MUltiple SIgnal Classification (MUSIC) is a standard technique for direction of arrival (DOA) estimation with high resolution. However, MUSIC cannot estimate DOAs accurately in the case of underdetermined conditions, where the number of sources exceeds the number of microphones. To overcome this drawback, an extension of MUSIC using cumulants called 2q-MUSIC has been proposed, but this method greatly suffers from the variance of the statistics, given as the temporal mean of the observation process, and requires long observation. In this paper, we propose a new approach for extending MUSIC that exploits higher-order moments of the signal for the underdetermined DOA estimation with smaller variance. We propose an estimation algorithm that nonlinearly maps the observed signal onto a space with expanded dimensionality and conducts MUSIC-based correlation analysis in the expanded space. Since the dimensionality of the noise subspace is increased by the mapping, the proposed method enables the estimation of DOAs in the case of underdetermined conditions. Furthermore, we describe the class of mapping that allows us to analyze the higher-order moments of the observed signal in the original space. We compare 2q-MUSIC and the proposed method through an experiment assuming that the true number of sources is known as prior information to evaluate in terms of the bias-variance tradeoff of the statistics and computational complexity. The results clarify that the proposed method has advantages for both computational complexity and estimation accuracy in short-time analysis, i.e., the time duration of the analyzed data is short.