NUMERICAL SOLUTION OF THE PROBLEM OF ESTIMATING THE EMPIRICAL DISTRIBUTION FUNCTION FOR SMALL SAMPLES WITH A GIVEN CONFIDENCE

Abstract

Due to the random nature of the functioning of real technical systems as physical objects, their state of operability, defined as the difference between resistance and external load, cannot be determined absolutely accurately. In practical tasks of analyzing the reliability of building structures, the estimation of resistance and load parameters is performed on the basis of relatively small samples of real measurement results, the statistical analysis of which is usually associated with the construction of empirical distribution functions. The well-known classical methods of constructing such a function do not allow us to determine, much less set, the confidence probability (statistical security) of the results obtained.

Two methods of estimating the empirical distribution function based on ordinal statistics are proposed: a reliable estimate of the quantiles of the desired selected level, as well as an estimate of the quantile level for available points (sample data). The first method is associated with the numerical reconstruction of the quantile distribution function, the second with the need for a numerical solution of the inverse nonparametric problem for the empirical distribution function itself. Both methods involve setting the required confidence level of the result. Their effectiveness is shown for small samples of empirical data.