Multivariate Statistics



Log likelihood function for θ based on observations z l(θ; z) = log fz(z; θ). Maximum likelihood estimate of θ Dθr l(θ; z) = 0. Observed information for θ and its matrix inverse ir, s = -Dr, s(θ^), ir, s. Re-parameterisation in terms of φ Dr* = θri Di, Dr, s* = θri θsj Di, j + θr, si Di. ir, s* = θri θsj ii, j, i*, r, s = φir φjs ii, j, θri = Dφr θi, θr, si = Dφr, φs θi. Fisher information for θ E(Dθr, s l(θ; z); θ). Mean E(xi). Covariance matrix cov(xi, xj) = e(xi xj) - e(xi) e(xj). Affine transformation from x to y yr = ar + air xi, mean ar + air E(xi), covariance air ajs cov(xi, xj), air = Dxi yr.

High-Dimensional Statistics