parlap: Estimate the Parameters of the Laplace Distribution

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

Estimate the Parameters of the Laplace Distribution

Description

This function computes the parameters of the Laplace distribution given the L-moments of the data in an L-moment object such as that returned by lmoms. Relations between distribution parameters and sample L-moments are simple, but there are two methods to discuss.

Basic theoretical formula of L-moment are simple for the Laplace and only use \lambda_1 and \lambda_2, but these are used if minvar=FALSE:

\xi = \lambda_1\mbox{\ and\ } \alpha = 4\lambda_2/3\mbox{.}

Hosking (1986, p. 56) reports these estimators as inefficient, and more efficient estimators can be obtained. Hosking (1986, p. 57) provides minimum-variance unbiased linear estimators with the best combination being shown below, and these are suggested for estimation the default implemented in lmomco. This minimum variance method jointly uses \lambda_1, \lambda_2, \lambda_3, and \lambda_4, and these are minimum variance unbiased estimators by Hosking (1986). The mathematical expressions are

\xi = \lambda_1 - 50/31\times\lambda_3 \mbox{\ and}

\alpha = 1.4741\lambda_2 - 0.5960\lambda_4 = 1309/888\times\lambda_2 - 149/250\times\lambda_4\mbox{.}

Astute readers will notice a fascinating aspect with these estimators. Consider that the distribution is symmetrical and therefore \lambda_3 = 0 \equiv \lambda_3/\lambda_2 = \tau_3 = 0, yet for a minimum-variance estimator, the \lambda_3 resides within the estimation scheme.

Usage

parlap(lmom, checklmom=TRUE, minvar=TRUE, ...)

Arguments

lmom	An L-moment object created by lmoms or vec2lmom.
checklmom	Should the lmom be checked for validity using the are.lmom.valid function. Normally, this should be left as the default and it is very unlikely that the L-moments will not be viable (particularly in the \tau_4 and \tau_3 inequality). However, for some circumstances or large simulation exercises then one might want to bypass this check.
minvar	A logical to use the simpler parameter estimation scheme shown above and this minvar will need to be set true if one where to check against output of theoLmoms.
...	Other arguments to pass.

Value

An R list is returned.

type	The type of distribution: lap.
para	The parameters of the distribution.
source	The source of the parameters: “parlap”.

Note

The decision to use only one of the two systems of equations for Laplace fitting is largely arbitrary, but it seems most fitting to use four L-moments instead of two. The Examples section provides evidence that the the minvar=TRUE is an appropriate default because we get smaller variance on the parameters at a potential cost in minor change in bias.

We can cross check the basic definition of the L-moments of the distribution (refer to minvar=FALSE) by the recipe in this section and noting (lmomlap) that \lambda_1 = \xi, \lambda_2 = 3\alpha/4, \tau_3 = 0, \tau_4 = 17/72, \tau_5 = 0, and \tau_6 = 31/360. The results show reversibility between parameters and L-moments, which does not occur by the relations for minvar=TRUE.

  lap  <- vec2par(c(1, 1), type="lap") # without loss of generalization
  lmomlap(lap) # L1=U, L2=3*A/4, T3=T5=0, T4=17/72=0.2361111, T6=31/360=0.0861111
  # $lambdas [1] 1.00000000 0.75000000 0.00000000 0.17708333 0.00000000 0.0645833
  # $ratios  [1] 0.00000000 0.75000000 0.00000000 0.23611111 0.00000000 0.0861111
  theoLmoms(lap, nmom=6) # by numerical integration of base L-moment definitions
  # $lambdas [1] 1.000e+00 0.7499999  -6.4*e-17  0.1770829  -1.0*e-15  0.06459555
  # $ratios  [1]        NA 0.7499999  -8.6*e-17  0.2361106  -1.4*e-15  0.08612742

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

lmomlap, cdflap, pdflap, qualap

Examples

lmr <- lmoms(rnorm(20))
parlap(lmr)

## Not run: 
  # Hosking (1986) shows by theory minimum variance estimators are preferred.
  # This is a general conclusion to be drawn from the simulations shown below.
  # plotlmrdia(lmrdia()) # commented out for more speed
  X <- 1; A <- 1; lap <- vec2par(c(1, 1), type="lap")
  nsim <- 50000; nsam <- c(10, 30)
  for(n in nsam) { set.seed(1)
    x1 <- x2 <- a1 <- a2 <- vector(mode="numeric", length=nsim)
    for(i in seq_len(nsim)) {
      q <- rlmomco(n, para=lap); lmr <- lmoms(q, no.stop=TRUE)
      points(lmr$ratios[3], lmr$ratios[4], pch=16, col=grey(0,0.2), cex=0.4)
      # Note, in very small samples, we can get situations of Tau4 beyond the
      # theoretical limits. In the setup here, we accept the Tau4 as is and
      # do not snap it to the boundary. For n=10, we see very few and for n=30,
      # see no such situations. Uncomment the plotting operations to see more.
      par1  <- parlap(lmr, checklmom=FALSE, minvar=TRUE )$para # minvar below
      par2  <- parlap(lmr, checklmom=FALSE, minvar=FALSE)$para # direct below
      x1[i] <- par1[1]; a1[i] <- par1[2]; x2[i] <- par2[1]; a2[i] <- par2[2]
    }
    message("n=", n)
    message("minvar: Bias[x1]=", mean(X - x1), " and Var[x1]=", var(x1))
    message("direct: Bias[x2]=", mean(X - x2), " and Var[x2]=", var(x2))
    message("minvar: Bias[a1]=", mean(A - a1), " and Var[a1]=", var(a1))
    message("direct: Bias[a2]=", mean(A - a2), " and Var[a2]=", var(a2))
  }
  # n=10
  # minvar: Bias[x1]=+0.00112636824968240 and Var[x1]=0.149853593366873
  # direct: Bias[x2]=+0.00147737388905255 and Var[x2]=0.198993001247171
  # minvar: Bias[a1]=+0.00025675625790722 and Var[a1]=0.106629132387976
  # direct: Bias[a2]=-0.00011732500233820 and Var[a2]=0.108899468215813
  # n=30
  # minvar: Bias[x1]=+0.00099647413073717 and Var[x1]=0.045870359664270
  # direct: Bias[x2]=+0.00178495637700070 and Var[x2]=0.066410718497819
  # minvar: Bias[a1]=+0.00074862540197578 and Var[a1]=0.033678344848655
  # direct: Bias[a2]=+0.00079661893306864 and Var[a2]=0.034749047433539 #
## End(Not run)

lmomco documentation built on Oct. 18, 2026, 5:07 p.m.