# L-moments of the Singh--Maddala Distribution 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 [REMOVE_ME]$$ \lambda_1 = R\times\biggl[\frac{a\Gamma(1q-b')}{\Gamma(1q)}\biggr] + \xi\mbox{.} [REMOVE_ME_2]$$ The second L-moment ($\lambda_2$) is [REMOVE_ME]$$ \lambda_2 = R\times\biggl[\frac{1\Gamma(1q - b')}{\Gamma(1q)} -\frac{1\Gamma(2q - b')}{\Gamma(2q)}\biggr]\mbox{.} [REMOVE_ME_2]$$ The third L-moment ($\lambda_3$) is [REMOVE_ME]$$ \lambda_3 = R\times\biggl[\frac{1\Gamma(1q - b')}{\Gamma(1q)} -\frac{3\Gamma(2q - b')}{\Gamma(2q)} +\frac{2\Gamma(3q - b')}{\Gamma(3q)}\biggr]\mbox{.} [REMOVE_ME_2]$$ The fourth L-moment ($\lambda_4$) is [REMOVE_ME]$$ \lambda_4 = R\times\biggl[\frac{ 1\Gamma(1q - b')}{\Gamma(1q)} -\frac{ 6\Gamma(2q - b')}{\Gamma(2q)} +\frac{10\Gamma(3q - b')}{\Gamma(3q)} -\frac{ 5\Gamma(4q - b')}{\Gamma(4q)}\biggr]\mbox{.} [REMOVE_ME_2]$$ The fifth L-moment ($\lambda_5$) (unique to `lmomco` development) is [REMOVE_ME]$$ \lambda_5 = R\times\biggl[\frac{ 1\Gamma(1q - b')}{\Gamma(1q)} -\frac{10\Gamma(2q - b')}{\Gamma(2q)} +\frac{30\Gamma(3q - b')}{\Gamma(3q)} -\frac{35\Gamma(4q - b')}{\Gamma(4q)} +\frac{14\Gamma(5q - b')}{\Gamma(5q)}\biggr]\mbox{.} [REMOVE_ME_2]$$ The sixth L-moment ($\lambda_6$) (unique to `lmomco` development) is [REMOVE_ME]$$ \lambda_6 = R\times\biggl[\frac{ 1\Gamma(1q - b')}{\Gamma(1q)} -\frac{ 15\Gamma(2q - b')}{\Gamma(2q)} +\frac{ 70\Gamma(3q - b')}{\Gamma(3q)} -\frac{140\Gamma(4q - b')}{\Gamma(4q)} + [REMOVE_ME_2]$$ [REMOVE_ME]$$ \frac{126\Gamma(5q - b')}{\Gamma(5q)} -\frac{ 42\Gamma(6q - b')}{\Gamma(6q)}\biggr]\mbox{.} [REMOVE_ME_2]$$ ## 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 $$ \lambda_6 = R\times\biggl[\frac{ 1\Gamma(1q - b')}{\Gamma(1q)} -\frac{ 15\Gamma(2q - b')}{\Gamma(2q)} +\frac{ 70\Gamma(3q - b')}{\Gamma(3q)} -\frac{140\Gamma(4q - b')}{\Gamma(4q)} + $$ $$ \frac{126\Gamma(5q - b')}{\Gamma(5q)} -\frac{ 42\Gamma(6q - b')}{\Gamma(6q)}\biggr]\mbox{.} $$ ```r lmomsmd(para) ``` ## Arguments - `para`: The parameters of the distribution. ## Returns An `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 . ## References Bhatti, F.A., Hamedani, G.G., Korkmaz, M. ., 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") . Shahzad, M.N., and Zahid, A., 2013, Parameter estimation of Singh Maddala distribution by moments: International Journal of Advanced Statistics and Probability, v. 1, no. 3, pp. 121--131, tools:::Rd_expr_doi("10.14419/ijasp.v1i3.1206") . ## Author(s) W.H. Asquith ## See Also `parsmd`, `cdfsmd`, `pdfsmd`, `quasmd` ## Examples ```r lmr <- lmoms(c(123, 34, 4, 654, 37, 78), nmom=6) lmr$source <- lmr$trim <- lmr$leftrim <- lmr$rightrim <-NULL # The parsmd() reports Tau4 is too big and snaps it to an empirical boundary. # "Tau4(~Tau3) snapped to upper limit, Tau4=0.65483 for Tau3=0.75126" bmr <- lmomsmd(parsmd(lmr, snap.tau4=TRUE)) dmr <- data.frame(bmr$lambdas, bmr$ratios) cbind(as.data.frame(lmr), dmr) # See in table that row 4 has different Tau4s # lambdas ratios bmr.lambdas bmr.ratios # 1 155.0 NA 155.00000 NA # 2 118.6 0.7651613 118.60000 0.7651613 # 3 89.1 0.7512648 89.18739 0.7520016 # 4 82.1 0.6922428 77.59904 0.6542921 # see different Tau4s (snapping) # 5 69.5 0.5860034 68.40150 0.5767411 # We are not fitting to these # 6 102.5 0.8642496 62.58792 0.5277228 # higher L-moments. # T3 and T4 of the Gumbel distribution, which is inside the SMD domain. gumt3t4 <- c(log(9/8)/log(2), (16 * log(2) - 10 * log(3))/log(2)) lmr <- theoLmoms(pargum(vec2lmom(c(155, 118.6, gumt3t4))), nmom=6) lmr$source <- lmr$trim <- lmr$leftrim <- lmr$rightrim <-NULL bmr <- lmomsmd(parsmd(lmr, snap.tau4=TRUE)) dmr <- data.frame(bmr$lambdas, bmr$ratios) cbind(as.data.frame(lmr), dmr) # lambdas ratios bmr.lambdas bmr.ratios # 1 155.000000 NA 155.000000 NA # 2 118.600005 0.76516132 118.600005 0.7651613 # 3 20.153103 0.16992498 20.153104 0.1699250 # 4 17.834464 0.15037490 17.834464 0.1503749 # see same Tau4s (no snapping) # 5 6.625972 0.05586823 7.688957 0.0648310 # We are not fitting to these # 6 6.891842 0.05810997 7.213039 0.0608182 # higher L-moments. ## Not run: # T3 and T4 of the Gumbel distribution, which is inside the SMD domain. gumt3t4 <- c(log(9/8)/log(2), (16 * log(2) - 10 * log(3))/log(2)) FF <- nonexceeds(); qFF <- qnorm(FF) gumx <- qlmomco(FF, pargum(vec2lmom(c(155, 118.6, gumt3t4)))) smdx <- qlmomco(FF, parsmd(lmr, snap.tau4=TRUE)) plot( qFF, gumx, col="blue", type="l", xlab="Standard normal variate", ylab="Quantile") lines(qFF, smdx, col="red") # ## End(Not run) ```

lmomsmd function