Methods and Techniques for Accurate and Reliable Measurement and Estimation of Distribution Functions in Real Time

Three methods for the probability distribution function F ( x ) estimation, based on the traditional Monte Carlo method, the Chebyshev inequality, and the theorems of Smirnov and Kolmogorov, are developed and compared. The methods are based on the functional dependence of the sample size n on the estimated value of F ( x ) with an accuracy factor as a parameter. Expressions for this dependence are derived for all methods and compared. It is shown that the accuracy of the estimation can be expressed and controlled by two parameters: the confidence and the confidence limits factor. Based on the methods developed, two techniques for F ( x ) estimation are proposed and demonstrated: a technique with prespecified accuracy and a technique with controlled accuracy. The technique with controlled accuracy allows the processing of a random sample starting with a limited initial sample of observations and then proceeding with the processing on an observation-by-observation basis. This way, the time that is needed to sort the sample has been minimized, and the estimation of distribution functions in real-time has become possible. This technique is demonstrated by estimating the distribution functions of chaotic and Gaussian sequences that are applied in code-division multiple-access (CDMA) systems.