# parnor: Estimate the Parameters of the Normal Distribution In lmomco: L-Moments, Censored L-Moments, Trimmed L-Moments, L-Comoments, and Many Distributions

## Description

This function estimates the parameters of the Normal distribution given the L-moments of the data in an L-moment object such as that returned by `lmoms`. The relation between distribution parameters and L-moments is seen under `lmomnor`.

There are interesting parallels between λ_2 (L-scale) and σ (standard deviation). The σ estimated from this function will not necessarily equal the output of the `sd` function of R, and in fact such equality is not expected. This disconnect between the parameters of the Normal distribution and the moments (sample) of the same name can be most confusing to young trainees in statistics. The Pearson Type III is similar. See the extended example for further illustration.

## Usage

 `1` ```parnor(lmom, checklmom=TRUE, ...) ```

## Arguments

 `lmom` An L-moment object created by `lmoms` or `vec2lmom`. `checklmom` Should the `lmom` be checked for validity using the `are.lmom.valid` function. Normally this should be left as the default and it is very unlikely that the L-moments will not be viable (particularly in the τ_4 and τ_3 inequality). However, for some circumstances or large simulation exercises then one might want to bypass this check. `...` Other arguments to pass.

## Value

An R `list` is returned.

 `type` The type of distribution: `nor`. `para` The parameters of the distribution. `source` The source of the parameters: “parnor”.

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.

`lmomnor`, `cdfnor`, `pdfnor`, `quanor`
 ``` 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26``` ```lmr <- lmoms(rnorm(20)) parnor(lmr) # A more extended example to explore the differences between an # L-moment derived estimate of the standard deviation and R's sd() true.std <- 15000 # select a large standard deviation std <- vector(mode = "numeric") # vector of sd() std.by.lmom <- vector(mode = "numeric") # vector of L-scale values sam <- 7 # number of samples to simulate sim <- 100 # perform simulation sim times for(i in seq(1,sim)) { Q <- rnorm(sam,sd=15000) # draw random normal variates std[i] <- sd(Q) # compute standard deviation lmr <- lmoms(Q) # compute the L-moments std.by.lmom[i] <- lmr\$lambdas[2] # save the L-scale value } # convert L-scale values to equivalent standard deviations std.by.lmom <- sqrt(pi)*std.by.lmom # compute the two biases and then output # see how the standard deviation estimated through L-scale # has a smaller bias than the usual (product moment) standard # deviation. The unbiasness of L-moments is demonstrated. std.bias <- true.std - mean(std) std.by.lmom.bias <- true.std - mean(std.by.lmom) cat(c(std.bias,std.by.lmom.bias,"\n")) ```