Let Xr , Xrs , Xrst , be a sequence of arrays of arbitrary random variables Such a sequence will be called triangular Let the joint moments and cumulants be denoted as in Section 7 2 1 by and so on Now write and for the sum over all partitions of the subscripts as follows and so on, with identical definitions for rs , rst and so on Show that r r , Hence show that the cumulants of the log likelihood derivatives satisfy 0, whatever the indices

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Question: Let Xr , Xrs , Xrst , be a sequence of arrays of arbitrary random variables. Such a sequence will be called triangular. Let the

Let Xr

, Xrs

, Xrst

,… be a sequence of arrays of arbitrary random variables.

Such a sequence will be called triangular. Let the joint moments and cumulants be denoted as in Section 7.2.1 by and so on. Now write μ[…] and κ[…] for the sum over all partitions of the subscripts as follows and so on, with identical definitions for κ[rs]

, κ[rst] and so on. Show that κ[r]

= μ[r]

, Hence show that the cumulants of the log likelihood derivatives satisfy κ[…] =

0, whatever the indices.

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