Uniform Mean Square Research Articles

Convergence of nonlinear filtering for multiscale systems with correlated Lévy noises

The work concerns nonlinear filtering problems of multiscale systems with Lévy noises in two cases-correlated noises and correlated sensor noises. First of all, we prove that the slow part of the origin system converges to the average system in the uniform mean square sense. Then based on the convergence result, in two cases of correlated Lévy noises and correlated Gaussian noises, we prove that the nonlinear filtering of the slow part converges to that of the average system in the weak sense and the [Formula: see text] sense, respectively. Finally, in the case of correlated sensor Gaussian noises, the nonlinear filtering for the slow part is shown to approximate that of the average system in the [Formula: see text] sense.

English

In this paper, we introduce a kernel-type version for the bi-dimensional extension of the Foster, Greer, and Thorbecke index that was introduced by Duclos et al. (2006a) for the purpose of a dominance approach to multidimensional poverty. The measure they used in their dominance exercise is essentially a generalization, from one to two dimensions, of the FGT index with separate poverty aversion parameters for each dimension. Our estimator is constructed by using a bidimensional Parzen-Rosenblatt kernel of a probability density function (pdf). We next provide its complete asymptotic behaviour by establishing its almost- sure uniform and its uniform mean square consistencies. A simulation study shows that it performs well for small samples comparatively to the empirical plug-in estimator. Our results are also extensions of those of Dia (2008) and of Ciss et al. (2014) in one dimension. R

Piecewise-Multilinear Interpolation of a Random Field

We consider a piecewise-multilinear interpolation of a continuous random field on a d-dimensional cube. The approximation performance is measured using the integrated mean square error. Piecewise-multilinear interpolator is defined by N-field observations on a locations grid (or design). We investigate the class of locally stationary random fields whose local behavior is like a fractional Brownian field, in the mean square sense, and find the asymptotic approximation accuracy for a sequence of designs for large N. Moreover, for certain classes of continuous and continuously differentiable fields, we provide the upper bound for the approximation accuracy in the uniform mean square norm.

Piecewise-Multilinear Interpolation of a Random Field

We consider a piecewise-multilinear interpolation of a continuous random field on a d-dimensional cube. The approximation performance is measured using the integrated mean square error. Piecewise-multilinear interpolator is defined by N-field observations on a locations grid (or design). We investigate the class of locally stationary random fields whose local behavior is like a fractional Brownian field, in the mean square sense, and find the asymptotic approximation accuracy for a sequence of designs for large N. Moreover, for certain classes of continuous and continuously differentiable fields, we provide the upper bound for the approximation accuracy in the uniform mean square norm.

Estimation nonparamétrique de la distribution des revenus et de l'indice de pauvreté

In this Note we propose an estimator of Foster, Greer and Thorbecke class of measures P ( z , α ) = ∫ 0 z ( z − x z ) α f ( x ) d x where z is the poverty line, f ( x ) the density of the income distribution and α the so-called poverty aversion. The estimator is constructed with Parzen–Rosenblatt kernel. Uniform almost sure consistency and uniform mean square consistency are established. A simulation study indicates that this new estimator performs well in finite samples. To cite this article: G. Dia, C. R. Acad. Sci. Paris, Ser. I 346 (2008).

Mean Square Summability of Solution of Stochastic Difference Second-Kind Volterra Equation with Small Nonlinearity

Stochastic difference second-kind Volterra equation with continuous time and small nonlinearity is considered. Via the general method of Lyapunov functionals construction, sufficient conditions for uniform mean square summability of solution of the considered equation are obtained.

The Order 1.5 Approximation for Solutions of Jump-Diffusion Equations

Using in Itô equation, only information from increments of a driving Wiener process make it possible to make such Runge-Kutta approximations in which the error in ℒ2-norm is of an order not greater than 1 in general case. This convergence order can be greater only when functional coefficients from the equation satisfy a certain additional condition. However, it is obvious that in general case information only from increments of the driving Wiener process does not allow to compute trajectories of the solution at discretization points, even though the analytical formula of this solution is known. In this article we show that if values of increments of the driving Wiener process are sufficient to calculate values of a solution trajectory at discretization points, then the condition mentioned above, which allows to make an order approximation, also is fulfilled. Moreover, for a certain class of scalar jump-diffusion equations a numerical method is described which converges with uniform mean square error of order .

Nonparametric least squares estimation of a regression function

The problem 01 nonparametric function iittmg with me observation model: is considered, where are uncorrelated random variables irith zero mean and finite variance are fixed design variables diile / is an unknown function with only requirements imposed. AsymptoticN{n) ehaviour of the estimator is studied for obtained by the least' nuares methodbeing a complete set of orthonormal functions, Sufficient conditions for consistency in the sense of integrated and uniform mean square rror are derived. Bounds obtained for mean integrated square error (MISE) are used o indicate that for smooth functions the convergence rate of MISE is the optimal one.These results are obtained under minimal requirements imposed on measurement errors. Tnder additiona assumotion that these variables are mutually independent also strong niform converg once of the estimator is shown.

Equivalence of uniform asymptotic unbiasedness, mean square and strong consistencies of recursive estimates of a density and its p-th derivative

Menon, Prasad and Singh have given nonparametric recursive kernel estimators of a probability density function f and its p-th order derivative f ( p) based on a random sample of size n from f. Denoting the estimates of f ( p) (where f (0) is simply the p.d.f. f) by f n, p they have shown that if f ( p) is bounded and uniformly continuous, then their recursive estimators f n, p are uniformly mean square as well as uniformly strongly consistent for f ( p) . In this note we will prove the converse. That is, we show that for uniform strong consistency or for uniform mean square consistency of f n, p it is necessary that f ( p) be bounded and uniformly continuous, whether it is the case of p = 0 or p ≥ 1. Under certain conditions on the window-width function, it is shown that each of the properties of uniform asymptotic unbiasedness, uniform mean square consistency and uniform strong consistency of f n, p is equivalent to uniform continuity of f ( p) along with its boundedness.

Nonparametric estimates of distribution functions

Let F be a distribution function with a density f. For non-negative integers r, statistics Fr based on a random sample of size n from F are exhibited. Under no assumption whatsoever on f, it is shown that sup with probability one as n If r > 1 and is integrable, then sup F(x) is at least of the order O(n-1 loglog n)1/2 W.p.1. The same conclusion holds with regard to is bounded. Uniform mean square consistencies of Fo and Fr are also proved and rates of convergence are investigated. Exact asymptotic values for the bias and variance of the estimators Fr are obtained. The central limit theorem with the rate of convergence for our estimators is established.

Properties of Hermite Series Estimation of Probability Density

An unknown density function $f(x)$, its derivatives, and its characteristic function are estimated by means of Hermite functions $\{h_j\}$. The estimates use the partial sums of series of Hermite functions with coefficients $\hat{a}_{jn} = (1/n) \sum^n_{i=1} h_j(X_i)$ where $X_1\cdots X_n$ represent a sequence of i.i.d. random variables with the unknown density function $f$. The integrated mean square rate of convergence of the $p$th derivative of the estimate is $O(n^{(p/r) + (5/6r)-1})$. The same is true for the Fourier transform of the estimate to the characteristic function. Here the assumption is made that $(x - D)^r f \in L^2$ and $p < r$. Similar results are obtained for other conditions on $f$ and uniform mean square convergence.

Nonparametric estimation of mixed partial derivatives of a multivariate density

On the basis of a random sample of size n on an m-dimensional random vector X, this note proposes a class of estimators f n ( p) of f ( p) , where f is a density of X w.r.t. a σ-finite measure dominated by the Lebesgue measure on R m , p = ( p 1,…, p m ), p j ≥ 0, fixed integers, and for x = ( x 1,…, x m ) in R m , f (p)(x) = ∂ p 1+…+p m f (x)/(∂ p 1 x 1 … ∂ p m x m ). Asymptotic unbiasedness as well as both almost sure and mean square consistencies of f n ( p) are examined. Further, a necessary and sufficient condition for uniform asymptotic unbisedness or for uniform mean square consistency of f n ( p) is given. Finally, applications of estimators of this note to certain statistical problems are pointed out.