Talk:L-moment

Changes

I have removed reference to a "L-variance" as I think this term is never used. I am not sure that L-mean is used, certainly not much if at all, but it seems just about acceptable . Perhaps someone knows better. Melcombe (talk) 11:00, 17 September 2009 (UTC)[reply]

The first (r=1) L-moment measures location, and is equivalent the (arithmetic) mean (unless it's a trimmed L-moment). But even so, I can't remember any authors that referred to it as the "L-mean". So calling it the L-mean, instead of the commonly accepted "L-location" (or L-loc for short), is rather unconventional, and could be a bit confusing.

I have never seen the term "L-variance" in any of the relevant literature that I have read. The second L-moment (r=2), the L-scale, is the L-moment analogue of the standard deviation. Unlike the L-location and the mean, the L-scale and standard deviation are different measures, but both measure the same thing: dispersion. The L-scale grows linearly w.r.t. the scale of the random variable, just like the standard deviation does. However, the variance scales quadratically. So, for a given random variable, the ratio of L-scale and standard deviation (assuming they exist) is constant for linear transformations of that random varianble.

TLDR; referring to the second L-moment, the L-scale, as "L-variance" is not only unconventional, it is inaccurate. Hammu666 (talk) 04:10, 13 December 2023 (UTC)[reply]

Binomial Transform?

What is the relevance of the binomial transform to l-moments? The article briefly mentions that l-moments are related to the binomial transform, then provides no additional information, context, or citations as to how. I was unable to find any references to a relationship between the two in Google Scholar. I'd be interested in hearing from User:Nbarth about why he added this. Closed Limelike Curves (talk) 22:07, 12 April 2022 (UTC)[reply]

The binomial transform can be used to e.g. convert lowest order statistics into the highest, and vice versa. I've used this quite often when deriving population L-moments of distributions.

If you consider the definition of the L-moments using order statistics, or probability-weighted moments, you'll see it's a linear sum involving binomial coefficients. Although I'm not very familiar with the details of the binomial transform, I wouldn't be suprised if these L-moment definitions are instances of some generalization of the transform. Hammu666 (talk) 04:29, 13 December 2023 (UTC)[reply]