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summation, our efficient algorithms described in Section 3 enable us to consider the penalized average score per m , 

<img src='content_image/320410.jpg'>

where SLR$_{m}$π ( I| m ) is S$_{m}$ divided by the number of partitions of size m for the K - sample test, and S ADP m × m (or S DDP m × $_{m}$) divided by the number of partitions of size m × m for the test of independence, using the likelihood ratio score per partition. The null distribution of these regularized statistics is computed by a permutation test, and they are distribution free. 

An extensive numerical investigation, partially summarized in Appendix G, led us to choose the minimum p -value as the preferred regularization method. Between the two combining functions, we preferred the minimum over Fisher, since Fisher was far more sensitive to the choice of the range of m for combining (see Table 6). Regularization using priors was less effective, except when the Poisson prior was used with parameter λ = √ N (see Table 7). We preferred the first type of regularization since it was at least as effective as regularizing by a Poisson prior, without requiring setting any additional parameters. This regularized statistic is consistent, as the next theorems show. 

Theorem 2.5. Let Y be continuous, and X categorical. Let N$_{g}$ be the total number of observations from distribution g ∈ { 1 ,...,K } , and N = ∑ K g $_{=1}$N$_{g}$ . If the distri- bution of Y differs at a continuous density point y$_{0}$ across values of X in at least two categories, label these 1 and 2, lim$_{N}$$_{→∞}$ min( N$_{1}$,N$_{2}$ ) N > 0 , then the permutation test based on min$_{m}$$_{∈{}$$_{2}$$_{,...,m}$$_{max}$$_{}}$ p$_{m}$ is consistent, if: 

<img src='content_image/320409.jpg'>

