lmomco (version 2.0.1)

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

$${\lambda }_1=\xi +\alpha (1/\kappa -\kappa )\frac{\mathrm{\Gamma }(2/h)}{\mathrm{\Gamma }(1/h)}$$

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

The L-scale $\lambda_2$ is

$${\lambda }_2=-\frac{\alpha \kappa (1/\kappa -\kappa {)}^2\mathrm{\Gamma }(2/h)}{(1+{\kappa }^2)\mathrm{\Gamma }(1/h)}+2\frac{\alpha {\kappa }^2(1/{\kappa }^3+{\kappa }^3)\mathrm{\Gamma }(2/h)I_{1/2}(1/h,2/h)}{(1+{\kappa }^2{)}^2\mathrm{\Gamma }(1/h)}$$

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

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

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

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

and where $\Delta$ is

$$\mathrm{\Delta }=\frac{1}{\beta (1/h,2/h)}{\int }_0^{1/2}{t}^{1/h-1}(1-t{)}^{2/h-1}{I}_{(1-t)/(2-t)}(1/h,3/h)\mathrm{d}t$$

The fourth L-moment $\lambda_4$ is

$${\lambda }_4=B_1+B_2+B_3+B_4$$

where the $B_i$ are

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

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

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

$$B_4=20\frac{\alpha {\kappa }^4(1/{\kappa }^5+{\kappa }^5)\mathrm{\Gamma }(2/h){\mathrm{\Delta }}_1}{(1+{\kappa }^2{)}^4\mathrm{\Gamma }(1/h)}$$

and where $\Delta_1$ is

$${\mathrm{\Delta }}_1=\frac{{\int }_0^{1/2}{\int }_0^{(1-y)/(2-y)}{y}^{1/h-1}(1-y{)}^{2/h-1}{z}^{1/h-1}(1-z{)}^{3/h-1}{I}^{\prime }\mathrm{d}z\mathrm{d}y}{\beta (1/h,2/h)\beta (1/h,3/h)}$$

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.
• 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
  Trim level = 0
• leftrim
  Left trimming level = 0
• rightrim
  Right trimming level = 0
• source
  An attribute identifying the computational source of the L-moments: lmomaep4.
• or an alternative Rlist is returned if t3t4only=TRUE
• T3
  L-skew, $\tau_3$
• T4
  L-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.

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.

See Also

paraep4, quaaep4, cdfaep4

Examples

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