# 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?

• Why would that be a problem?
– Tim
Commented Sep 21, 2022 at 11:12
• It might not be; I'm just unclear on this in a similar way I was unclear about what it meant for you to be able to consider samples "paired" at one point. The decoupling from the samples feels extreme, and particularly in this case involves data beyond the samples. Commented Sep 21, 2022 at 11:14
• I am happy with the idea that a statistic is any result that varies from sample to sample. It's just a question of what space you're in. So, a scatter plot is a statistic and a sampling distribution of scatter plots is an idea easy in principle and sometimes useful in practice. Naturally, a broad idea like this doesn't undermine narrower and customary definitions, e..g that a statistic is a scalar quantity. I don't worry about whether a scatter plot is a biased estimator. Commented Sep 22, 2022 at 8:58
• In my understanding a statistic is everything calculated from a drawn sample, regardless how much processed it is. If the population is the basis of the calculation, then it is a parameter. To me, there is just that difference: statistic vs parameter. Commented Sep 23, 2022 at 8:59

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.

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.

• This is somewhat trivial, but does it need to change with the data? Like, say I had a function f:$\mathbb{R}^n->\mathbb{R}$ s.t. f(x)=2. Commented Sep 21, 2022 at 11:25
• @DavidMcKnight mathematical definition of a function does not restrict it from being constant.
– Tim
Commented Sep 21, 2022 at 11:29
• Are there possibly restrictions to the type of function $T(X^n)$? Would something still be a statistic if it is computed using a non-deterministic function? E.g. when $f(x) = \bar{x} + \text{some value from RNG}$. Commented Sep 22, 2022 at 8:16
• @SextusEmpiricus AFAIK no. I added one more definition. It is a function of sample, any function, as far as it is not a function of the parameter. But your example with RNG doesn't match the definition of a function in mathematics, as it is not a mapping anymore.
– Tim
Commented Sep 22, 2022 at 8:26
• @Tim it is not clear but 'a massive machine-learned matrix and subsequent nonlinear processing' might involve randomness. For instance when cross validation is involved, or stochastic descent, or fitting methods that use random starting conditions (e.g. GLM). Commented Sep 22, 2022 at 16:42

This answer is a theoretical supplement to Tim's more practical answer (+1).

There is an axiomatic and mathematical side of statistics. Random variables are measurable functions on the outcome space $$\Omega$$ of a probability space $$(\Omega, \mathcal{F}, P)$$. Data are modelled to be instances of random variables, that is subsets of the image of the random variable. In this theoretical context, a statistic is any function of one or more random variables; a functional. Functions of data are then thought of as instances of a statistic. It sometimes creates confusion that the word "statistic" is often used either as a function of random variables or as a function of data, with the former being a model of the latter.

I am not suggesting you think in these terms all the time, or even most of the time depending on what it is you are into doing, but you might like to know that such a perspective exists.

If you are ever curious about this theoretical perspective applied to bootstrapping, see Chapter 17 of Measure Theory and Probability Theory by Athreya and Lahiri 2006.

• Your distinction is like that between estimator and estimate, for example where the mean $\bar X$ of a sample from the distribution of $X$ is an estimator of $\mathbb E[X]$, while the mean of $18, 29, 34$ sampled from the distribution of $X$ being $27$ is an estimate of the expectation of $X$. In both cases we can usefully say the sample mean is a statistic of the sample, and I am not sure that highlighting the obvious point that the second case is an instance of the first adds much to understanding. Commented Sep 22, 2022 at 0:37
• @Henry Yes, we could further consider the distinctions of estimate, estimator, and estimand where they apply. And I see the correspondence between sample and model of a sample can be so close that it is tempting to simply conclude they are the same, but as a matter of principle I separate the map from the territory. On occasion the distinction is even useful, and the obviousness in this case is a good sign of close correspondence between the two. Commented Sep 22, 2022 at 0:58