  Multiparametric statistical model providing stable reconstruction of
parameters by observations is considered. The only general method of this kind
is the root model based on the representation of the probability density as a
squared absolute value of a certain function, which is referred to as a
psi-function in analogy with quantum mechanics. The psi-function is represented
by an expansion in terms of an orthonormal set of functions. It is shown that
the introduction of the psi-function allows one to represent the Fisher
information matrix as well as statistical properties of the estimator of the
state vector (state estimator) in simple analytical forms. The chi-square test
is considered to test the hypotheses that the estimated vector converges to the
state vector of a general population. The method proposed may be applied to its
full extent to solve the statistical inverse problem of quantum mechanics (root
estimator of quantum states). In order to provide statistical completeness of
the analysis, it is necessary to perform measurements in mutually complementing
experiments (according to the Bohr terminology). The maximum likelihood
technique and likelihood equation are generalized in order to analyze quantum
mechanical experiments. It is shown that the requirement for the expansion to
be of a root kind can be considered as a quantization condition making it
possible to choose systems described by quantum mechanics from all statistical
models consistent, on average, with the laws of classical mechanics.
