The formula for obtaining the $\lambda_i$ linear moment is:
$$\lambda_i=i^{-1} \sum_{j=0}^{i-1}\binom{i-1}{j}E(X_{i-j:i})$$
where $X_{a:b}$ denotes the $a^{th}$ order statistic of the $b$ sized sample.
I am trying to find linear moments using this formula:
$$\lambda_1=1^{-1} \sum_{j=0}^{0}\binom{0}{j}E(X_{1-j:1})= E(X_{1:1})$$
So if I understand well, the result is the expected value of the minimum of sample of size $1$. As far as I know the result should be equal to the mean value of the sample, so I take each element of the sample, find its minimum and find the average of those? Is my interpretation proper?
I would be grateful for derivation of the second moment.
