For estimating the direction of arrival (DOA) s of non-stationary source signals such as speech and audio, a constrained optimization problem (COP) that exploits the spatial diversity provided by an array of sensors is formulated in terms of a noise-eliminated local 2 rho th-order cumulant matrix. The COP solution provides a weight vector to the look direction such that it is constrained to the 2 rho th-order source-signal subspace when the look direction is in alignment with the true DOA; otherwise, it is constrained to the 2 rho th-order noise subspace. This weight vector is incorporated into the spatial spectrum to determine the degree of orthogonality between itself and either the 2 rho th-order source-signal subspace when the number of sources is unknown, or the 2 rho th-order noise subspace when the number of sources is known. For a uniform linear array (ULA) of M sensors, the spatial spectrum for known number of sources can theoretically be shown to identify up to 2 rho(M - 1) sources. Realizing the difficulty in identifying stationarity in the received sensor signals, the estimate of the noise-eliminated local 2 rho th-order cumulant matrix is marginalized over various possible stationary segmentations, for a more robust DOA estimation. In this paper, we focus on the use of local second and fourth order cumulants (rho = 1, 2), and the proposed algorithms when rho = 1 outperformed the KR subspace-based algorithms and also the 4-MUSIC for globally non-stationary, non-Gaussian synthetic data and also for speech/audio in various adverse environments. We verified that the identifiability for rho = 2 is improved by two-folds compared to that for rho = 1 with an ULA.