Assume that we have some count data $x_{1}, \dots, x_{n}$, which take values $\{1, \dots, m\}$ and we have some estimator of the probability mass function, $\hat{\mathbf{p}} = (\hat{p}_{1}, \dots, \hat{p}_{m})$. The Brier score originally defined as $$ BS = \frac{1}{n}\sum_{i=1}^{n}||\mathbf{I}_{i} - \hat{\mathbf{p}}||_{2}^{2} = \frac{1}{n}\sum_{i=1}^{n} \sum_{j=1}^{m}(I_{i,j} - \hat{p}_{j})^2, $$ where $\mathbf{I}_{i} = (0, \dots, 1, \dots, 0)$ is a vector in $\mathbb{R}^{m}$, with value $1$ is at the $t$-th position from the beginning of the vector, if $x_{i} = t$, for $t \in \{1, \dots, m\}$ and all $i=1,\dots, n$.
Is it possible and does it improve something if we use $L_1$ distance instead of $L_2$?
I mean the following modification: $$ \tilde{BS} = \frac{1}{n}\sum_{i=1}^{n}||\mathbf{I}_{i} - \hat{\mathbf{p}}||_{1} = \frac{1}{n}\sum_{i=1}^{n} \sum_{j=1}^{m}|I_{i,j} - \hat{p}_{j}|. $$
