Table 2 The first six cumulants l κ l (...

Table 2

The first six cumulants

l	$κ_{l} (Y)$	$κ_{l - 1,1} (Y_{M}, Y)$
1	$E [Y]$	$E [Y]$
2	$E [{\hat{Y}}^{2}]$	$E [{\hat{Y}}_{M} \hat{Y}]$
3	$E [{\hat{Y}}^{3}]$	$E [{\hat{Y}}_{M}^{2} \hat{Y}]$
4	$E [{\hat{Y}}^{4}] - 3 E {[{\hat{Y}}^{2}]}^{2}$	$E [{\hat{Y}}_{M}^{3} \hat{Y}] - 3 E [{\hat{Y}}_{M}^{2}] E [{\hat{Y}}_{M} \hat{Y}]$
5	$E [{\hat{Y}}^{5}] - 10 E [{\hat{Y}}^{3}] E [{\hat{Y}}^{2}]$	$E [{\hat{Y}}_{M}^{4} \hat{Y}] - 6 E [{\hat{Y}}_{M}^{2}] E [{\hat{Y}}_{M}^{2} \hat{Y}] - 4 E [{\hat{Y}}_{M}^{3}] E [{\hat{Y}}_{M} \hat{Y}]$
6	$E [{\hat{Y}}^{6}] - 15 E [{\hat{Y}}^{4}] E [{\hat{Y}}^{2}] - 10 E {[{\hat{Y}}^{3}]}^{2} + 30 E {[{\hat{Y}}^{2}]}^{3}$	$E [{\hat{Y}}_{M}^{5} \hat{Y}] - 10 E [{\hat{Y}}_{M}^{3} \hat{Y}] E [{\hat{Y}}_{M}^{2}] - 5 E [{\hat{Y}}_{M}^{4}] E [{\hat{Y}}_{M} \hat{Y}] - 10 E [{\hat{Y}}_{M}^{3}] E [{\hat{Y}}_{M}^{2} \hat{Y}] + 30 E {[{\hat{Y}}_{M}^{2}]}^{2} E [{\hat{Y}}_{M} \hat{Y}]$

Note(s): The second column presents cumulants of a random variable Y. The third column presents joint cumulants $κ_{l - 1,1} (Y_{M}, Y)$ for l = 1,2, …,6. $\hat{Y}$ and ${\hat{Y}}_{M}$ in the row 3–7 are $Y - E [Y]$ and $Y_{M} - E [Y_{M}]$ ⁠, respectively

[ViewLarge]