This function computes the L-moments of the 4-parameter Asymmetric Exponential Power distribution given the parameters (\(\xi\), \(\alpha\), \(\kappa\), and \(h\)) from paraep4. The first four L-moments are complex. The mean \(\lambda_1\) is
$$\lambda_1 = \xi + \alpha(1/\kappa - \kappa)\frac{\Gamma(2/h)}{\Gamma(1/h)}\mbox{,}$$
where \(\Gamma(x)\) is the complete gamma function or gamma() in R.
The L-scale \(\lambda_2\) is
$$\lambda_2 = -\frac{\alpha\kappa(1/\kappa - \kappa)^2\Gamma(2/h)}
{(1+\kappa^2)\Gamma(1/h)}
+ 2\frac{\alpha\kappa^2(1/\kappa^3 + \kappa^3)\Gamma(2/h)I_{1/2}(1/h,2/h)}
{(1+\kappa^2)^2\Gamma(1/h)}\mbox{,}$$
where \(I_{1/2}(1/h,2/h)\) is the cumulative distribution function of the Beta distribution (\(I_x(a,b)\)) or pbeta(1/2, shape1=1/h, shape2=2/h) in R. This function is also referred to as the normalized incomplete beta function (Delicado and Goria, 2008) and defined as
$$I_x(a,b) = \frac{\int_0^x t^{a-1} (1-t)^{b-1}\; \mathrm{d}t}{\beta(a,b)}\mbox{,}$$
where \(\beta(1/h, 2/h)\) is the complete beta function or beta(1/h, 2/h) in R.
The third L-moment \(\lambda_3\) is $$\lambda_3 = A_1 + A_2 + A_3\mbox{,}$$ where the \(A_i\) are $$A_1 = \frac{\alpha(1/\kappa - \kappa)(\kappa^4 - 4\kappa^2 + 1)\Gamma(2/h)} {(1+\kappa^2)^2\Gamma(1/h)}\mbox{,}$$ $$A_2 = -6\frac{\alpha\kappa^3(1/\kappa - \kappa)(1/\kappa^3 + \kappa^3)\Gamma(2/h)I_{1/2}(1/h,2/h)} {(1+\kappa^2)^3\Gamma(1/h)}\mbox{,}$$ $$A_3 = 6\frac{\alpha(1+\kappa^4)(1/\kappa - \kappa)\Gamma(2/h)\Delta} {(1+\kappa^2)^2\Gamma(1/h)}\mbox{,}$$ and where \(\Delta\) is $$\Delta = \frac{1}{\beta(1/h, 2/h)}\int_0^{1/2} t^{1/h - 1} (1-t)^{2/h - 1} I_{(1-t)/(2-t)}(1/h, 3/h) \; \mathrm{d}t\mbox{.}$$
The fourth L-moment \(\lambda_4\) is
$$\lambda_4 = B_1 + B_2 + B_3 + B_4\mbox{,}$$
where the \(B_i\) are
$$B_1 = -\frac{\alpha\kappa(1/\kappa - \kappa)^2(\kappa^4 - 8\kappa^2 + 1)\Gamma(2/h)}
{(1+\kappa^2)^3\Gamma(1/h)}\mbox{,}$$
$$B_2 = 12\frac{\alpha\kappa^2(\kappa^3 + 1/\kappa^3)(\kappa^4 - 3\kappa^2 + 1)\Gamma(2/h)I_{1/2}(1/h,2/h)}
{(1+\kappa^2)^4\Gamma(1/h)}\mbox{,}$$
$$B_3 = -30\frac{\alpha\kappa^3(1/\kappa - \kappa)^2(1/\kappa^2 + \kappa^2)\Gamma(2/h)\Delta}
{(1+\kappa^2)^3\Gamma(1/h)}\mbox{,}$$
$$B_4 = 20\frac{\alpha\kappa^4(1/\kappa^5 + \kappa^5)\Gamma(2/h)\Delta_1}
{(1+\kappa^2)^4\Gamma(1/h)}\mbox{,}$$
and where \(\Delta_1\) is
$$\Delta_1 = \frac{\int_0^{1/2} \int_0^{(1-y)/(2-y)} y^{1/h - 1} (1-y)^{2/h - 1}
z^{1/h - 1} (1-z)^{3/h - 1}
\;I'\; \mathrm{d}z\,\mathrm{d}y}{\beta(1/h, 2/h)\beta(1/h, 3/h)}\mbox{,}$$
for which \(I' = I_{(1-z)(1-y)/(1+(1-z)(1-y))}(1/h, 2/h)\) is the cumulative distribution function of the beta distribution (\(I_x(a,b)\)) or pbeta((1-z)(1-y)/(1+(1-z)(1-y)), shape1=1/h, shape2=2/h) in R.
lmomaep4(para, paracheck=TRUE, t3t4only=FALSE)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: “lmomaep4”.
or an alternative R list is returned if t3t4only=TRUE
L-skew, \(\tau_3\).
L-kurtosis, \(\tau_4\).
The parameters of the distribution.
Should the parameters be checked for validity by the are.paraep4.valid function.
Return only the \(\tau_3\) and \(\tau_4\) for the parameters \(\kappa\) and \(h\). The \(\lambda_1\) and \(\lambda_2\) are not explicitly used although numerical values for these two L-moments are required only to avoid computational errors. Care is made so that the \(\alpha\) parameter that is in numerator of \(\lambda_{2,3,4}\) is not used in the computation of \(\tau_3\) and \(\tau_4\). Hence, this option permits the computation of \(\tau_3\) and \(\tau_4\) when \(\alpha\) is unknown. This features is largely available for research and development purposes. Mostly this feature was used for the \(\{\tau_3, \tau_4\}\) trajectory for lmrdia
.
W.H. Asquith
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.
paraep4, cdfaep4, pdfaep4, quaaep4
if (FALSE) {
para <- vec2par(c(0, 1, 0.5, 4), type="aep4")
lmomaep4(para)
}
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