lmomsmd: L-moments of the Singh--Maddala Distribution

Description

This function computes the L-moments of the Singh--Maddala (Burr Type XII) distribution given the parameters ($\xi$, $a$, $b$, and $q$) from parsmd. The first L-moment ($\lambda_1$) for $b' = 1/b$ and $R = a\Gamma(1 + b')$ is $$\lambda_1 = R\times\biggl[\frac{a\Gamma(1q-b')}{\Gamma(1q)}\biggr] + \xi\mbox{.}$$

The second L-moment ($\lambda_2$) is $$\lambda_2 = R\times\biggl[\frac{1\Gamma(1q - b')}{\Gamma(1q)} - \frac{1\Gamma(2q - b')}{\Gamma(2q)}\biggr]\mbox{.}$$

The third L-moment ($\lambda_3$) is $$\lambda_3 = R\times\biggl[\frac{1\Gamma(1q - b')}{\Gamma(1q)} - \frac{3\Gamma(2q - b')}{\Gamma(2q)} + \frac{2\Gamma(3q - b')}{\Gamma(3q)}\biggr]\mbox{.}$$

The fourth L-moment ($\lambda_4$) is $$\lambda_4 = R\times\biggl[\frac{ 1\Gamma(1q - b')}{\Gamma(1q)} - \frac{ 6\Gamma(2q - b')}{\Gamma(2q)} + \frac{10\Gamma(3q - b')}{\Gamma(3q)} - \frac{ 5\Gamma(4q - b')}{\Gamma(4q)}\biggr]\mbox{.}$$

The fifth L-moment ($\lambda_5$) (unique to lmomco development) is $$\lambda_5 = R\times\biggl[\frac{ 1\Gamma(1q - b')}{\Gamma(1q)} - \frac{10\Gamma(2q - b')}{\Gamma(2q)} + \frac{30\Gamma(3q - b')}{\Gamma(3q)} - \frac{35\Gamma(4q - b')}{\Gamma(4q)} + \frac{14\Gamma(5q - b')}{\Gamma(5q)}\biggr]\mbox{.}$$

The sixth L-moment ($\lambda_6$) (unique to lmomco development) is $$\lambda_6 = R\times\biggl[\frac{ 1\Gamma(1q - b')}{\Gamma(1q)} - \frac{ 15\Gamma(2q - b')}{\Gamma(2q)} + \frac{ 70\Gamma(3q - b')}{\Gamma(3q)} - \frac{140\Gamma(4q - b')}{\Gamma(4q)} +$$ $$\frac{126\Gamma(5q - b')}{\Gamma(5q)} - \frac{ 42\Gamma(6q - b')}{\Gamma(6q)}\biggr]\mbox{.}$$

Usage

lmomsmd(para)

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: “lmomsmd”.

Arguments

para: The parameters of the distribution.

Author

W.H. Asquith

References

Bhatti, F.A., Hamedani, G.G., Korkmaz, M.C., and Munir Ahmad, M., 2019, New modified Singh--Maddala distribution---Development, properties, characterizations, and applications: Journal of Data Science, v. 17, no. 3, pp. 551--574, tools:::Rd_expr_doi("10.6339/JDS.201907_17(3).0006").