lmomaep4 {lmomco}

L-moments of the 4-Parameter 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 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. Finally, if the \tau_3 of the distribution is zero (symmetrical), then the distribution is known as the Exponential Power (see lmrdia46).

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 \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

.

Value

An R list is returned.

lambdas

Vector of the L-moments. First element is \lambda_1, second element is \lambda_2, and so on.

ratios

Vector of the L-moment ratios. Second element is \tau, third element is \tau_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: “lmomaep4”.

or an alternative R list is returned if t3t4only=TRUE

T3

L-skew, \tau_3.

T4

L-kurtosis, \tau_4.

Author(s)

W.H. Asquith

References

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.

See Also

paraep4, cdfaep4, pdfaep4, quaaep4

Examples

## Not run: 
para <- vec2par(c(0, 1, 0.5, 4), type="aep4")
lmomaep4(para)

## End(Not run)