Exact Distribution of the Total Run Count

Computes the exact probability mass and cumulative distribution of the total number of runs in mutually independent Bernoulli trials. The trials need not have a common success probability. The supplied probabilities are treated as fixed and known; this function performs no estimation.

Usage

runs_distribution(prob)

Arguments

prob

A non-empty numeric vector. Its ith element is the success probability of the ith Bernoulli trial and must be finite and in the closed interval [0, 1].

Value

A data frame with one row for every possible run count and columns:

runs

The possible total run counts from 1 to length(prob).

probability

The exact probability mass at each run count.

cumulative

The inclusive lower cumulative probability.

Details

The calculation uses a finite-state dynamic program indexed by the number of observations processed, the current run count, and the last binary value. Its time complexity is quadratic and its working memory is linear in length(prob).

References

Fu, J. C. and Koutras, M. V. (1994). Distribution theory of runs: A Markov chain approach. Journal of the American Statistical Association, 89(427), 1050–1058. doi:10.1080/01621459.1994.10476841 .

Examples

runs_distribution(c(0.15, 0.40, 0.75, 0.60))