Axiomatically based risk measures have been the object of numerous studies and generalizations in recent years. In the literature we find two main schools: coherent risk measures (Artzner et al. [1]) and insurance risk measures (Wang et al. [3]). In this note, we set to study yet another extension motivated by a third axiomatically based risk measure that has been recently introduced. In Heyde et al. [2], the concept of * natural risk statistics* is discussed as a data-based risk measure, i.e. as an axiomatic risk measure defined in the space $\mathbb R^n$. One drawback of these kind of risk measures is their dependence on the space dimension $n$. In order to circumvent this issue, we propose a way to define a family $\{\rho_n\}_{n=1,2,\dots}$ of *natural risk statistics* whose members are defined on $\mathbb{R}^n$ and related in an appropriate way. This construction requires the generalization of *natural risk statistics* to the space of infinite sequences $l^\infty$. We also look at the problem of extending this construction to a bi-variate sequence. Both of these problems are relevant in finance because they give a theoretical framework to study estimators of risk measures in the univariate and bi-variate setting. In fact, these axiomatic risk measures can be seen as defined over the set of univariate and bivariate data sets. In a risk management application this is important because typically one would have a data base of a financial position without any *a-priori* knowledge of the model itself.

**References**

[1] Artzner, Philippe; Delbaen, Freddy; Eber, Jean-Marc; Heath, David (2002) Coherent measures of risk. Risk management: value at risk and beyond (Cambridge, 1998), 145–175, Cambridge Univ. Press, Cambridge.

[2] Heyde, C. C.; Kou, S.G.; Peng, X. H. (2007). What Is a Good Risk Measure: Bridging the Gaps between Data, Coherent Risk Measures, and Insurance Risk Measures. Working Paper, Columbia University.

[3] Wang, S. S., V. R. Young, and H. H. Panjer (1997). Axiomatic characterization of insurance prices. Insurance: Mathematics and Economics, 21, 173-183.