lmomaep4 {lmomco} | R Documentation |
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 λ1
is
λ1=ξ+α(1/κ−κ)Γ(1/h)Γ(2/h)\mbox,
where Γ(x)
is the complete gamma function or gamma()
in R.
The L-scale λ2
is
λ2=−(1+κ2)Γ(1/h)ακ(1/κ−κ)2Γ(2/h)+2(1+κ2)2Γ(1/h)ακ2(1/κ3+κ3)Γ(2/h)I1/2(1/h,2/h)\mbox,
where I1/2(1/h,2/h)
is the cumulative distribution function of the Beta distribution (Ix(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
Ix(a,b)=β(a,b)∫0xta−1(1−t)b−1dt\mbox,
where β(1/h,2/h)
is the complete beta function or beta(1/h, 2/h)
in R.
The third L-moment λ3
is
λ3=A1+A2+A3\mbox,
where the Ai
are
A1=(1+κ2)2Γ(1/h)α(1/κ−κ)(κ4−4κ2+1)Γ(2/h)\mbox,
A2=−6(1+κ2)3Γ(1/h)ακ3(1/κ−κ)(1/κ3+κ3)Γ(2/h)I1/2(1/h,2/h)\mbox,
A3=6(1+κ2)2Γ(1/h)α(1+κ4)(1/κ−κ)Γ(2/h)Δ\mbox,
and where Δ
is
Δ=β(1/h,2/h)1∫01/2t1/h−1(1−t)2/h−1I(1−t)/(2−t)(1/h,3/h)dt\mbox.
The fourth L-moment λ4
is
λ4=B1+B2+B3+B4\mbox,
where the Bi
are
B1=−(1+κ2)3Γ(1/h)ακ(1/κ−κ)2(κ4−8κ2+1)Γ(2/h)\mbox,
B2=12(1+κ2)4Γ(1/h)ακ2(κ3+1/κ3)(κ4−3κ2+1)Γ(2/h)I1/2(1/h,2/h)\mbox,
B3=−30(1+κ2)3Γ(1/h)ακ3(1/κ−κ)2(1/κ2+κ2)Γ(2/h)Δ\mbox,
B4=20(1+κ2)4Γ(1/h)ακ4(1/κ5+κ5)Γ(2/h)Δ1\mbox,
and where Δ1
is
Δ1=β(1/h,2/h)β(1/h,3/h)∫01/2∫0(1−y)/(2−y)y1/h−1(1−y)2/h−1z1/h−1(1−z)3/h−1I′dzdy\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 (Ix(a,b)
) or pbeta((1-z)(1-y)/(1+(1-z)(1-y)), shape1=1/h, shape2=2/h)
in R. Finally, if the τ3
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 τ3 and τ4 for the parameters κ and h . The λ1 and λ2 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 λ2,3,4 is not used in the computation of τ3 and τ4 . Hence, this option permits the computation of τ3 and τ4 when α is unknown. This features is largely available for research and development purposes. Mostly this feature was used for the {τ3,τ4} trajectory for lmrdia
|
.
Value
An R list
is returned.
lambdas |
Vector of the L-moments. First element is
λ1 , second element is λ2 , and so on.
|
ratios |
Vector of the L-moment ratios. Second element is
τ , third element is τ3 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, τ3 .
|
T4 |
L-kurtosis, τ4 .
|
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)
[Package
lmomco version 2.5.1
Index]