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

Abstract

k1, k2)-runs have received a special attention in the literature and its distribution can be obtained using combinatorial method (Huang and Tsai [19]) and Markov chain approach (Dafnis et al. [13]). 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.

On Discrete Gibbs Measure Approximation to Runs

“…Runs and patterns is an important topic in the areas related to probability and statistics such as, reliability theory, meteorology and agriculture, statistical testing and quality control among many others (see Balakrishnan and Koutras [5], Kumar and Upadhye [21] and Dafnis et al [14]). The research in this topic initiated with the runs related to success/failure (see Philippou et al [24] and Philippou and Makri [25]).…”

“…However, the distribution of the event of first type (B n k1,k2 ) is studied by Huang and Tsai [17] in 1991 where probability generating function (PGF), recursive relations for probability mass function (PMF), Poisson convergence and an extension of this distribution is given. Recently, approximation problem related B n k1,k2 is studied widely, for example, Poisson approximation to B n k1,k2 is given by Vellaisamy [29], binomial convoluted Poisson approximation to B n 1,1 is studied by Upadhye et al [28], negative binomial approximation to waiting time for B n k1,k2 and pseudo-binomial approximation to B n k1,k2 are given by Kumar and Upadhye [20,21].…”

“…)[(l p− q)p m,n−k−s−i+1 − pa(p)p ⋆ m,n,k+s+i−1 ] g(m + w)p m,n−k−s−i+1 − qa(p) g(m + j)p m,n−k−s−i+1Using(15), the expression leads toE U 3 g M n s (l) 1 − 1(s ≤ n − 2k − i + 1u ≤ n − k − 2)) − 1(u = n − k) + q1(u = n − k − 1− q)B s (l) ⌊n/k⌋ m=0 ∆g(m + 1)(p ⋆ m,n,k+s+i−1 − p ⋆ m−1,n,k+s+i−1 ) g(m + w)p m,n−k−s−i+1 − qa(p) g(m + j)p m,n−k−s−i+1 From (31), it is easy to see that s (l) 1 − 1(s ≤ n − 2k − i + 1) + k+s+i−qp1(u ≤ n − k − 2)) −1(u=n−k)+q1(u=n−k−1)) +a(p) g(m + w)p m,n−k−s−i+1 − qa(p) g(m + j)p m,n−k−s−i+1 Hence, for g ∈ G X3 ∩ G M n k 1 ,k 2and using(21), we getE U 3 g M n k1,k2 ≤ 2 ∆g (2 + qp)a(p) 2 4(n − k)δ 1 + (2δ + qp)c pa(p) + (n − k)δ + c(3)n,k + qpc(6)n,k q d T V M n−3k−…”