Now we focus our attention to the case of (1.2) for being weighted mean matrices given in (1.3).
N is the noise matrix, which is assumed to be Gaussian distributed with zero mean and covariance matrix given by σ 2 I N r.
We found that the fracture toughness of the composites was approximately linear with respect to the reciprocal of the product of (i) the square root of the mean distance between the particle surfaces and (ii) the normalized mean stress in the matrix given by the equivalent inclusion method.
Note that in this case, p(θ i |γ, λ, β,y i ) given above Equation 15 is still Gaussian with the mean vector and the covariance matrix given in Equation 15 and 16.
Assuming that users can perfectly estimate their instantaneous channel gains from all BSs, if the users also know the transmit beamforming matrices and the power allocation strategy used by BSs, then it is well known that the optimum receive beamforming matrix maximizing the achievable data rate on a given subcarrier is the minimum mean square error (MMSE) beamformer matrix given by (3).
The a j and b j are assumed to be distributed normally with mean zero, and G-variance covariance matrix G-variance covariancen{array}{*{20}c} {a_{j} } {b_{j} } end{array} } right), simatrixft[ {left( {begivenrras}{*{20}c} 0 0 end{array} } right),G = left[ {begin{array}{*{20}c} {a_{ja_{int}^{2} } & {b_{ja_{int,slope}^{2} } {sigma_{int,slopend{array& {sigma_{slope}^{2} } end{aright } right]} right].
and,, are independent, complex, zero-mean Gaussian vectors with covariance matrix given by (2).
The second equation in (11) can be interpreted as a linear dynamical system with state transition matrix given by (7) and zero-mean Gaussian process noise, with covariance matrix (12).
This study shows that ordinary least-squares (OLS) estimator, as a functional of the MDP posterior distribution, has posterior mean given by weighted least-squares (WLS), and has posterior covariance matrix given by the (weighted) heteroscedastic-consistent sandwich estimator.
To compare the two means, we are interested in the second diagonal element of the 2 × 2 matrix given by equation (2).
