Relations between moments and cumulants

Question

From the definition of KGF (cumulant generating function) we can write: \begin{align} K_x(t) &= \log_eM_x(t) \\ &= \log_e\left[1+\sum_{r=1}^{\infty}\frac{t^r}{r!}\mu_r^{'}\right] \\ &= k_1t+k_2\frac{t^2}{2!}+\cdots+k_r\frac{t^r}{r!}+\cdots \\ &= \log_e\left[ 1+t\mu_1^{'}+\frac{t^2}{2!}\mu_2^{'}+\cdots+\frac{t^r}{r!}\mu_r^{'}+\cdots \right] \end{align} But now I am confused how to get $r$^th cumulant expressed in raw moments. My book shows that: $$ k_1=\mu_1^{'},\quad k_2=\mu_2^{'}-(\mu_1^{'})^2,\quad k_3=\mu_3^{'}-3\mu_2^{'}\mu_1^{'}+2(\mu_1^{'})^3,\quad \ldots $$ How do they get cumulants in term of moments?

Answer 1

We can use the series expansion of $\ln\left(1 + x\right)$ to write $$\ln\left(1 + t\mu_{1}' + \dfrac{t^{2}}{2!}\mu_{2}' + \dots\right) = \sum_{j = 1}^{\infty}\left(-1\right)^{j - 1}\dfrac{\left(\frac{\mu_{1}'t}{1!} + \frac{\mu_{2}'t^{2}}{2!} + \dots\right)^{j}}{j} $$ The general technique is to then collect for powers of $t$ in $$k_{1}t + k_{2}\dfrac{t^{2}}{2!} + \dots = \sum_{j = 1}^{\infty}\left(-1\right)^{j - 1}\dfrac{\left(\frac{\mu_{1}'t}{1!} + \frac{\mu_{2}'t^{2}}{2!} + \dots\right)^{j}}{j}$$ and equate on both sides to solve for the cumulants in terms of the moments. It's relatively straightforward to do for the first few cumulants but becomes more tedious to do by hand for higher cumulants.

A good reference is 2.2.5 of Expansions and Asymptotics for Statistics by Small.