1 Sn lim log P( ∈ A). n→∞ n n Given θ ∈ Rd, define M(θ) = E[exp((θ, X1))] where (·, ·) represents the inner p product of two vectors: (a, b) = i aibi. Define I(x) = supθ∈Rd ((θ, x) − log M(θ)), where again I(x) = ∞ is a possibility. Theorem 1 (Cram ́ Suppose M(θ) < ∞ er’s theorem in multiple dimensions). for all θ ∈ Rd. Then (a) for all closed set F ⊂ Rd , 1 Sn lim sup log P( ∈ F ) ≤ − inf I(x) ...

) | ( j k Y prob pkj = = = θ The jth column of P gives the probability distribution associated with the parameter value j = θ . We consider the problem of estimatingθ , based on an observed sample of Y. The m values of θ are called the hypotheses, and finding the best value of θ is hypothesis testing. A randomized detector of θ is a random variable , with a distribution dependent on the observe...

The Sharma–Mittal entropies generalize the celebrated Shannon, Rényi and Tsallis entropies. We report a closed-form formula for the Sharma–Mittal entropies and relative entropies for arbitrary exponential family distributions. We explicitly instantiate the formula for the case of the multivariate Gaussian distributions and discuss on its estimation. Q1 Q2 PACS numbers: 03.65.Ta, 03.67.−a (Some ...

co(A): the convex hull of a set A, supp(Q): the support of a measure Q ∈ M, suppP(g(X θ)): the support of g(X θ) when X is distributed according to P ∈ M, s(Q θ): the dimension of the co(suppP(g(X θ))). The principal challenge in deriving our optimality result is establishing part (a) of Theorem 3.1. For ease of exposition, we provide an outline of the proof of this claim before its formal deri...

Abstract. Maximum likelihood (ML) estimation based on bivariate record data is considered as the general inference problem. Assume that the process of observing k records is repeated m times, independently. The asymptotic properties including consistency and asymptotic normality of the Maximum Likelihood (ML) estimates of parameters of the underlying distribution is then established, when m is ...

co(A): the convex hull of a set A, supp(Q): the support of a measure Q ∈ M, suppP(g(X θ)): the support of g(X θ) when X is distributed according to P ∈ M, s(Q θ): the dimension of the co(suppP(g(X θ))). The principal challenge in deriving our optimality result is establishing part (a) of Theorem 3.1. For ease of exposition, we provide an outline of the proof of this claim before its formal deri...

The concepts of fuzzy θ-open (θ-closed)sets and fuzzy θ-closure operator are introduced and discussed in intuitionistic fuzzy topological spaces. As applications of these concepts, certain functions are characterized in terms of intuitionistic fuzzy θ-closure operator.

Let M(A, θ) be a free partially commutative monoid. We give here a necessary and sufficient condition on a subalphabet B ⊂ A such that the right factor of a bisection M(A, θ) = M(B, θB).T be also partially commutative free. This extends strictly the (classical) elimination theory on partial commutations and allows to construct new factorizations of M(A, θ) and associated bases of LK(A, θ).

Bagging is a device intended for reducing the prediction error of learning algorithms. In its simplest form, bagging draws bootstrap samples from the training sample, applies the learning algorithm to each bootstrap sample, and then averages the resulting prediction rules. We extend the definition of bagging from statistics to statistical functionals and study the von Mises expansion of bagged ...