# lmompe3: L-moments of the Pearson Type III Distribution

### Description

This function estimates the L-moments of the Pearson Type III distribution given the parameters (μ, σ, and γ) from `parpe3` as the product moments: mean, standard deviation, and skew. The first three L-moments in terms of these parameters are complex and numerical methods are required. For simplier expression of the distribution functions (`cdfpe3`, `pdfpe3`, and `quape3`) the “moment parameters” are expressed differently.

The Pearson Type III distribution is of considerable theoretical interest because the parameters, which are estimated via the L-moments, are in fact the product moments. Although, these values fitted by the method of L-moments will not be numerically equal to the sample product moments. Further details are provided in the Examples section of the `pmoms` function documentation.

### Usage

 `1` ```lmompe3(para) ```

### Arguments

 `para` The parameters of the distribution.

### Value

An R `list` is returned.

 `lambdas` Vector of the L-moments. First element is λ_1, second element is λ_2, and so on. `ratios` Vector of the L-moment ratios. Second element is τ, third element is τ_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: “lmompe3”.

W.H. Asquith

### References

Hosking, J.R.M., 1990, L-moments—Analysis and estimation of distributions using linear combinations of order statistics: Journal of the Royal Statistical Society, Series B, v. 52, pp. 105–124.

Hosking, J.R.M., 1996, FORTRAN routines for use with the method of L-moments: Version 3, IBM Research Report RC20525, T.J. Watson Research Center, Yorktown Heights, New York.

Hosking, J.R.M., and Wallis, J.R., 1997, Regional frequency analysis—An approach based on L-moments: Cambridge University Press.

`parpe3`, `cdfpe3`, `pdfpe3`, `quape3`

### Examples

 ```1 2 3``` ```lmr <- lmoms(c(123,34,4,654,37,78)) lmr lmompe3(parpe3(lmr)) ```

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