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Fourth-order (FO) and, a short while ago, 2qth-order, q ges 2, high-resolution methods exploiting the information contained in the FO and the 2qth-order, q ges 2, statistics of the data, respectively, are now available for direction finding of non-Gaussian signals. Among these methods, the 2q-MUSIC methods, q ges 2, are the most popular. These methods are asymptotically robust to a Gaussian background noise whose spatial coherence is unknown and offer increasing resolution and robustness to modeling errors jointly with an increasing processing capacity as q increases. However, these methods have been mainly developed for arrays with identical sensors only and cannot put up with arrays of diversely polarized sensors in the presence of diversely polarized sources. In this context, the purpose of this paper is to introduce, for arbitrary values of q, q ges 1, three extensions of the 2q-MUSIC method, able to put up with arrays having diversely polarized sensors for diversely polarized sources. This gives rise to the so-called polarization diversity 2q-MUSIC (PD-2q-MUSIC) algorithms. For a given value of q, these algorithms are shown to increase the resolution, the robustness to modeling errors, and the processing capacity of the 2q-MUSIC method in the presence of diversely polarized sources. Besides, some PD-2q-MUSIC algorithms are shown to offer increasing performances with q when resolution in both direction of arrival and polarization is required.