This function estimates the L-moments of the Laplace distribution given the parameters (\(\xi\) and \(\alpha\)) from parlap. The L-moments in terms of the parameters are
\(\lambda_1 = \xi\),
\(\lambda_2 = 3\alpha/4\),
\(\tau_3 = 0\),
\(\tau_4 = 17/22\),
\(\tau_5 = 0\), and
\(\tau_6 = 31/360\).
For \(r\) odd and \(r \ge 3\), \(\lambda_r = 0\), and for \(r\) even and \(r \ge 4\), the L-moments using the hypergeometric function \({}_2F_1()\) are $$\lambda_r = \frac{2\alpha}{r(r-1)}[1 - {}_2F_1(-r, r-1, 1, 1/2)]\mbox{,}$$ where \({}_2F_1(a, b, c, z)\) is defined as $${}_2F_1(a, b, c, z) = \sum_{n=0}^\infty \frac{(a)_n(b)_n}{(c)_n}\frac{z^n}{n!}\mbox{,}$$ where \((x)_n\) is the rising Pochhammer symbol, which is defined by $$(x)_n = 1 \mbox{\ for\ } n = 0\mbox{, and}$$ $$(x)_n = x(x+1)\cdots(x+n-1) \mbox{\ for\ } n > 0\mbox{.}$$
lmomlap(para)An R
list is returned.
Vector of the L-moments. First element is \(\lambda_1\), second element is \(\lambda_2\), and so on.
Vector of the L-moment ratios. Second element is \(\tau\), third element is \(\tau_3\) and so on.
Level of symmetrical trimming used in the computation, which is 0.
Level of left-tail trimming used in the computation, which is NULL.
Level of right-tail trimming used in the computation, which is NULL.
An attribute identifying the computational source of the L-moments: “lmomlap”.
The parameters of the distribution.
W.H. Asquith
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
lmr <- lmoms(c(123,34,4,654,37,78))
lmr
lmomlap(parlap(lmr))
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