# Sample Trimmed L-moments to Fitted Distribution Parameter estimation of a distribution given initial estimate of the parameters of the distribution to the sample trimmed L-moment (TL-moment) using numerical optimization. Thought the TL-moments can be used with substantial depth into either tail and need not be symmetrically trimmed, the TL-moments do not appear as useful when substantial tail trimming is needed, say for mix population mitigation. Then censored or truncation methods might be preferred. The `x2xlo` family of operations can be used for conditional left-tail truncation, which is not uncommon in frequency analyses of rail-tail interest water resources phenomena. UTF-8 ```r tlmr2par(x, type, init.para=NULL, trim=NULL, leftrim=NULL, rightrim=NULL, ...) ``` ## Arguments - `x`: A vector of data values. - `type`: Three character (minimum) distribution type (for example, `type="gev"`, see `dist.list`. - `init.para`: Initial parameters as a vector $\Theta$ or as an `lmomco` parameter object from say `vec2par`. If a vector is given, then internally `vec2par` is called with distribution equal to `type`. - `trim`: Level of symmetrical trimming to use in the computations. Although `NULL` is in the argument list, the default is 0---the usual L-moment is returned. - `leftrim`: Level of trimming of the left-tail of the sample, which should be left to `NULL` if no or symmetrical trimming is used. - `rightrim`: Level of trimming of the right-tail of the sample, which should be left to `NULL` if no or symmetrical trimming is used. - `...`: Other arguments to pass to the `optim()` function. ## Returns An `list` is returned. This list should contain at least the following items, but some distributions such as the `revgum` have extra. - **type**: The type of distribution in three character (minimum) format. - **para**: The parameters of the distribution. - **text**: Optional material. If the solution fails but the optimization appears to converge, then this element is inserted into the list and the `para` will be all `NA`. - **source**: Attribute specifying source of the parameters. - **rt**: The list from the `optim()` function. - **init.para**: A copy of the initial parameters given. ## References Elamir, E.A.H., and Seheult, A.H., 2003, Trimmed L-moments: Computational Statistics and Data Analysis, v. 43, pp. 299--314. ## Author(s) W.H. Asquith ## See Also `theoTLmoms`, `TLmoms`, `lmr2par` ## Examples ```r # (1) An example to check that trim(0,0) should recover whole sample the.data <- rlmomco(140, vec2par(c(3, 0.4, -0.1), type="pe3")) wild.guess <- vec2par(c(mean(the.data), 1, 0), type="pe3") pe3whole <- lmom2par(lmoms(the.data), type="pe3") pe3trimA <- tlmr2par(the.data, "pe3", init.para=wild.guess, leftrim=0, rightrim=0) pe3trimB <- tlmr2par(the.data, "pe3", init.para=wild.guess, leftrim=10, rightrim=3) message("PE3 parent = ", paste0(pe3whole$para, sep=" ")) message("PE3 whole sample = ", paste0(pe3whole$para, sep=" ")) message("PE3 trim( 0, 0) = ", paste0(pe3trimA$para, sep=" ")) message("PE3 trim(10, 3) = ", paste0(pe3trimB$para, sep=" ")) # # (2) An example with "real" outliers FF <- lmomco::nonexceeds(); qFF <- qnorm(FF); type <- "gev" the.data <- c(3.064458, 3.139879, 3.167317, 3.225309, 3.324282, 3.330414, 3.3304140, 3.340444, 3.357935, 3.376577, 3.378398, 3.392697, 3.4149730, 3.421604, 3.424882, 3.434569, 3.448706, 3.451786, 3.4517860, 3.462398, 3.465383, 3.469822, 3.491362, 3.501059, 3.5224440, 3.523746, 3.527630, 3.527630, 3.531479, 3.546543, 3.5932860, 3.597695, 3.600973, 3.614897, 3.620136, 3.660865, 3.6848450, 3.820858, 4.708421) the.data <- sort(the.data) # though already sorted, backup for plotting needs # visually, looks like 4 outliers to the left and one outlier to the right # perhaps the practical situation is that we do not wan the left tail to # mess up the right when fitting a distribution because maybe the practical # aspects are the that right tail is of engineering interest, but then we # have some idea that the one very large event is of questionable suitability t1 <- 4; t2 <- 1 # see left and right trimming and then estimation parameters whole.para <- lmom2par(lmoms(the.data), type=type) trim.para <- tlmr2par(the.data, type, init.para=whole.para, leftrim=t1, rightrim=t2) n <- length(the.data) cols <- rep(grey(0.5), n) pchs <- rep(1, n) if(t1 != 0) { cols[ 1 :t1] <- "red" cols[(n-t2+1):n ] <- "purple" } if(t2 != 0) { pchs[ 1 :t1] <- 16 pchs[(n-t2+1):n ] <- 16 } plot( qFF, qlmomco(FF, whole.para), type="l", lwd=2, ylim=c(3.1,4.8), xlab="Standard normal variate", ylab="Some phenomena, log10(cfs)") lines(qFF, qlmomco(FF, trim.para), col=4, lwd=3) points(qnorm(pp(the.data)), sort(the.data), pch=pchs, col=cols) legend("topleft", c("L-moments", paste0("TL-moments(", t1, ",", t2,")")), bty="n", lty=c(1,1), lwd=c(2,3), col=c(1,4)) # see the massive change from the whole sample to the trim(t1,t2) curve ```

tlmr2par function