**8. Conclusions**

The problem of randomized maximum entropy estimation of a probability density function based on real available data has been formulated and solved. The developed estimation algorithm (the RME algorithm) finds the conditional maximum of an information entropy functional on a set of admissible probability density functions characterized by the empirical balance equations for Lagrange multipliers. These equations define an implicit dependence of the Lagrange multipliers on the data collection. The existence of such an implicit function for any values in a data collection has been established. The function's behavior for a data collection of a greater size has been studied, and the asymptotic efficiency of the RME estimates has been proved.

The positive features of RME estimates have been illustrated with an example of estimation and testing a linear dynamic regression model of the evolution of the thermokarst lake area in Western Siberia with real data.

**Funding:** This research was funded by the Ministry of Science and Higher Education of the Russian Federation, project no. 075-15-2020-799.

**Institutional Review Board Statement:** Not applicable.

**Informed Consent Statement:** Not applicable.

**Conflicts of Interest:** The author declares no conflict of interest.

## **References**

