The method of quasi-optimal weights is applied to constructing (quasi-)optimal criteria for various anomalous contributions in experimental spectra. Anomalies in the spectra could indicate physics beyond the Standard Model (additional interactions an
d neutrino flavours, Lorenz violation etc.). In particular the cumulative tritium $beta$-decay spectrum (for instance, in Troitsk-$ u$-mass, Mainz Neutrino Mass and KATRIN experiments) is analysed using the derived special criteria. Using the power functions we show that the derived quasi-optimal criteria are efficient statistical instruments for detecting the anomalous contributions in the spectra.
We review the methods of constructing confidence intervals that account for a priori information about one-sided constraints on the parameter being estimated. We show that the so-called method of sensitivity limit yields a correct solution of the pro
blem. Derived are the solutions for the cases of a continuous distribution with non-negative estimated parameter and a discrete distribution, specifically a Poisson process with background. For both cases, the best upper limit is constructed that accounts for the a priori information. A table is provided with the confidence intervals for the parameter of Poisson distribution that correctly accounts for the information on the known value of the background along with the software for calculating the confidence intervals for any confidence levels and magnitudes of the background (the software is freely available for download via Internet).
