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My question is simple, but I could not find any references for this particular case.

When we have a random vector, we usually use $\boldsymbol{X}$ to denote the random vector and $\boldsymbol{x}$ to denote an observation of this random vector.

Now, when we use matrices, the usual notation for a matrix is a non-bold upper-case letter (e.g. $X$). In this case, what is the proper (or most usual) notation in order to distinguish the random matrix and the matrix of observations?

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  • $\begingroup$ I think by "non-bold lower-case letter" you mean "non-bold upper-case letter" so I edited your question. If I'm wrong feel free to update it accordingly. $\endgroup$
    – utobi
    Commented Nov 9, 2022 at 12:24
  • $\begingroup$ Thanks @utobi. My mistake. : ) $\endgroup$
    – Renato Fernandes
    Commented Nov 9, 2022 at 12:27

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In the case of an IID sample ${\bf X}_1, \ldots, {\bf X}_n$, where ${\bf X}_i$ is a $p\times 1$ random vector, to denote a random matrix of all samples I prefer to use the Blackboard bold typeface

$$ \mathbb{X} = [{\bf X}_1|\cdots |{\bf X}_n]. $$

Yes, this matrix has dimension $p\times n$ and not $n\times p$, but it has its advantages, at least in multivariate analysis. It is used also by others, e.g. Koch (2014) Analysis of Multivariate and High-Dimensional Data, ISBN: 9780521887939.

In the case of an observed sample ${\bf x}_1, \ldots, {\bf x}_n$, I use

$$ \mathbb{x} = [{\bf x}_1|\cdots |{\bf x}_n]. $$

If you use LaTeX, to produce lower-case letters in Blackboard bold typeface, e.g. $\mathbb{x}$, you'll have to use the bbm package.

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  • $\begingroup$ Thanks @utobi. In my case, the columns are independents, but not identically distributed. the notation $\mathbb{X}$ still applies? And more important (for me): It is an recognized and accepted notation in statistical journals? $\endgroup$
    – Renato Fernandes
    Commented Nov 9, 2022 at 12:53
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    $\begingroup$ It does work also for non identically cases, although I'm not sure how this is useful there. In mean, in multivariate analysis $\mathbb{X}$ is useful because, .e.g., I can express the sample average say as $\bar x = (1/n)\mathbb{X}^\top {\bf1}_n$. So journals do have very strict rules on this (e.g. Biometrika) others (say Bayesian Analysis) are not that strict. My advice is to check your journal instructions for this. $\endgroup$
    – utobi
    Commented Nov 9, 2022 at 13:04

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