Pseudo-binomial Approximation to $(k_1, k_2)$-runs

TL;DR

This paper proposes a pseudo-binomial approximation for the distribution of $(k_1, k_2)$-runs in Bernoulli trials, using Stein's method, and demonstrates its effectiveness through theoretical bounds and real-life applications.

Contribution

It introduces the pseudo-binomial distribution as a suitable approximation for $(k_1, k_2)$-runs and derives bounds using Stein's method, improving upon existing results.

Findings

Pseudo-binomial approximation provides accurate estimates.

Bounds are comparable or better than existing literature.

Application to real-life problems demonstrates practical utility.

Abstract

($k_1,k_2$)-runs have received a special attention in the literature and its distribution can be obtained using combinatorial method (Huang and Tsai) and Markov chain approach (Dafnis et al). But the formulae are difficult to use when the number of Bernoulli trials is too large under identical setup and is generally intractable under non-identical setup. So, it is useful to approximate it with a suitable random variable. In this paper, it is demonstrated that pseudo-binomial is most suitable distribution for approximation and the approximation results are derived using Stein's method. Also, application of these results is demonstrated through real-life problems. It is shown that the bounds obtained are either comparable to or improvement over bounds available in the literature.