lmomaep4: L-moments of the 4-p Asymmetric Exponential Power Distribution

Description

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 of the distribution are complex.

The mean $\lambda_1$ is

$$\lambda_1 = \xi + \alpha(1/\kappa - \kappa)\frac{\Gamma(2/h)}{\Gamma(1/h)}$$

where $\Gamma(x)$ is the complete gamma function or gamma() in R.

The L-scale $\lambda_2$ is

$$\lambda_2 = -\frac{\alpha\kappa(1/\kappa - \kappa)^2\Gamma(2/h)} {(1+\kappa^2)\Gamma(1/h)} + 2\frac{\alpha\kappa^2(1/\kappa^3 + \kappa^3)\Gamma(2/h)I_{1/2}(1/h,2/h)} {(1+\kappa^2)^2\Gamma(1/h)}$$

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

$$I_x(a,b) = \frac{\int_0^x t^{a-1} (1-t)^{b-1}\; \mathrm{d}t}{\beta(a,b)}$$

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

$$\lambda_3 = A_1 + A_2 + A_3$$

where the $A_i$ are

$$A_1 = \frac{\alpha(1/\kappa - \kappa)(\kappa^4 - 4\kappa^2 + 1)\Gamma(2/h)} {(1+\kappa^2)^2\Gamma(1/h)}$$

$$A_2 = -6\frac{\alpha\kappa^3(1/\kappa - \kappa)(1/\kappa^3 + \kappa^3)\Gamma(2/h)I_{1/2}(1/h,2/h)} {(1+\kappa^2)^3\Gamma(1/h)}$$

$$A_3 = 6\frac{\alpha(1+\kappa^4)(1/\kappa - \kappa)\Gamma(2/h)\Delta} {(1+\kappa^2)^2\Gamma(1/h)}$$

and where $\Delta$ is

$$\Delta = \frac{1}{\beta(1/h, 2/h)}\int_0^{1/2} t^{1/h - 1} (1-t)^{2/h - 1} I_{(1-t)/(2-t)}(1/h, 3/h) \; \mathrm{d}t$$

The fourth L-moment $\lambda_4$ is

$$\lambda_4 = B_1 + B_2 + B_3 + B_4$$

where the $B_i$ are

$$B_1 = -\frac{\alpha\kappa(1/\kappa - \kappa)^2(\kappa^4 - 8\kappa^2 + 1)\Gamma(2/h)} {(1+\kappa^2)^3\Gamma(1/h)}$$

$$B_2 = 12\frac{\alpha\kappa^2(\kappa^3 + 1/\kappa^3)(\kappa^4 - 3\kappa^2 + 1)\Gamma(2/h)I_{1/2}(1/h,2/h)} {(1+\kappa^2)^4\Gamma(1/h)}$$

$$B_3 = -30\frac{\alpha\kappa^3(1/\kappa - \kappa)^2(1/\kappa^2 + \kappa^2)\Gamma(2/h)\Delta} {(1+\kappa^2)^3\Gamma(1/h)}$$

$$B_4 = 20\frac{\alpha\kappa^4(1/\kappa^5 + \kappa^5)\Gamma(2/h)\Delta_1} {(1+\kappa^2)^4\Gamma(1/h)}$$

and where $\Delta_1$ is

$$\Delta_1 = \frac{\int_0^{1/2} \int_0^{(1-y)/(2-y)} y^{1/h - 1} (1-y)^{2/h - 1} z^{1/h - 1} (1-z)^{3/h - 1} \;I'\; \mathrm{d}z\,\mathrm{d}y}{\beta(1/h, 2/h)\beta(1/h, 3/h)}$$

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.

Usage

lmomaep4(para, paracheck=TRUE, t3t4only=FALSE)

Arguments

para

The parameters of the distribution.

paracheck

Should the parameters be checked for validity by the are.paraep4.valid function.

t3t4only

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

Value

An Rlist is returned.
lambdasVector of the L-moments. First element is $\lambda_1$, second element is $\lambda_2$, and so on.
ratiosVector of the L-moment ratios. Second element is $\tau$, third element is $\tau_3$ and so on.
trimTrim level = 0
leftrimLeft trimming level = 0
rightrimRight trimming level = 0
sourceAn attribute identifying the computational source of the L-moments: lmomaep4.
or an alternative Rlist is returned if t3t4only=TRUE
T3L-skew, $\tau_3$
T4L-kurtosis, $\tau_4$

References

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.

Examples

Run this code

para <- vec2par(c(0, 1, 0.5, 4), type="aep4")
lmomaep4(para)

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