We consider an exact sequential conditional test for three-way conditional test of no interaction. At each time τ, the test uses as the... Show moreWe consider an exact sequential conditional test for three-way conditional test of no interaction. At each time τ, the test uses as the conditional inference frame the set F(Hτ) of all tables with the same three two-way marginal tables as the obtained table Hτ . For 3 × 3 × K tables, we propose a method to construct F(Hτ) from F(Hτ−1). This enables us to perform efficiently the sequential exact conditional test. The subset Sτ of F (Hτ ) consisting of s + Hτ − Hτ −1 for s ∈ F(Hτ−1) contains Hτ , where the operations + and − are defined elementwise. Our argument is based on the minimal Markov basis for 3 × 3 × K contingency tables and we give a minimal subset M of some Markov basis which has the property that F (Hτ ) = {s − m | s ∈ Sτ , m ∈ M}. Show less

Focusing on the discrete probabilistic setting we generalize the combinatorial definition of cumulants to L-cumulants. This generalization... Show moreFocusing on the discrete probabilistic setting we generalize the combinatorial definition of cumulants to L-cumulants. This generalization keeps all the desired properties of the classical cumulants like semi-invariance and vanishing for independent blocks of random variables. These properties make L-cumulants useful for the algebraic analysis of statistical models. We illustrate this for general Markov models and hidden Markov processes in the case when the hidden process is binary. The main motivation of this work is to understand cumulant-like coordinates in alge- braic statistics and to give a more insightful explanation why tree cumulants give such an elegant description of binary hidden tree models. Moreover, we argue that L-cumulants can be used in the analysis of certain classical algebraic varieties. Show less