This paper gives optimal algorithms for determining realvalued univariate unimodal regressions, that is, for determining the optimal regression which is increasing and then decreasing. Such regressions arise in a wide variety of applications. They are a form of shape-constrained nonparametric regression, closely related to isotonic regression. For the L2 metric our algorithm requires only (n) t...

The mean and the variance value of a random variable play an important role in statistical analysis. Sometimes the available observations are not precise, and we want to test the hypotheses of mean and variance in such environment. Hence, in this paper, we first extend and introduced L2−metric based on imprecise (fuzzy) observations, and then, the concepts of fuzzy test statistics are defined b...

Let M be a metric space with a finite diameter D and a finite normalized measure μM. Let the Hilbert space L2(M) of complex-valued functions be decomposed into a countable (whenM is infinite) or finite (with D+ 1 members whenM is finite) direct sum of mutually orthogonal subspaces L2(M) = V0 ⊕ V1 ⊕ · · · (V0 is the space of constant functions). We denote N = D + 1 ifM is finite and N = ∞ otherw...

We examine some differential geometric approaches to finding approximate solutions to the continuous time nonlinear filtering problem. Our primary focus is a new projection method for the optimal filter infinite dimensional Stochastic Partial Differential Equation (SPDE), based on the direct L2 metric and on a family of normal mixtures. We compare this method to earlier projection methods based...

in statistical inference, the point estimation problem is very crucial and has a wide range of applications. when, we deal with some concepts such as random variables, the parameters of interest and estimates may be reported/observed as imprecise. therefore, the theory of fuzzy sets plays an important role in formulating such situations. in this paper, we rst recall the crisp uniformly minimum ...

We apply the L2 based Fisher-Rao vector-field projection by Brigo, Hanzon and LeGland (1999) to finite dimensional approximations of the Fokker Planck equation on exponential families. We show that if the sufficient statistics are chosen among the diffusion eigenfunctions the finite dimensional projection or the equivalent assumed density approximation provide the exact maximum likelihood densi...

The l2 flattening lemma of Johnson and Lindenstrauss [JL84] is a powerful tool for dimension reduction. It has been conjectured that the target dimension bounds can be refined and bounded in terms of the intrinsic dimensionality of the data set (for example, the doubling dimension). One such problem was proposed by Lang and Plaut [LP01] (see also [GKL03, Mat02, ABN08, CGT10]), and is still open...

We show that the Yao graph Y4 in the L2 metric is a spanner with stretch factor 8 √ 2(29 + 23 √ 2).

We introduce a gradient operator that generalizes the Euclidean and Riemannian gradients. This operator acts on sections of vector bundles and is determined by three geometric data: a Riemannian metric on the base manifold, a Riemannian metric and a covariant derivative on the vector bundle. Under the assumption that the covariant derivative is compatible with the metric of the vector bundle, w...

Let (X,d,μ) be a space of homogeneous type in the sense Coifman and Weiss, i.e. d is quasi metric on X μ positive measure satisfying doubling condition. Suppose that u v are two locally finite Borel measures (X,d,μ). Subject to pair weights side condition, we characterize boundedness Calderón–Zygmund operator T from L2(u) L2(v) terms A2 condition testing conditions. For every cube B⊂X, have fol...