Abstract
We show that in finite settings with identical firms and consumers, asymmetric pure price equilibria with positive profits exist. We consider a price competition duopoly for a homogeneous product. Demand stems from a second-stage consumption game at posted prices, with consumers' behavior impacted by negative network effects. We characterize equilibrium prices and demand. In all subgame-perfect pure price equilibria, both firms have positive profits, and in some, firms charge different prices.

Abstract
An important problem in directional statistics is to test the null hypothesis of a common mean direction for several populations. The Analysis of Variance (ANOVA) test for vectorial data may be used to test the hypothesis of the equality of the mean directions for several von Mises-Fisher populations. As this test is valid only for large concentrations, we propose in this paper to apply the resampling techniques of bootstrap and permutation to the ANOVA test. We carried out an extensive simulation study in order to evaluate the performance of the ANOVA test with the resampling techniques, for several sphere dimensions and different sample sizes and we compare with the usual ANOVA test for data from von Mises-Fisher populations. The purpose of this simulation study is also to investigate whether the proposed tests are preferable to the ANOVA test, for low concentrations and small samples. Finally, we present an example with spherical data.

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