To estimate the parameters of a truncated distribution (lognormal for example), we can use the Maximum Likelihood Estimation or Method of Moments.
For the Method of Moments Estimation, one needs to write down the mathematical expression of the expected value of the truncated lognormal distribution. Is it possible to do so?
Or can we use a numerical method?
There are some formulas which may simplify the numerical work, but they use the error function repeatedly.
Let $f$ be the pdf for $LN(\mu,\sigma)$. If we define $$g(b,c):=\int_0^b x^c f(x)\,dx = \frac{\exp(c\mu+c^2\sigma^2/2)}{2}\left(1-\text{erf}(\frac{\mu+c\sigma^2-\log b}{\sqrt{2}\sigma})\right)$$ then the $c^{th}$ moment of that distribution truncated between $a$ and $b$ is $$\frac{g(b,c)-g(a,c)}{g(b,0)-g(a,0)}$$ For example, the case $c=1$ gives the expectation.
