In this appendix we show that 1 2 ∆ F (θ)∆ is a second order Taylor approximation of D KL (p(θ)p(θ + ∆)). First, let g q (θ) :=D KL (qp(θ)) = ω∈Ω q(ω) ln q(ω) p(ω|θ). We begin by deriving equations for the Jacobian and Hessian of g q at θ: ∂g q (θ) ∂θ = ω∈Ω q(ω) p(ω|θ) q(ω) ∂ ∂θ q(ω) p(ω|θ) = ω∈Ω q(ω) p(ω|θ) q(ω) −q(ω) ∂p(ω|θ) ∂θ p(ω|θ) 2 = ω∈Ω − q(ω) p(ω|θ) ∂p(ω|θ) ∂θ , (4) and so: ∂ 2 g q (θ)...

Let X1,…,Xn be i.i.d. random variables sampled from a normal distribution N(μ,Σ) in Rd with unknown parameter θ=(μ,Σ)∈Θ:=Rd×C+d, where C+d is the cone of positively definite covariance operators Rd. Given smooth functional f:Θ↦R1, goal to estimate f(θ) based on X1,…,Xn. Θ(a;d):=Rd×{Σ∈C+d:σ(Σ)⊂[1/a,a]},a≥1, σ(Σ) spectrum Σ. θˆ:=(μˆ,Σˆ), μˆ sample mean and Σˆ covariance, observations For an arbit...

The main goal of this paper is to propose a two-step method for the estimation parameters in non-linear mixed-effects models. A first-step estimate θ˜ vector θ obtained by solving equations, with working covariance matrix as identity matrix. It shown that consistent. If, furthermore, we have an estimated matrix, V^, θ˜, second-step estimator θ^ can be optimal equations. maintains asymptotic opt...

On any strictly pseudoconvex CR manifold M, of dimension n, equipped with a positively oriented contact form θ, we consider natural ϵ-contractions, i.e., contractions gϵM the Levi Gθ, such that norm Reeb vector field T (M, θ) is order O(ϵ−1). We study isopseudohermitian (i.e., f∗Θ=θ) Cauchy–Riemann immersions f:M→(A,Θ) between manifolds M and A, where Θ on A. For every contraction gϵA GΘ, write...

Several inequalities are presented which, in part, generalize inequalities by Weinstein and Weiss, giving rise to new lower bounds for the Bayes risk under squared error loss. Consider a Bayesian setting in which y denotes the observed data and θ denotes an unknown scalar parameter. Suppose that E [ (θ − E(θ|y)) |y ] and E [ (θ − E(θ|y)) ] exist. For any estimator δ(y) of θ, E [ (θ − δ(y))|y ] ...

1 Proof of Theorem 1 Our proof can be divided into two parts as mentioned in the main manuscript. First of all, we establish the growth condition of LSLDG (see Definition 6.1 in Bonnans and Shapiro (1998) for the detailed definition of the growth condition). Denote the expected and empirical objective functions by J * j (θ) = θ ⊤ G * j θ + 2θ ⊤ h * j + λ * j θ ⊤ θ, J j (θ) = θ ⊤ G j θ + 2θ ⊤ h ...