The Chernoff Modification of the Fisher Exact Test for the Difference of Two Binomial Probabilities

To test the difference of two binomial probabilities, we will apply the Fisher exact test modified by Chernoff (2002), which will be called CMF test. The key idea is to use the P-value of Fisher exact test as a test statistic and this makes it possible to apply the Fisher exact test for more general problems. The numerical comparison with the likelihood ratio test and the Bayes test are also conducted and we find that the performance of CMF test is close to that of the likelihood ratio test and it is satisfactory in various situations. We also give a Kullback-Leibler information number of the margins to for the testing problems which justify the stability of P-values of CMF test.Fisher Exact Test, Binomial Probabilities, Chernoff Modification, Test of Difference of Probabilities

The Influence of Paulo Freire on Critical Pedagogy : In case of Henry Giroux, Michael Apple, and Peter McLaren

The purpose of this study is to examine the influence of Paulo Freire on distinguished scholars of critical pedagogy, such as Henry A. Giroux, Michael W. Apple, and Peter L. McLaren. The main findings are as follows: (1) All the three scholars supported Freire’s utopian view of society. His view was fundamental to a movement toward social justice. (2) Giroux appreciated that Freire offered not only “the language of critique” but also “the language of possibility”, and he proposed border pedagogy. It is necessary to close the gap between the real teachers and ”transformative intellectuals” or “border crossers”. (3) Apple stressed that not only Freire contributed to educational theory by writing articles or books but also he engaged in practice. Apple suggested decentered unity, so it is necessary to examine more practices in schools, communities, unions and so on. (4) McLaren focused particularly on class in Freire’s thought and he aimed for a revolution. However, Freire insisted that multiple differences must have been considered in his later years, so it doesn’t seem to be advanced from Freire’s original theory

On Theoretically Optimal Ranking Functions in Bipartite Ranking

<p>This article investigates the theoretical relation between loss criteria and the optimal ranking functions driven by the criteria in bipartite ranking. In particular, the relation between area under the ROC curve (AUC) maximization and minimization of ranking risk under a convex loss is examined. We characterize general conditions for ranking-calibrated loss functions in a pairwise approach, and show that the best ranking functions under convex ranking-calibrated loss criteria produce the same ordering as the likelihood ratio of the positive category to the negative category over the instance space. The result illuminates the parallel between ranking and classification in general, and suggests the notion of consistency in ranking when convex ranking risk is minimized as in the RankBoost algorithm for instance. For a certain class of loss functions including the exponential loss and the binomial deviance, we specify the optimal ranking function explicitly in relation to the underlying probability distribution. In addition, we present an in-depth analysis of hinge loss optimization for ranking and point out that the RankSVM may produce potentially many ties or granularity in ranking scores due to the singularity of the hinge loss, which could result in ranking inconsistency. The theoretical findings are illustrated with numerical examples. Supplementary materials for this article are available online.</p