1. First, assume a scalar unknown parameter θ (a) Min classical MSE estimator: ˆ θ(X) = arg minˆθ E X [(θ − ˆ θ(X)) 2 2 ] (b) Often the resulting estimator is not realizable, i.e. it depends on θ.

With an optimal design one wants to determine the ideal allocation of observations for the estimation of the unknown parameter vector θ of a given model ( ) ( ) θ ; i i x f y E = , with [ ] p T θ θ θ L 1 = , ( ) n T x x X ,..., 1 = , the xi from a given experimental region ER.. Let ( ) X V ; θ be the asymptotical variance-covariance matrix of the least squares estimatorθ̂ of θ . We consider the ...

Conditions on the existence of self-dual θ-codes defined over a finite field IFq are studied for θ automorphism of IFq. When q ≡ 1 (mod 4) it is proven that there always exists a self-dual θ-code in any dimension and that self-dual θ-codes of a given dimension are either all θ-cyclic or all θ-negacyclic. When q ≡ 3 (mod 4), there does not exist a selfdual θ-cyclic code and a necessary and suffi...

We give a two-parameter quantum deformation of the exterior plane and its differential calculus without the use of any R-matrix and relate it to the differential calculus with the R-matrix. We prove that there are two types of solutions of the Yang-Baxter equation whose symmetry group is GLp,q(2). We also give a two-parameter deformation of the fermionic oscillator algebra. Quantum groups are a...

CHAPTER 4. ASYMPTOTICS OF POSTERIORS AND MODEL SELECTION 4.1 Consistency of posteriors. Given a measurable family {P θ , θ ∈ Θ}, dominated by a σ-finite measure v, for a measurable space (Θ, T), a prior π on Θ, and observations X 1 , X 2 ,. .. i.i.d. (P θ 0) for some θ 0 ∈ Θ, we have posteriors π x,n on Θ where x = (X 1 ,. .. , X n) for each n. Recall that we have defined the posteriors by mult...

We study skew product systems driven by a hyperbolic base map Ŝ : Θ → Θ (for example, a baker map or an Anosov surface diffeomorphism) and with simple concave fibre maps on R+ like x → ĝ(θ) arctan(x), where θ ∈ Θ is a parameter driven by the base map. The fibrewise attractor is the graph of an upper semicontinuous function θ → φ̂∞(θ) ∈ R+. For many choices of ĝ, φ̂∞ has a residual set of zeros, b...

We present tight bounds on the spanning ratio of a large family of ordered θ-graphs. A θ-graph partitions the plane around each vertex into m disjoint cones, each having aperture θ = 2π/m. An ordered θ-graph is constructed by inserting the vertices one by one and connecting each vertex to the closest previously-inserted vertex in each cone. We show that for any integer k ≥ 1, ordered θ-graphs w...

Gradient ascent and stochastic gradient ascent provide attractive approaches in learning parameters, Θ of DPP kernel L(Θ) because of their theoretical guarantees, but require knowledge of the gradient of the log-likelihood L(Θ). In the discrete DPP setting, this gradient can be computed straightforwardly and we provide examples for discrete Gaussian and polynomial kernels here. L(Θ) = T t=1 log...

where k = m1 + . . . + mn is the total number of parts, θ > 0 is a parameter of the distribution, and θ = θ(θ + 1) . . . (θ + n − 1). For background and applications to genetics see [11, 12, 13, 6, 5, 19]. This note presents a family of distributions Pn,θ,α for random partitions of n, with two parameters θ ≥ 0 and 0 ≤ α ≤ 1. For α = 0, θ > 0, Pn,θ,0 = Pn,θ as in (1). As indicated at the end of ...

The pair 〈Θ, P 〉 of a pointset Θ ⊂ IR and a polynomial space P on IR is correct if the restriction map P → IR : p 7→ p|Θ is invertible, i.e., if there is, for any f defined on Θ, a unique p ∈ P which matches f on Θ. We discuss here a particular assignment Θ 7→ ΠΘ, introduced in [3], for which 〈Θ,ΠΘ〉 is always correct, and provide an algorithm for the contruction of a basis for ΠΘ which is relat...