  Though widely used in applications, reinforced random walk on graphs have
never been the subject of a valid statistical inference. We develop in this
paper a statistical framework for a general two-colored urn model. The
probability to draw a ball at each step depends on the number of balls of each
color and on a multidimensional parameter $\theta$ through a function $f$,
called a choice function. We introduce two estimators of $\theta$: the maximum
likelihood estimator and a weighted least squares estimator which is less
efficient, but is closer to the calibration techniques used in the applied
literature. In general, the model is an inhomogeneous Markov chain and this
property makes the estimation of the parameter impossible on a single path,
even if it were infinite. Therefore we assume that we observe i.i.d.
experiments, each of a predetermined finite length. This is coherent with the
usual experimental set-ups. We apply the statistical framework to a real life
experiment: the selection of a path among pre-existing channels by an ant
colony. We performed experiments, which consisted of letting ants pass through
the branches of a fork. We consider the particular urn model proposed by J.-L.
Deneubourg in 1990 to describe this phenomenon. We simulate this model for
several parameter values in order to assess the accuracy of the MLE and the
WLSE. Then we estimate the parameter from the experimental data and evaluate
confident regions with Bootstrap algorithms. The findings of this paper do not
contradict the biological literature, but give statistical significance to the
values of the parameter found therein.
