# lmomlap: L-moments of the Laplace Distribution

### Description

This function estimates the L-moments of the Laplace distribution given the parameters (ξ and α) from parlap. The L-moments in terms of the parameters are λ_1 = ξ, λ_2 = 3α/4, τ_3 = 0, τ_4 = 17/22, τ_5 = 0, and τ_6 = 31/360.

For r odd and r ≥ 3, λ_r = 0, and for r even and r ≥ 4, the L-moments using the hypergeometric function {}_2F_1() are

λ_r = \frac{2α}{r(r-1)}[1 - {}_2F_1(-r, r-1, 1, 1/2)]\mbox{,}

where {}_2F_1(a, b, c, z) is defined as

{}_2F_1(a, b, c, z) = ∑_{n=0}^∞ \frac{(a)_n(b)_n}{(c)_n}\frac{z^n}{n!}\mbox{,}

where (x)_n is the rising Pochhammer symbol, which is defined by

(x)_n = 1 \mbox{\ for\ } n = 0\mbox{, and}

(x)_n = x(x+1)\cdots(x+n-1) \mbox{\ for\ } n > 0\mbox{.}

### Usage

 1 lmomlap(para) 

### Arguments

 para The parameters of the distribution.

### Value

An R list is returned.

 lambdas Vector of the L-moments. First element is λ_1, second element is λ_2, and so on. ratios Vector of the L-moment ratios. Second element is τ, third element is τ_3 and so on. trim Level of symmetrical trimming used in the computation, which is 0. leftrim Level of left-tail trimming used in the computation, which is NULL. rightrim Level of right-tail trimming used in the computation, which is NULL. source An attribute identifying the computational source of the L-moments: “lmomlap”.

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.

parlap, cdflap, pdflap, qualap

### Examples

 1 2 3 lmr <- lmoms(c(123,34,4,654,37,78)) lmr lmomlap(parlap(lmr)) 

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