When is a statistic not a statistic? I have a fairly involved ``statistic'' that involves transformation of a set of samples $\mathbf{x}$ by a massive machine-learned matrix and subsequent nonlinear processing into said statistic $\mathbf{x}_s$. However, it still is reflective of the choice of $\mathbf{x}$.
I am trying to perform a bootstrapped test on the differences between two populations by using two samples, $\mathbf{x}_1$ and $\mathbf{x}_2$. This test calls for using the standard deviation of the samples; however, this data is a set of transformed three-hot vector inputs and one-hot vector selections.
While the inputs and outputs are linked, is it still fair to call $\mathbf{x}_s$ a statistic of $\mathbf{x}$, given how much processing is involved?

The statistic is defined as

A statistic is a function $T (X^n )$ of the data.

(Larry Wasserman All of Statistics, p. 137)

A statistic (singular) or sample statistic is any quantity computed from values in a sample which is considered for a statistical purpose.

(Wikipedia)


Definition 5.2.1 Let $X_1,\dots,X_n$ be a random sample
of size $n$ from a population and let $T(X_1,\dots,X_n)$ be
a real-valued or vector-valued function whose domain includes
the sample space of $(X_1,\dots,X_n)$. Then the random variable
or random vector $Y = T(X_1,\dots,X_n)$ is called a statistic.
The probability distribution of statistic $Y$ is called the
sampling distribution of $Y$.

The definition of a statistic is very broad, with the only restriction
being that a statistic cannot be a function of a parameter. [...]

(Casella and Berger Statistical inference, p. 211)
TL;DR If you calculate something from the data, "for statistical purposes", it's a statistic.
As noticed by @SextusEmpiricus in the comments, another restriction that may not be instantly obvious to everyone is that $T$ is a function in a mathematical sense, that is, a mapping

a function from a set $X$ to a set $Y$ assigns to each element of $X$ exactly one element of $Y$

So something generating the results at random wouldn't fit the definition.
