This function estimates the L-moments of the Generalized Exponential Poisson (GEP) distribution given the parameters (\(\beta\), \(\kappa\), and \(h\)) from pargep. The L-moments in terms of the parameters are best expressed in terms of the expectations of order statistic maxima \(\mathrm{E}[X_{n:n}]\) for the distribution. The fundamental relation is
$$\lambda_r = \sum_{k=1}^r (-1)^{r-k}k^{-1}{r-1 \choose k-1}{r+k-2 \choose k-1}\mathrm{E}[X_{k:k}]\mbox{.}$$
The L-moments do not seem to have been studied for the GEP. The challenge is the solution to \(\mathrm{E}[X_{n:n}]\) through an expression by Barreto-Souza and Cribari-Neto (2009) that is
$$\mathrm{E}[X_{n:n}] = \frac{\beta\,h\,\Gamma(\kappa+1)\,\Gamma(n\kappa + 1)}{n\,\Gamma(n)\,(1 - \exp(-h))^{n\kappa}}\sum_{j=0}^{\infty} \frac{(-1)^j\exp(-h(j+1))}{\Gamma(n\kappa - j)\,\Gamma(j+1)}\;F^{12}_{22}(h(j+1))\mbox{,}$$
where \(F^{12}_{22}(h(j+1))\) is the Barnes Extended Hypergeometric function with arguments reflecting those needed for the GEP (see comments under BEhypergeo).
lmomgep(para, byqua=TRUE)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: “lmomgep”.
The parameters of the distribution.
A logical triggering the theoLmoms.max.ostat instead of using the mathematics of Barreto-Souza and Cribari-Neto (2009) (see Details).
W.H. Asquith
The mathematics (not of L-moments but \(\mathrm{E}[X_{n:n}]\)) shown by Barreto-Souza and Cribari-Neto (2009) are correct but are apparently subject to considerable numerical issues even with substantial use of logarithms and exponentiation in favor of multiplication and division in the above formula for \(\mathrm{E}[X_{n:n}]\). Testing indicates that numerical performance is better if the non-\(j\)-dependent terms in the infinite sum remain inside it. Testing also indicates that the edges of performance can be readily hit with large \(\kappa\) and less so with large \(h\). It actually seems superior to not use the above equation for L-moment computation based on \(\mathrm{E}[X_{n:n}]\) but instead rely on expectations of maxima order statistics (expect.max.ostat) from numerical integration of the quantile function (quagep) as is implementated in theoLmoms.max.ostat. This is the reason that the byqua argument is available and set to the shown default. Because the GEP is experimental, this function provides two approaches for \(\lambda_r\) computation for research purposes.
Barreto-Souza, W., and Cribari-Neto, F., 2009, A generalization of the exponential-Poisson distribution: Statistics and Probability, 79, pp. 2493--2500.
pargep, cdfgep, pdfgep, quagep