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This article is concerned with the nonparametric statistical analysis of multiple binary sequences, a commonly occurring data structure. One example that we consider comes from dairy science, where each of a number of cows is tested for the presence of a pathogen infection throughout the lactation cycle. Sports statistics provide our second example, in which each of many baseball players produces a sequence of hits/no hits over the course of a season (Albright 1993). A third example comes from the National Longitudinal Survey of Youth (NLSY), in which the employment and health status of a group of young people are monitored yearly by questionnaire (see Borus 1984). It is natural to allow correlation within each binary sequence when formulating models for such data. Among the various approaches, a particularly simple model says that each binary sequence is the realization of a Markov chain having some order of serial dependence. Zeroth-order chains correspond to independence, first-order chains exhibit serial dependence on the most recent binary variable, second-order chains depend on the most recent pair of variables, and so on.