The paper integrates and adapts a range of advanced computational, mathematical and statistical tools for the purpose of analysis of neonate sleep stages based on extensive electroencephalogram (EEG) recordings. The level of brain dysmaturity of a neonate is difficult to assess by direct physical or cognitive examination, but dysmaturity is known to be directly related to the structure of neonatal sleep as reflected in the nonstationary time series produced by EEG signals which, importantly, can be collected trough a noninvasive procedure. In the past, the assessment of sleep EEG structure has often been done manually by experienced clinicians. The goal of this paper is to develop rigorous algorithmic tools for the same purpose by providing a formal scheme to separate different sleep stages corresponding to different stationary segments of the EEG signal based on statistical analysis of the spectral and nonlinear characteristics of the sleep EEG recordings. The methods developed in this paper can, potentially, be translated to other areas of biomedical research.