TITLE:
Numerical Procedures for Calculating the Probabilities of Recurrent Runs
AUTHORS:
M. P. Silverman
KEYWORDS:
Recurrent Events; Theory of Runs; Time Series Analysis; Generating Functions; Probability Distributions
JOURNAL NAME:
Open Journal of Statistics,
Vol.4 No.2,
February
27,
2014
ABSTRACT: Run count statistics serve
a central role in tests of non-randomness of stochastic processes of interest
to a wide range of disciplines within the physical sciences, social sciences,
business and finance, and other endeavors involving intrinsic uncertainty. To
carry out such tests, it is often necessary to calculate two kinds of run count
probabilities: 1) the
probability that a certain number of trials results in a specified multiple
occurrence of an event, or 2) the probability that a specified number of occurrences of an event
take place within a fixed number of trials. The use of appropriate generating
functions provides a systematic procedure for obtaining the distribution
functions of these probabilities. This paper examines relationships among the
generating functions applicable to recurrent runs and discusses methods,
employing symbolic mathematical software, for implementing numerical extraction
of probabilities. In addition, the asymptotic form of the cumulative
distribution function is derived, which allows accurate runs statistics to be
obtained for sequences of trials so large that computation times for extraction
of this information from the generating functions could be impractically long.