The estimation of L-moments is based on a sample of size \(n\), arranged in ascending order.
Let \(x_{1:n} \le x_{2:n} \le \dots \le x_{n:n}\) be the ordered sample.
An unbiased estimator of the probability weighted moments \(\beta_r\) is:
$$b_r = n^{-1} \sum_{j=r+1}^n \frac{(j-1)(j-2)\dots(j-r)}{(n-1)(n-2)\dots(n-r)} x_{j:n}$$
The sample L-moments are defined by:
$$l_1 = b_0$$
$$l_2 = 2b_1 - b_0$$
$$l_3 = 6b_2 - 6b_1 + b_0$$
$$l_4 = 20b_3-30b_2+12b_1-b_0$$
and in general
$$l_{r+1} = \sum_{k=0}^r \frac{(-1)^{r-k}(r+k)!}{(k!)^2(r-k)!} b_k$$
where \(r=0,1,\dots,n-1\).
The sample L-moment ratios are defined by
$$t_r=l_r/l_2$$
and the sample L-CV by
$$t=l_2/l_1$$
Sample regional L-CV, L-skewness and L-kurtosis coefficients are defined as
$$t^R = \frac{\sum_{i=1}^k n_i t^{(i)}}{ \sum_{i=1}^k n_i}$$
$$t_3^R =\frac{ \sum_{i=1}^k n_i t_3^{(i)}}{ \sum_{i=1}^k n_i}$$
$$t_4^R =\frac{ \sum_{i=1}^k n_i t_4^{(i)}}{\sum_{i=1}^k n_i}$$