In the present paper we construct a new, simple and powerful test for independence by using symbolic dynamics and permutation entropy as a measure of serial dependence. We also give the asymptotic distribution of an affine transformation of the permutation entropy under the null hypothesis of independence. An application to several daily financial time series illustrates our approach.2

RESUMENIn the present paper, we construct a new, simple, consistent and powerful test for spatial independence, called the SG test, by using symbolic dynamics and symbolic entropy as a measure of spatial dependence. We also give a standard asymptotic distribution of an affine transformation of the symbolic entropy under the null hypothesis of independence in the spatial process. The test statistic and its standard limit distribution, with the proposed symbolization, are invariant to any monotonuous transformation of the data.The test applies to discrete or continuous distributions. Given that the test is based on entropy measures, it avoids smoothed nonparametric estimation. We include a Monte Carlo study of our test, together with the well-known Moran's I, the SBDS (de Graaff

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BackgroundThe etiology of complex diseases is due to the combination of genetic and environmental factors, usually many of them, and each with a small effect. The identification of these small-effect contributing factors is still a demanding task. Clearly, there is a need for more powerful tests of genetic association, and especially for the identification of rare effectsResultsWe introduce a new genetic association test based on symbolic dynamics and symbolic entropy. Using a freely available software, we have applied this entropy test, and a conventional test, to simulated and real datasets, to illustrate the method and estimate type I error and power. We have also compared this new entropy test to the Fisher exact test for assessment of association with low-frequency SNPs. The entropy test is generally more powerful than the conventional test, and can be significantly more powerful when the genotypic test is applied to low allele-frequency markers. We have also shown that both the Fisher and Entropy methods are optimal to test for association with low-frequency SNPs (MAF around 1-5%), and both are conservative for very rare SNPs (MAF<1%)ConclusionsWe have developed a new, simple, consistent and powerful test to detect genetic association of biallelic/SNP markers in case-control data, by using symbolic dynamics and symbolic entropy as a measure of gene dependence. We also provide a standard asymptotic distribution of this test statistic. Given that the test is based on entropy measures, it avoids smoothed nonparametric estimation. The entropy test is generally as good or even more powerful than the conventional and Fisher tests. Furthermore, the entropy test is more computationally efficient than the Fisher's Exact test, especially for large number of markers. Therefore, this entropy-based test has the advantage of being optimal for most SNPs, regardless of their allele frequency (Minor Allele Frequency (MAF) between 1-50%). This property is quite beneficial, since many researchers tend to discard low allele-frequency SNPs from their analysis. Now they can apply the same statistical test of association to all SNPs in a single analysis., which can be especially helpful to detect rare effects.

a b s t r a c tWe propose a novel test to determine, given a time series, if the dynamics are generated by a deterministic (including low dimensional chaos), rather than a stochastic, process. In addition, we introduce a new nonparametric bootstrap test for independence which is consistent against a broad class of alternatives. The conditions under which the tests can be applied are very weak. The advantages of the presented methods are simplicity, invariance with respect to monotonic transformations and the applicability of the tests regardless of the discrete or continuous nature of the data generating process. We conduct several simulation studies to evaluate the performance of our tests on well-known dynamic processes. Finally, our tests are applied to several sets of financial returns that have been recently studied.