caa a22 4500 475826752 CHVBK 20180406123830.0 cr unu---uuuuu 170329e20001201xx s 000 0 eng 10.1023/A:1017585512412 doi (NATIONALLICENCE)springer-10.1023/A:1017585512412 Numbers of Success-Runs of Specified Length Until Certain Stopping Time Rules and Generalized Binomial Distributions of Order k [Elektronische Daten] [Sigeo Aki, Katuomi Hirano] A new distribution called a generalized binomial distribution of order k is defined and some properties are investigated. A class of enumeration schemes for success-runs of a specified length including non-overlapping and overlapping enumeration schemes is rigorously studied. For each nonnegative integer μ less than the specified length of the runs, an enumeration scheme called μ-overlapping way of counting is defined. Let k and ℓ be positive integers satisfying ℓ < k. Based on independent Bernoulli trials, it is shown that the number of (ℓ− 1)-overlapping occurrences of success-run of length k until the n-th overlapping occurrence of success-run of length ℓ follows the generalized binomial distribution of order (k−ℓ). In particular, the number of non-overlapping occurrences of success-run of length k until the n-th success follows the generalized binomial distribution of order (k− 1). The distribution remains unchanged essentially even if the underlying sequence is changed from the sequence of independent Bernoulli trials to a dependent sequence such as higher order Markov dependent trials. A practical example of the generalized binomial distribution of order k is also given. The Institute of Statistical Mathematics, 2000 Binomial distribution of order k nationallicence Markov chain nationallicence probability generating function nationallicence stopping time nationallicence success-run nationallicence waiting time nationallicence Aki Sigeo Department of Informatics and Mathematical Science, Graduate School of Engineering Science, Osaka University, 1-3 Machikaneyama-cho, 560-8531, Toyonaka, Japan aut Hirano Katuomi The Institute of Statistical Mathematics, 4-6-7 Minani-Azabu, Minato-ku, 106-8569, Tokyo, Japan aut Annals of the Institute of Statistical Mathematics Kluwer Academic Publishers 52/4(2000-12-01), 767-777 0020-3157 52:4<767 2000 52 10463 https://doi.org/10.1023/A:1017585512412 text/html Onlinezugriff via DOI 1 research-article jats NATIONALLICENCE 856 40 https://doi.org/10.1023/A:1017585512412 text/html Onlinezugriff via DOI NATIONALLICENCE 700 1- Aki Sigeo Department of Informatics and Mathematical Science, Graduate School of Engineering Science, Osaka University, 1-3 Machikaneyama-cho, 560-8531, Toyonaka, Japan aut NATIONALLICENCE 700 1- Hirano Katuomi The Institute of Statistical Mathematics, 4-6-7 Minani-Azabu, Minato-ku, 106-8569, Tokyo, Japan aut NATIONALLICENCE 773 0- Annals of the Institute of Statistical Mathematics Kluwer Academic Publishers 52/4(2000-12-01), 767-777 0020-3157 52:4<767 2000 52 10463 Metadata rights reserved Springer special CC-BY-NC licence nationallicence BK010053 XK010053 XK010000 NATIONALLICENCE NATIONALLICENCE NL-springer