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I'm reading Tensor Methods in Statistics by McCullagh 1987, (P209 for this question) and I can't understand one step he uses.

He begins with the usual log-likelihood \begin{equation*} l(\theta; Y) = \sum_i l(\theta;Y_i). \end{equation*} And then writes "It is convenient in the calculations that follow to make the dependence on n explicit by writing \begin{equation*} \frac{\partial\mathcal{L}(\beta)}{\partial\beta^r}= U_r = n^\frac{1}{2} Z_r, \end{equation*} \begin{equation*} \frac{\partial^2\mathcal{L}(\beta)}{\partial\beta^r \partial\beta^s} = U_{rs} = n\kappa_{rs} + n^\frac{1}{2} Z_{rs}, \end{equation*} \begin{equation*} \frac{\partial^3\mathcal{L}(\beta)}{\partial\beta^r \partial\beta^s\partial^t} = U_{rst} = n\kappa_{rst} + n^\frac{1}{2} Z_{rst}, \end{equation*} and so on for the higher-order derivatives."

My question is where does this relationship come from? $\kappa$ are the cumulants, but he doesn't explicitly say what $Z$ is, but I believe they might be observations. Please can someone verify/explain how he's written the partial derivatives in this way?

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  • $\begingroup$ It just means you rescale. For example, $U_r$ is some function, so we introduces a factor of $\sqrt{n}$ and rather writes it as $\sqrt{n} \cdot (U_r/\sqrt{n})$. The new resulting function, $U_r/\sqrt{n}$ is now denoted by $Z_r$. Therefore, $Z_r$ is simply $U_r$ but rescaled by a factor of $\sqrt{n}$. $\endgroup$
    – Nicolas Bourbaki
    Commented Nov 19, 2023 at 21:38
  • $\begingroup$ Please can you explain the $n\kappa_{rs}$ in the 2nd (and further) derivations then? Why are they of magnitude n? Possibly the question I really want to know is the relationship between the partial derivatives and the cumulants and then how they depend on n? $\endgroup$
    – Nick Green
    Commented Nov 19, 2023 at 21:54
  • $\begingroup$ @Nick It's likely McCullagh wrote more than one thing in 1987; I'm guessing you might mean to refer to a book (but little more than a guess; 209 can certainly occur as an article page number), in which case is this Tensor Methods in Statistics you're talking about? $\endgroup$
    – Glen_b
    Commented Nov 19, 2023 at 22:54
  • $\begingroup$ @Glen_b, yes that's the book I'm referring to. Apologies, I might have underestimated its "street cred" so to speak. $\endgroup$
    – Nick Green
    Commented Nov 20, 2023 at 7:27
  • $\begingroup$ I suspect these equations are defining the kappas and Z's: they are the coefficients in the asymptotic expansions of the derivatives. It's unclear that the kappas must be cumulants. Indeed, what exactly would they be the cumulants of? A look at the subscript indicates they would have to be cumulants of a $p$-variate random variable $\theta:$ but those parameters are not always modeled as random variables. $\endgroup$
    – whuber ♦
    Commented Dec 20, 2023 at 23:37

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