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 (ξ, α, κ, and h) from paraep4. The first four L-moments are complex. The mean λ_1 is

λ_1 = ξ + α(1/κ - κ)\frac{Γ(2/h)}{Γ(1/h)}\mbox{,}

where Γ(x) is the complete gamma function or gamma() in R.

The L-scale λ_2 is

λ_2 = -\frac{ακ(1/κ - κ)^2Γ(2/h)} {(1+κ^2)Γ(1/h)} + 2\frac{ακ^2(1/κ^3 + κ^3)Γ(2/h)I_{1/2}(1/h,2/h)} {(1+κ^2)^2Γ(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}{β(a,b)}\mbox{,}

where β(1/h, 2/h) is the complete beta function or beta(1/h, 2/h) in R.

The third L-moment λ_3 is

λ_3 = A_1 + A_2 + A_3\mbox{,}

where the A_i are

A_1 = \frac{α(1/κ - κ)(κ^4 - 4κ^2 + 1)Γ(2/h)} {(1+κ^2)^2Γ(1/h)}\mbox{,}

A_2 = -6\frac{ακ^3(1/κ - κ)(1/κ^3 + κ^3)Γ(2/h)I_{1/2}(1/h,2/h)} {(1+κ^2)^3Γ(1/h)}\mbox{,}

A_3 = 6\frac{α(1+κ^4)(1/κ - κ)Γ(2/h)Δ} {(1+κ^2)^2Γ(1/h)}\mbox{,}

and where Δ is

Δ = \frac{1}{β(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 λ_4 is

λ_4 = B_1 + B_2 + B_3 + B_4\mbox{,}

where the B_i are

B_1 = -\frac{ακ(1/κ - κ)^2(κ^4 - 8κ^2 + 1)Γ(2/h)} {(1+κ^2)^3Γ(1/h)}\mbox{,}

B_2 = 12\frac{ακ^2(κ^3 + 1/κ^3)(κ^4 - 3κ^2 + 1)Γ(2/h)I_{1/2}(1/h,2/h)} {(1+κ^2)^4Γ(1/h)}\mbox{,}

B_3 = -30\frac{ακ^3(1/κ - κ)^2(1/κ^2 + κ^2)Γ(2/h)Δ} {(1+κ^2)^3Γ(1/h)}\mbox{,}

B_4 = 20\frac{ακ^4(1/κ^5 + κ^5)Γ(2/h)Δ_1} {(1+κ^2)^4Γ(1/h)}\mbox{,}

and where Δ_1 is

Δ_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}{β(1/h, 2/h)β(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.

Usage

1
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 τ_3 and τ_4 for the parameters κ and h. The λ_1 and λ_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 α parameter that is in numerator of λ_{2,3,4} is not used in the computation of τ_3 and τ_4. Hence, this option permits the computation of τ_3 and τ_4 when α is unknown. This features is largely available for research and development purposes. Mostly this feature was used for the \{τ_3, τ_4\} trajectory for lmrdia

.

Value

An R list is returned.

lambdas

Vector of the L-moments. First element is λ_1, second element is λ_2, and so on.

ratios

Vector of the L-moment ratios. Second element is τ, third element is τ_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, τ_3.

T4

L-kurtosis, τ_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

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

## End(Not run)

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