parameter object and the intent of the user.

Author(s)

W.H. Asquith

References

Barreto-Souza, W., and Cribari-Neto, F., 2009, A generalization of the exponential-Poisson distribution: Statistics and Probability, 79, pp. 2493–2500.

See Also

is.gep, pargep

Examples

#para <- pargep(lmoms(c(123,34,4,654,37,78)))
#if(are.pargep.valid(para)) Q <- quagep(0.5,para)

Description

Is the distribution parameter object consistent with the corresponding distribution? The distribution functions (cdfgev, pdfgev, quagev, and lmomgev) require consistent parameters to return the cumulative probability (nonexceedance), density, quantile, and L-moments of the distribution, respectively. These functions internally use the are.pargev.valid function.

Usage

are.pargev.valid(para, nowarn=FALSE)

Arguments

Value

Note

This function calls is.gev to verify consistency between the distribution parameter object and the intent of the user.

Author(s)

W.H. Asquith

References

Hosking, J.R.M., 1996, FORTRAN routines for use with the method of L-moments: Version 3, IBM Research Report RC20525, T.J. Watson Research Center, Yorktown Heights, New York.

Hosking, J.R.M., and Wallis, J.R., 1997, Regional frequency analysis—An approach based on L-moments: Cambridge University Press.

See Also

is.gev, pargev

Examples

para <- pargev(lmoms(c(123, 34, 4, 654, 37, 78)))
if(are.pargev.valid(para)) Q <- quagev(0.5, para)

Description

Is the distribution parameter object consistent with the corresponding distribution? The distribution functions (cdfgld, pdfgld, quagld, and lmomgld) require consistent parameters to return the cumulative probability (nonexceedance), density, quantile, and L-moments of the distribution, respectively. These functions internally use the are.pargld.valid function.

Usage

are.pargld.valid(para, verbose=FALSE, nowarn=FALSE)

Arguments

Details

Karian and Dudewicz (2000) outline valid parameter space of the Generalized Lambda distribution. First, according to Theorem 1.3.3 the distribution is valid if and only if

$\alpha(\kappa F^{\kappa - 1} + h(1-F)^{h -1 }) \ge 0 \mbox{.}$

for all $F \in [0,1]$. The are.pargld.valid function tests against this condition by incrementing through $[0,1]$ by $dF = 0.0001$. This is a brute force method of course. Further, Karian and Dudewicz (2002) provide a diagrammatic representation of regions in $\kappa$ and $h$ space for suitable $\alpha$ in which the distribution is valid. The are.pargld.valid function subsequently checks against the 6 valid regions as a secondary check on Theorem 1.3.3. The regions of the distribution are defined for suitably choosen $\alpha$ by

$\mbox{Region 1: } \kappa \le -1 \mbox{ and } h \ge 1 \mbox{,}$

$\mbox{Region 2: } \kappa \ge 1 \mbox{ and } h \le -1 \mbox{,}$

$\mbox{Region 3: } \kappa \ge 0 \mbox{ and } h \ge 0 \mbox{,}$

$\mbox{Region 4: } \kappa \le 0 \mbox{ and } h \le 0 \mbox{,}$

$\mbox{Region 5: } h \ge (-1/\kappa) \mbox{ and } -1 \ge \kappa \le 0 \mbox{, and}$

$\mbox{Region 6: } h \le (-1/\kappa) \mbox{ and } h \ge -1 \mbox{ and } \kappa \ge 1 \mbox{.}$

Value

Note

This function calls is.gld to verify consistency between the distribution parameter object and the intent of the user.

Author(s)

W.H. Asquith

References

Asquith, W.H., 2007, L-moments and TL-moments of the generalized lambda distribution: Computational Statistics and Data Analysis, v. 51, no. 9, pp. 4484–4496.

Karian, Z.A., and Dudewicz, E.J., 2000, Fitting statistical distributions—The generalized lambda distribution and generalized bootstrap methods: CRC Press, Boca Raton, FL, 438 p.

See Also

is.gld, pargld

Examples

## Not run: 
para <- vec2par(c(123,34,4,3),type='gld')
if(are.pargld.valid(para)) Q <- quagld(0.5,para)

# The following is an example of inconsistent L-moments for fitting but
# prior to lmomco version 2.1.2 and untrapped error was occurring.
lmr <- lmoms(c(33, 37, 41, 54, 78, 91, 100, 120, 124))
para <- pargld(lmr); are.pargld.valid(para)
## End(Not run)

Description

Is the distribution parameter object consistent with the corresponding distribution? The distribution functions (cdfglo, pdfglo, quaglo, and lmomglo) require consistent parameters to return the cumulative probability (nonexceedance), density, quantile, and L-moments of the distribution, respectively. These functions internally use the are.parglo.valid function.

Usage

are.parglo.valid(para, nowarn=FALSE)

Arguments

Value

Note

This function calls is.glo to verify consistency between the distribution parameter object and the intent of the user.

Author(s)

W.H. Asquith

References

Hosking, J.R.M., 1996, FORTRAN routines for use with the method of L-moments: Version 3, IBM Research Report RC20525, T.J. Watson Research Center, Yorktown Heights, New York.

Hosking, J.R.M., and Wallis, J.R., 1997, Regional frequency analysis—An approach based on L-moments: Cambridge University Press.

See Also

is.glo, parglo

Examples

para <- parglo(lmoms(c(123,34,4,654,37,78)))
if(are.parglo.valid(para)) Q <- quaglo(0.5,para)

Description

Is the distribution parameter object consistent with the corresponding distribution? The distribution functions (cdfgno, pdfgno, quagno, and lmomgno) require consistent parameters to return the cumulative probability (nonexceedance), density, quantile, and L-moments of the distribution, respectively. These functions internally use the are.pargno.valid function.

Usage

are.pargno.valid(para, nowarn=FALSE)

Arguments

Value

Note

This function calls is.gno to verify consistency between the distribution parameter object and the intent of the user.

Author(s)

W.H. Asquith

References

Hosking, J.R.M., 1996, FORTRAN routines for use with the method of L-moments: Version 3, IBM Research Report RC20525, T.J. Watson Research Center, Yorktown Heights, New York.

Hosking, J.R.M., and Wallis, J.R., 1997, Regional frequency analysis—An approach based on L-moments: Cambridge University Press.

See Also

is.gno, pargno

Examples

para <- pargno(lmoms(c(123,34,4,654,37,78)))
if(are.pargno.valid(para)) Q <- quagno(0.5,para)

Description

Is the distribution parameter object consistent with the corresponding distribution? The distribution functions (cdfgov, pdfgov, quagov, and lmomgov) require consistent parameters to return the cumulative probability (nonexceedance), density, quantile, and L-moments of the distribution, respectively. These functions internally use the are.pargov.valid function.

Usage

are.pargov.valid(para, nowarn=FALSE)

Arguments

Value

Note

This function calls is.gov to verify consistency between the distribution parameter object and the intent of the user.

Author(s)

W.H. Asquith

References

Gilchrist, W.G., 2000, Statistical modelling with quantile functions: Chapman and Hall/CRC, Boca Raton.

Nair, N.U., Sankaran, P.G., and Balakrishnan, N., 2013, Quantile-based reliability analysis: Springer, New York.

See Also

is.gov, pargov

Examples

para <- pargov(lmoms(c(123,34,4,654,37,78)))
if(are.pargov.valid(para)) Q <- quagov(0.5,para)

Description

Is the distribution parameter object consistent with the corresponding distribution? The distribution functions (cdfgpa, pdfgpa, quagpa, and lmomgpa) require consistent parameters to return the cumulative probability (nonexceedance), density, quantile, and L-moments of the distribution, respectively. These functions internally use the are.pargpa.valid function.

Usage

are.pargpa.valid(para, nowarn=FALSE)

Arguments

Value

Note

This function calls is.gpa to verify consistency between the distribution parameter object and the intent of the user.

Author(s)

W.H. Asquith

References

Hosking, J.R.M., 1996, FORTRAN routines for use with the method of L-moments: Version 3, IBM Research Report RC20525, T.J. Watson Research Center, Yorktown Heights, New York.

Hosking, J.R.M., and Wallis, J.R., 1997, Regional frequency analysis—An approach based on L-moments: Cambridge University Press.

See Also

is.gpa, pargpa

Examples

para <- pargpa(lmoms(c(123,34,4,654,37,78)))
if(are.pargpa.valid(para)) Q <- quagpa(0.5,para)

Description

Is the distribution parameter object consistent with the corresponding distribution? The distribution functions (cdfgum, pdfgum, quagum, and lmomgum) require consistent parameters to return the cumulative probability (nonexceedance), density, quantile, and L-moments of the distribution, respectively. These functions internally use the are.pargum.valid function.

Usage

are.pargum.valid(para, nowarn=FALSE)

Arguments

Value

Note

This function calls is.gum to verify consistency between the distribution parameter object and the intent of the user.

Author(s)

W.H. Asquith

References

Hosking, J.R.M., 1996, FORTRAN routines for use with the method of L-moments: Version 3, IBM Research Report RC20525, T.J. Watson Research Center, Yorktown Heights, New York.

Hosking, J.R.M., and Wallis, J.R., 1997, Regional frequency analysis—An approach based on L-moments: Cambridge University Press.

See Also

is.gum, pargum

Examples

para <- pargum(lmoms(c(123,34,4,654,37,78)))
if(are.pargum.valid(para)) Q <- quagum(0.5,para)

Description

Is the distribution parameter object consistent with the corresponding distribution? The distribution functions (cdfkap, pdfkap, quakap, and lmomkap) require consistent parameters to return the cumulative probability (nonexceedance), density, quantile, and L-moments of the distribution, respectively. These functions internally use the are.parkap.valid function.

Usage

are.parkap.valid(para, nowarn=FALSE)

Arguments

Value

Note

This function calls is.kap to verify consistency between the distribution parameter object and the intent of the user.

Author(s)

W.H. Asquith

References

Hosking, J.R.M., 1994, The four-parameter kappa distribution: IBM Journal of Reserach and Development, v. 38, no. 3, pp. 251–258.

Hosking, J.R.M., and Wallis, J.R., 1997, Regional frequency analysis—An approach based on L-moments: Cambridge University Press.

See Also

is.kap, parkap

Examples

para <- parkap(lmoms(c(123,34,4,654,37,78)))
if(are.parkap.valid(para)) Q <- quakap(0.5,para)

Description

Is the distribution parameter object consistent with the corresponding distribution? The distribution functions (pdfkmu, cdfkmu, quakmu, and lmomkmu) require consistent parameters to return the cumulative probability (nonexceedance), density, quantile, and L-moments of the distribution, respectively. These functions internally use the are.parkmu.valid function. The documentation for pdfkmu provides the conditions for valid parameters.

Usage

are.parkmu.valid(para, nowarn=FALSE)

Arguments

Value

Note

This function calls is.kmu to verify consistency between the distribution parameter object and the intent of the user.

Author(s)

W.H. Asquith

See Also

is.kmu, parkmu

Examples

para <- vec2par(c(0.5, 1.5), type="kmu")
if(are.parkmu.valid(para)) Q <- quakmu(0.5,para)

Description

Is the distribution parameter object consistent with the corresponding distribution? The distribution functions (cdfkur, pdfkur, quakur, and lmomkur) require consistent parameters to return the cumulative probability (nonexceedance), density, quantile, and L-moments of the distribution, respectively. These functions internally use the are.parkur.valid function.

Usage

are.parkur.valid(para, nowarn=FALSE)

Arguments

Value

Note

This function calls is.kur to verify consistency between the distribution parameter object and the intent of the user.

Author(s)

W.H. Asquith

References

Jones, M.C., 2009, Kumaraswamy's distribution—A beta-type distribution with some tractability advantages: Statistical Methodology, v. 6, pp. 70–81.

See Also

is.kur, parkur

Examples

para <- parkur(lmoms(c(0.25, 0.4, 0.6, 0.65, 0.67, 0.9)))
if(are.parkur.valid(para)) Q <- quakur(0.5,para)

Description

Is the distribution parameter object consistent with the corresponding distribution? The distribution functions (cdflap, pdflap, qualap, and lmomlap) require consistent parameters to return the cumulative probability (nonexceedance), density, quantile, and L-moments of the distribution, respectively. These functions internally use the are.parlap.valid function.

Usage

are.parlap.valid(para, nowarn=FALSE)

Arguments

Value

Note

This function calls is.lap to verify consistency between the distribution parameter object and the intent of the user.

Author(s)

W.H. Asquith

References

Hosking, J.R.M., 1986, The theory of probability weighted moments: IBM Research Report RC12210, T.J. Watson Research Center, Yorktown Heights, New York.

See Also

is.lap, parlap

Examples

para <- parlap(lmoms(c(123,34,4,654,37,78)))
if(are.parlap.valid(para)) Q <- qualap(0.5,para)

Description

Is the distribution parameter object consistent with the corresponding distribution? The distribution functions (cdflmrq, pdflmrq, qualmrq, and lmomlmrq) require consistent parameters to return the cumulative probability (nonexceedance), density, quantile, and L-moments of the distribution, respectively. These functions internally use the are.parlmrq.valid function. The constraints on the parameters are listed under qualmrq. The documentation for qualmrq provides the conditions for valid parameters.

Usage

are.parlmrq.valid(para, nowarn=FALSE)

Arguments

Value

Note

This function calls is.lmrq to verify consistency between the distribution parameter object and the intent of the user.

Author(s)

W.H. Asquith

References

Midhu, N.N., Sankaran, P.G., and Nair, N.U., 2013, A class of distributions with linear mean residual quantile function and it's generalizations: Statistical Methodology, v. 15, pp. 1–24.

See Also

is.lmrq, parlmrq

Examples

para <- parlmrq(lmoms(c(3, 0.05, 1.6, 1.37, 0.57, 0.36, 2.2)))
if(are.parlmrq.valid(para)) Q <- qualmrq(0.5,para)

Description

Is the distribution parameter object consistent with the corresponding distribution? The distribution functions (cdfln3, pdfln3, qualn3, and lmomln3) require consistent parameters to return the cumulative probability (nonexceedance), density, quantile, and L-moments of the distribution, respectively. These functions internally use the are.parln3.valid function.

Usage

are.parln3.valid(para, nowarn=FALSE)

Arguments

Value

Note

This function calls is.ln3 to verify consistency between the distribution parameter object and the intent of the user.

Author(s)

W.H. Asquith

References

Asquith, W.H., 2011, Distributional analysis with L-moment statistics using the R environment for statistical computing: Createspace Independent Publishing Platform, ISBN 978–146350841–8.

See Also

is.ln3, parln3

Examples

para <- parln3(lmoms(c(123,34,4,654,37,78)))
if(are.parln3.valid(para)) Q <- qualn3(0.5,para)

Description

Is the distribution parameter object consistent with the corresponding distribution? The distribution functions (cdfnor, pdfnor, quanor, and lmomnor) require consistent parameters to return the cumulative probability (nonexceedance), density, quantile, and L-moments of the distribution, respectively. These functions internally use the are.parnor.valid function.

Usage

are.parnor.valid(para, nowarn=FALSE)

Arguments

Value

Note

This function calls is.nor to verify consistency between the distribution parameter object and the intent of the user.

Author(s)

W.H. Asquith

References

Hosking, J.R.M., 1996, FORTRAN routines for use with the method of L-moments: Version 3, IBM Research Report RC20525, T.J. Watson Research Center, Yorktown Heights, New York.

Hosking, J.R.M., and Wallis, J.R., 1997, Regional frequency analysis—An approach based on L-moments: Cambridge University Press.

See Also

is.nor, parnor

Examples

para <- parnor(lmoms(c(123,34,4,654,37,78)))
if(are.parnor.valid(para)) Q <- quanor(0.5,para)

Description

Is the distribution parameter object consistent with the corresponding distribution? The distribution functions (cdfpdq3, pdfpdq3, quapdq3, and lmompdq3) require consistent parameters to return the cumulative probability (nonexceedance), density, quantile, and L-moments of the distribution, respectively. These functions internally use the are.parpdq3.valid function.

Usage

are.parpdq3.valid(para, nowarn=FALSE)

Arguments

Value

Note

This function calls is.pdq3 to verify consistency between the distribution parameter object and the intent of the user.

Author(s)

W.H. Asquith

See Also

is.pdq3, parpdq3

Examples

para <- parpdq3(lmoms(c(46, 70, 59, 36, 71, 48, 46, 63, 35, 52)))
if(are.parpdq3.valid(para)) Q <- quapdq3(0.5, para)

Description

Is the distribution parameter object consistent with the corresponding distribution? The distribution functions (cdfpdq4, pdfpdq4, quapdq4, and lmompdq4) require consistent parameters to return the cumulative probability (nonexceedance), density, quantile, and L-moments of the distribution, respectively. These functions internally use the are.parpdq4.valid function.

Usage

are.parpdq4.valid(para, nowarn=FALSE)

Arguments

Value

Note

This function calls is.pdq4 to verify consistency between the distribution parameter object and the intent of the user.

Author(s)

W.H. Asquith

See Also

is.pdq4, parpdq4

Examples

para <- parpdq4(lmoms(c(46, 70, 59, 36, 71, 48, 46, 63, 35, 52)))
if(are.parpdq4.valid(para)) Q <- quapdq4(0.5, para)

Description

Is the distribution parameter object consistent with the corresponding distribution? The distribution functions (cdfpe3, pdfpe3, quape3, and lmompe3) require consistent parameters to return the cumulative probability (nonexceedance), density, quantile, and L-moments of the distribution, respectively. These functions internally use the are.parpe3.valid function.

Usage

are.parpe3.valid(para, nowarn=FALSE)

Arguments

Value

Note

This function calls is.pe3 to verify consistency between the distribution parameter object and the intent of the user.

Author(s)

W.H. Asquith

References

Hosking, J.R.M., 1996, FORTRAN routines for use with the method of L-moments: Version 3, IBM Research Report RC20525, T.J. Watson Research Center, Yorktown Heights, New York.

Hosking, J.R.M., and Wallis, J.R., 1997, Regional frequency analysis—An approach based on L-moments: Cambridge University Press.

See Also

is.pe3, parpe3

Examples

para <- parpe3(lmoms(c(123,34,4,654,37,78)))
if(are.parpe3.valid(para)) Q <- quape3(0.5,para)

Description

Is the distribution parameter object consistent with the corresponding distribution? The distribution functions (cdfray, pdfray, quaray, and lmomray) require consistent parameters to return the cumulative probability (nonexceedance), density, quantile, and L-moments of the distribution, respectively. These functions internally use the are.parray.valid function.

Usage

are.parray.valid(para, nowarn=FALSE)

Arguments

Value

Note

This function calls is.ray to verify consistency between the distribution parameter object and the intent of the user.

Author(s)

W.H. Asquith

References

Hosking, J.R.M., 1986, The theory of probability weighted moments: Research Report RC12210, IBM Research Division, Yorkton Heights, N.Y.

See Also

is.ray, parray

Examples

para <- parray(lmoms(c(123,34,4,654,37,78)))
if(are.parray.valid(para)) Q <- quaray(0.5,para)

Description

Is the distribution parameter object consistent with the corresponding distribution? The distribution functions (cdfrevgum, pdfrevgum, quarevgum, and lmomrevgum) require consistent parameters to return the cumulative probability (nonexceedance), density, quantile, and L-moments of the distribution, respectively. These functions internally use the are.parrevgum.valid function.

Usage

are.parrevgum.valid(para, nowarn=FALSE)

Arguments

Value

Note

This function calls is.revgum to verify consistency between the distribution parameter object and the intent of the user.

Author(s)

W.H. Asquith

References

Hosking, J.R.M., 1995, The use of L-moments in the analysis of censored data, in Recent Advances in Life-Testing and Reliability, edited by N. Balakrishnan, chapter 29, CRC Press, Boca Raton, Fla., pp. 546–560.

See Also

is.revgum, parrevgum

Examples

para <- vec2par(c(.9252, .1636, .7),type='revgum')
if(are.parrevgum.valid(para)) Q <- quarevgum(0.5,para)

Description

Is the distribution parameter object consistent with the corresponding distribution? The distribution functions (cdfrice, pdfrice, quarice, and lmomrice) require consistent parameters to return the cumulative probability (nonexceedance), density, quantile, and L-moments of the distribution, respectively. These functions internally use the are.parrice.valid function.

Usage

are.parrice.valid(para, nowarn=FALSE)

Arguments

Value

Note

This function calls is.rice to verify consistency between the distribution parameter object and the intent of the user.

Author(s)

W.H. Asquith

References

Asquith, W.H., 2011, Distributional analysis with L-moment statistics using the R environment for statistical computing: Createspace Independent Publishing Platform, ISBN 978–146350841–8.

See Also

is.rice, parrice

Examples

#para <- parrice(lmoms(c(123,34,4,654,37,78)))
#if(are.parrice.valid(para)) Q <- quarice(0.5,para)

Description

Is the distribution parameter object consistent with the corresponding distribution? The distribution functions (cdfsla, pdfsla, quasla, and lmomsla) require consistent parameters to return the cumulative probability (nonexceedance), density, quantile, and L-moments of the distribution, respectively. These functions internally use the are.parsla.valid function.

Usage

are.parsla.valid(para, nowarn=FALSE)

Arguments

Value

Note

This function calls is.sla to verify consistency between the distribution parameter object and the intent of the user.

Author(s)

W.H. Asquith

References

Rogers, W.H., and Tukey, J.W., 1972, Understanding some long-tailed symmetrical distributions: Statistica Neerlandica, v. 26, no. 3, pp. 211–226.

See Also

is.sla, parsla

Examples

para <- vec2par(c(12,1.2),type='sla')
if(are.parsla.valid(para)) Q <- quasla(0.5,para)

Description

Is the distribution parameter object consistent with the corresponding distribution? The distribution functions (cdfsmd, pdfsmd, quasmd, and lmomsmd) require consistent parameters to return the cumulative probability (nonexceedance), density, quantile, and L-moments of the distribution, respectively. These functions internally use the are.parsmd.valid function. The parameter constraints are simple $a > 0$ (scale), $b > 0$ (shape), and $q > 0$ (shape).

Usage

are.parsmd.valid(para, nowarn=FALSE)

Arguments

Value

Note

This function calls is.smd to verify consistency between the distribution parameter object and the intent of the user.

Author(s)

W.H. Asquith

References

Shahzad, M.N., and Zahid, A., 2013, Parameter estimation of Singh Maddala distribution by moments: International Journal of Advanced Statistics and Probability, v. 1, no. 3, pp. 121–131, doi:10.14419/ijasp.v1i3.1206.

See Also

is.smd, parsmd

Examples

#para <- parsmd(lmoms(c(123, 34, 4, 654, 37, 78)))
#if(are.parsmd.valid(para)) Q <- quasmd(0.5, para)

Description

Is the distribution parameter object consistent with the corresponding distribution? The distribution functions (cdfst3, pdfst3, quast3, and lmomst3) require consistent parameters to return the cumulative probability (nonexceedance), density, quantile, and L-moments of the distribution, respectively. These functions internally use the are.parst3.valid function.

Usage

are.parst3.valid(para, nowarn=FALSE)

Arguments

Value

Note

This function calls is.st3 to verify consistency between the distribution parameter object and the intent of the user.

Author(s)

W.H. Asquith

References

Asquith, W.H., 2011, Distributional analysis with L-moment statistics using the R environment for statistical computing: Createspace Independent Publishing Platform, ISBN 978–146350841–8.

See Also

is.st3, parst3

Examples

para <- parst3(lmoms(c(90,134,100,114,177,378)))
if(are.parst3.valid(para)) Q <- quast3(0.5,para)

Description

Is the distribution parameter object consistent with the corresponding distribution? The distribution functions (cdftexp, pdftexp, quatexp, and lmomtexp) require consistent parameters to return the cumulative probability (nonexceedance), density, quantile, and L-moments of the distribution, respectively. These functions internally use the are.partexp.valid function.

Usage

are.partexp.valid(para, nowarn=FALSE)

Arguments

Value

Note

This function calls is.texp to verify consistency between the distribution parameter object and the intent of the user.

Author(s)

W.H. Asquith

References

Vogel, R.M., Hosking, J.R.M., Elphick, C.S., Roberts, D.L., and Reed, J.M., 2008, Goodness of fit of probability distributions for sightings as species approach extinction: Bulletin of Mathematical Biology, DOI 10.1007/s11538-008-9377-3, 19 p.

See Also

is.texp, partexp

Examples

para <- partexp(lmoms(c(90,134,100,114,177,378)))
if(are.partexp.valid(para)) Q <- quatexp(0.5,para)

Description

Is the distribution parameter object consistent with the corresponding distribution? The distribution functions (cdftri, pdftri, quatri, and lmomtri) require consistent parameters to return the cumulative probability (nonexceedance), density, quantile, and L-moments of the distribution, respectively. These functions internally use the are.partri.valid function.

Usage

are.partri.valid(para, nowarn=FALSE)

Arguments

Value

Note

This function calls is.tri to verify consistency between the distribution parameter object and the intent of the user.

Author(s)

W.H. Asquith

See Also

is.tri, partri

Examples

para <- partri(lmoms(c(46, 70, 59, 36, 71, 48, 46, 63, 35, 52)))
if(are.partri.valid(para)) Q <- quatri(0.5,para)

Description

Is the distribution parameter object consistent with the corresponding distribution? The distribution functions (cdfwak, pdfwak, quawak, and lmomwak) require consistent parameters to return the cumulative probability (nonexceedance), density, quantile, and L-moments of the distribution, respectively. These functions internally use the are.parwak.valid function.

Usage

are.parwak.valid(para, nowarn=FALSE)

Arguments

Value

Note

This function calls is.wak to verify consistency between the distribution parameter object and the intent of the user.

Author(s)

W.H. Asquith

References

Hosking, J.R.M., 1996, FORTRAN routines for use with the method of L-moments: Version 3, IBM Research Report RC20525, T.J. Watson Research Center, Yorktown Heights, New York.

Hosking, J.R.M., and Wallis, J.R., 1997, Regional frequency analysis—An approach based on L-moments: Cambridge University Press.

See Also

is.wak, parwak

Examples

para <- parwak(lmoms(c(123,34,4,654,37,78)))
if(are.parwak.valid(para)) Q <- quawak(0.5,para)

Description

Is the distribution parameter object consistent with the corresponding distribution? The distribution functions (cdfwei, pdfwei, quawei, and lmomwei) require consistent parameters to return the cumulative probability (nonexceedance), density, quantile, and L-moments of the distribution, respectively. These functions internally use the are.parwei.valid function.

Usage

are.parwei.valid(para, nowarn=FALSE)

Arguments

Value

Note

This function calls is.wei to verify consistency between the distribution parameter object and the intent of the user.

Author(s)

W.H. Asquith

References

Hosking, J.R.M., and Wallis, J.R., 1997, Regional frequency analysis—An approach based on L-moments: Cambridge University Press.

See Also

is.wei, parwei

Examples

para <- parwei(lmoms(c(123,34,4,654,37,78)))
if(are.parwei.valid(para)) Q <- quawei(0.5,para)

Description

This function computes the Barnes Extended Hypergeometric function, which in lmomco is useful in applications involving expectations of order statistics for the Generalized Exponential Poisson (GEP) distribution (see lmomgep). The function is

$F_{p,q}(\bm{\mathrm{n}},\bm{\mathrm{d}}; \lambda) = \sum_{k=0}^\infty \frac{\lambda^k}{\Gamma(k+1)}\frac{\Pi_{i=1}^{p} \Gamma(n_i + k)\Gamma^{-1}{(n_i)}}{\Pi_{i=1}^{q} \Gamma(d_i + k)\Gamma^{-1}{(d_i)}}\mbox{,}$

where $\bm{\mathrm{n}} = [n_1, n_2, \ldots, n_p]$ for $p$ operands and $\bm{\mathrm{d}} = [d_1, d_2, \ldots, d_q]$ for $q$ operands, and $\lambda > 0$ is a parameter.

Usage

BEhypergeo(p,q, N,D, lambda, eps=1E-12, maxit=500)

Arguments

Details

For the GEP both $\bm{\mathrm{n}}$ and $\bm{\mathrm{d}}$ are vectors of the same value, such as $\bm{\mathrm{n}} = [1, \ldots, 1]$ and $\bm{\mathrm{d}} = [2, \ldots, 2]$. This implementation is built around this need by the GEP and if the length of either vector is not equal to the operand then the first value of the vector is repeated the operand times. For example for $\bm{\mathrm{n}}$, if n = 1, then n = rep(n[1], length(p)) and so on for $\bm{\mathrm{d}}$. Given that n and d are vectorized for the GEP, then a shorthand is used for the GEP mathematics shown herein:

$F^{12}_{22}(h(j+1)) \equiv F_{2,2}([1,\ldots,1], [2,\ldots,2]; h(j+1))\mbox{,}$

for the $h$ parameter of the distribution.

Lastly, for lmomco and the GEP the arguments only involve $p = q = 2$ and $N = 1$, $D = 2$, so the function is uniquely a function of the $h$ parameter of the distribution:

  H <- 10^seq(-10,10, by=0.01)
  F22 <- sapply(1:length(H), function(i) BEhypergeo(2,2,1,1, H[i])$value
  plot(log10(H), log10(F22), type="l")

For this example, the solution increasingly wobbles towards large $h$, which is further explored by

  plot(log10(H[1:(length(H)-1)]), diff(log10(F22)), type="l", xlim=c(0,7))
  plot(log10(H[H > 75 & H < 140]), c(NA,diff(log10(F22[H > 75 & H < 140]))),
       type="b"); lines(c(2.11,2.11), c(0,10))

It can be provisionally concluded that the solution to $F^{12}_{22}(\cdot)$ begins to be suddenly questionable because of numerical difficulties beyond $\log(h) = 2.11$. Therefore, it is given that $h < 128$ might be an operational numerical upper limit.

Value

An R list is returned.

Author(s)

W.H. Asquith

References

Kus, C., 2007, A new lifetime distribution: Computational Statistics and Data Analysis, v. 51, pp. 4497–4509.

See Also

lmomgep

Examples

BEhypergeo(2,2,1,2,1.5)

Description

This function computes the Bonferroni Curve for quantile function $x(F)$ (par2qua, qlmomco). The function is defined by Nair et al. (2013, p. 179) as

$B(u) = \frac{1}{\mu u}\int_0^u x(p)\; \mathrm{d}p\mbox{,}$

where $B(u)$ is Bonferroni curve for quantile function $x(F)$ and $\mu$ is the conditional mean for quantile $u=0$ (cmlmomco). The Bonferroni curve is related to the Lorenz curve ($L(u)$, lrzlmomco) by

$B(u) = \frac{L(u)}{u}\mbox{.}$

Usage

bfrlmomco(f, para)

Arguments

Value

Bonferroni curve value for $F$.

Author(s)

W.H. Asquith

References

Nair, N.U., Sankaran, P.G., and Balakrishnan, N., 2013, Quantile-based reliability analysis: Springer, New York.

See Also

qlmomco, lrzlmomco

Examples

# It is easiest to think about residual life as starting at the origin, units in days.
A <- vec2par(c(0.0, 2649, 2.11), type="gov") # so set lower bounds = 0.0

"afunc" <- function(u) { return(par2qua(u,A,paracheck=FALSE)) }
f <- 0.65 # Both computations report: 0.5517342
Bu1 <- 1/(cmlmomco(f=0,A)*f) * integrate(afunc, 0, f)$value
Bu2 <- bfrlmomco(f, A)

Description

This function converts “B”-type probability-weighted moments (PWMs, $\beta^B_r$) to the “A”-type $\beta^A_r$. The $\beta^A_r$ are the ordinary PWMs for the $m$ left noncensored or observed values. The $\beta^B_r$ are more complex and use the $m$ observed values and the $m-n$ right-tailed censored values for which the censoring threshold is known. The “A”- and “B”-type PWMs are described in the documentation for pwmRC.

This function uses the defined relation between to two PWM types when the $\beta^B_r$ are known along with the parameters (para) of a right-tail censored distribution inclusive of the censoring fraction $\zeta=m/n$. The value $\zeta$ is the right-tail censor fraction or the probability $\mathrm{Pr}\lbrace \rbrace$ that $x$ is less than the quantile at $\zeta$ nonexceedance probability ($\mathrm{Pr}\lbrace x < X(\zeta) \rbrace$). The relation is

$\beta^A_{r-1} = \frac{r\beta^B_{r-1} - (1-\zeta^r)X(\zeta)}{r\zeta^r} \mbox{,}$

where $1 \le r \le n$ and $n$ is the number of moments, and $X(\zeta)$ is the value of the quantile function at nonexceedance probability $\zeta$. Finally, the RC in the function name is to denote Right-tail Censoring.

Usage

Bpwm2ApwmRC(Bpwm,para)

Arguments

Value

An R list is returned.

Author(s)

W.H. Asquith

References

Hosking, J.R.M., 1995, The use of L-moments in the analysis of censored data, in Recent Advances in Life-Testing and Reliability, edited by N. Balakrishnan, chapter 29, CRC Press, Boca Raton, Fla., pp. 546–560.

See Also

Apwm2BpwmRC and pwmRC

Examples

# Data listed in Hosking (1995, table 29.2, p. 551)
H <- c(3,4,5,6,6,7,8,8,9,9,9,10,10,11,11,11,13,13,13,13,13,
             17,19,19,25,29,33,42,42,51.9999,52,52,52)
      # 51.9999 was really 52, a real (noncensored) data point.
z <-  pwmRC(H,52)
# The B-type PMWs are used for the parameter estimation of the
# Reverse Gumbel distribution. The parameter estimator requires
# conversion of the PWMs to L-moments by pwm2lmom().
para <- parrevgum(pwm2lmom(z$Bbetas),z$zeta) # parameter object
Abetas <- Bpwm2ApwmRC(z$Bbetas,para)
Bbetas <- Apwm2BpwmRC(Abetas$betas,para)
# Assertion that both of the vectors of B-type PWMs should be the same.
str(Bbetas)   # B-type PWMs of the distribution
str(z$Bbetas) # B-type PWMs of the original data

Description

Annual maximum precipitation data for Canyon, Texas

Usage

data(canyonprecip)

Format

An R data.frame with

YEAR

The calendar year of the annual maxima.

DEPTH

The depth of 7-day annual maxima rainfall in inches.

References

Asquith, W.H., 1998, Depth-duration frequency of precipitation for Texas: U.S. Geological Survey Water-Resources Investigations Report 98–4044, 107 p.

Examples

data(canyonprecip)
summary(canyonprecip)

Description

Compute a single L-moment from a cumulative distribution function. This function is sequentially called by cdf2lmoms to mimic the output structure for multiple L-moments seen by other L-moment computation functions in lmomco.

For $r = 1$, the quantile function is actually used for numerical integration to compute the mean. The expression for the mean is

$\lambda_1 = \int_0^1 x(F)\; \mathrm{d} F\mbox{,}$

for quantile function $x(F)$ and nonexceedance probability $F$. For $r \ge 2$, the L-moments can be computed from the cumulative distribution function $F(x)$ by