Diffusion weighted magnetic resonance imaging (dMRI) is used to measure, in vivo, the self-diffusion of water molecules in biological tissues. High order tensors (HOTs) are used to model the apparent diffusion coefficient (ADC) profile at each voxel from the dMRI data. In this paper we propose: (i) A new method for estimating HOTs from dMRI data based on weighted least squares (WLS) optimization; and (ii) A new expression for computing the fractional anisotropy from a HOT that does not suffer from singularities and spurious zeros. We also present an empirical evaluation of the proposed method relative to the two existing methods based on both synthetic and real human brain dMRI data. The results show that the proposed method yields more accurate estimation than the competing methods.