  In this work we propose a robust methodology to mitigate the undesirable
effects caused by outliers to generate reliable physical models. In this way,
we formulate the inverse problems theory in the context of Kaniadakis
statistical mechanics (or $\kappa$-statistics), in which the classical approach
is a particular case. In this regard, the errors are assumed to be distributed
according to a finite-variance $\kappa$-generalized Gaussian distribution.
Based on the probabilistic maximum-likelihood method we derive a
$\kappa$-objective function associated with the finite-variance
$\kappa$-Gaussian distribution. To demonstrate our proposal's
outlier-resistance, we analyze the robustness properties of the
$\kappa$-objective function with help of the so-called influence function. In
this regard, we discuss the role of the entropic index ($\kappa$) associated
with the Kaniadakis $\kappa$-entropy in the effectiveness in inferring physical
parameters by using strongly noisy data. In this way, we consider a classical
geophysical data-inverse problem in two realistic circumstances, in which the
first one refers to study the sensibility of our proposal to uncertainties in
the input parameters, and the second is devoted to the inversion of a seismic
data set contaminated by outliers. The results reveal an optimum $\kappa$-value
at the limit $\kappa \rightarrow 2/3$, which is related to the best results.
