The lower bound for the variance of unbiased estimators for one-directional family of distributions

Summary In this paper we introduce the concept of one-directionality which includes both cases of location (and scale) parameter and selection parameter and also other cases, and establish some theorems for shapr lower bounds and for the existence of zero variance unbiased estimator for this class of non-regular distributions.     

A basic family of iteration functions for polynomial root finding and its characterizations

Let p(x) be a polynomial of degree n?2 with coefficients in a subfield K of the complex numbers. For each natural number m?2, let L_m(x) be the m×m lower triangular matrix whose diagonal entries are p(x) and for each j=1,…,m−1, its jth subdiagonal entries are . For i=1,2, let L_m^i)(x) be the matrix obtained from L_m(x) by deleting its first i rows and its last i columns. L₁⁽¹⁾(x)≡1. Then, the function B_m(x)=x−p(x) defines a fixed-point iteration function having mth order convergence rate for simple roots of p(x). For m=2 and 3, B_m(x) coincides with Newton's and Halley's, respectively. The function B_m(x) is a member of S(m,m+n−2), where for any M?m, S(m,M) is the set of all rational iteration functions g(x) ∈ K(x) such that for all roots θ of p(x), then g(x)=θ+∑_i=m^Mγ_i(x)(θ−x)ⁱ, with γ_i(x) ∈ K(x) and well-defined at any simple root θ. Given g ∈ S(m,M), and a simple root θ of p(x), gⁱ(θ)=0, i=1, …, m−1 and the asymptotic constant of convergence of the corresponding fixed-point iteration is . For B_m(x) we obtain . If all roots of p(x) are simple, B_m(x) is the unique member of S(m,m + n − 2). By making use of the identity , we arrive at two recursive formulas for constructing iteration functions within the S(m,M) family. In particular, the family of B_m(x) can be generated using one of these formulas. Moreover, the other formula gives a simple scheme for constructing a family of iteration functions credited to Euler as well as Schröder, whose mth order member belongs to S(m,mn), m>2. The iteration functions within S(m,M) can be extended to any arbitrary smooth function f, with the uniform replacement of p^(j) with f^(j) in g as well as in γ_m(θ).     

An analytic approximation to the cardinal functions of Gaussian radial basis functions on an infinite lattice

Gaussian radial basis functions (RBFs) have been very useful in computer graphics and for numerical solutions of partial differential equations where these RBFs are defined, on a grid with uniform spacing h, as translates of the “master” function (x;α,h)≡exp(-[α²/h²]x²) where α is a user-choosable constant. Unfortunately, computing the coefficients of (x-jh;α,h) requires solving a linear system with a dense matrix. It would be much more efficient to rearrange the basis functions into the equivalent “Lagrangian” or “cardinal” basis because the interpolation matrix in the new basis is the identity matrix; the cardinal basis C_j(x;α,h) is defined by the set of linear combinations of the Gaussians such that C_j(kh)=1 when k=j and C_j(kh)=0 for all integers . We show that the cardinal functions for the uniform grid are C_j(x;h,α)=C(x/h-j;α) where C(X;α)≈(α²/π)sin(πX)/sinh(α²X). The relative error is only about 4exp(-2π²/α²) as demonstrated by the explicit second order approximation. It has long been known that the error in a series of Gaussian RBFs does not converge to zero for fixed α as h→0, but only to an “error saturation” proportional to exp(-π²/α²). Because the error in our approximation to the master cardinal function C(X;α) is the square of the error saturation, there is no penalty for using our new approximations to obtain matrix-free interpolating RBF approximations to an arbitrary function f(x). The master cardinal function on a uniform grid in d dimensions is just the direct product of the one-dimensional cardinal functions. Thus in two dimensions . We show that the matrix-free interpolation can be extended to non-uniform grids by a smooth change of coordinates.     

Bounds on differences of adjacent zeros of Bessel functions and iterative relations between consecutive zeros

Bounds for the distance between adjacent zeros of cylinder functions are given; and are such that ; stands for the th positive zero of the cylinder (Bessel) function , , .

These bounds, together with the application of modified (global) Newton methods based on the monotonic functions and , give rise to forward ( ) and backward ( ) iterative relations between consecutive zeros of cylinder functions.

The problem of finding all the positive real zeros of Bessel functions for any real and inside an interval , 0$">, is solved in a simple way.

     

An application of the branch,bound, and remember algorithm to a new simple assembly line balancing dataset

The simple assembly line balancing problem (SALBP) is a well-studied NP-complete problem for which a new problem database of generated instances was published in 2013. This paper describes the application of a branch, bound, and remember (BB&R) algorithm using the cyclic best-first search strategy to this new database to produce provably exact solutions for 86% of the unsolved problems in this database. A new backtracking rule to save memory is employed to allow the BB&R algorithm to solve many of the largest problems in the database.     

Elementary density bounds for self-similar sets and application

Falconer[1] used the relationship between upper convex density and upper spherical density to obtain elementary density bounds for s-sets at H S-almost all points of the sets. In this paper, following Falconer[1], we first provide a basic method to estimate the lower bounds of these two classes of set densities for the self-similar s-sets satisfying the open set condition （OSC）, and then obtain elementary density bounds for such fractals at all of their points. In addition, we apply the main results to the famous classical fractals and get some new density bounds.     

Lower and upper bounds of condition number for Vandermonde-wise matrices and method of fundamental solutions using pseudo radial-lines

Consider the method of fundamental solutions (MFS) for 2D Laplace's equation in a bounded simply connected domain $S$. In the standard MFS, the source nodes are located on a closed contour outside the domain boundary $\mathrm{\Gamma }(=\partial S)$, which is called pseudo-boundary. For circular, elliptic, and general closed pseudo-boundaries, analysis and computation have been studied extensively. New locations of source nodes are proposed along two pseudo radial-lines outside $\mathrm{\Gamma }$. Numerical results are very encouraging and promising. Since the success of the MFS mainly depends on stability, our efforts are focused on deriving the lower and upper bounds of condition number (Cond). The study finds stability properties of new Vandermonde-wise matrices on nodes $x_i\in [a,b]$ with . The Vandermonde-wise matrix is called in this article if it can be decomposed into the standard Vandermonde matrix. New lower and upper bounds of Cond are first derived for the standard Vandermonde matrix, and then for new algorithms of the MFS using two pseudo radial-lines. Both lower and upper bounds of Cond are intriguing in the stability study for the MFS. Numerical experiments are carried out to verify the stability analysis made. Since the fundamental solutions (as $\{\ln |\overline{{PQ}_i}|\}$) are the basis functions of the MFS, new Vandermonde-wise matrices are found. Since the nodes $x_i\in [a,b]$ with may come from approximations and interpolations by the Laurent polynomials with singular part, the conclusions in this article are important not only to the MFS but also to matrix analysis.     

Simple geometric constructions of quadratically and cubically convergent iterative functions to solve nonlinear equations

In this paper, we derive one-parameter families of Newton, Halley, Chebyshev, Chebyshev-Halley type methods, super-Halley, C-methods, osculating circle and ellipse methods respectively for finding simple zeros of nonlinear equations, permitting f ′ (x) = 0 at some points in the vicinity of the required root. Halley, Chebyshev, super-Halley methods and, as an exceptional case, Newton method are seen as the special cases of the family. All the methods of the family and various others are cubically convergent to simple roots except Newton’s or a family of Newton’s method.      

Stochastic bounds on distributions of optimal value functions with applications to pert,network flows and reliability

In various network models the quantities of interest are optimal value functions of the form max X _i , min X _i , min maxX _i , max minX _i , where the inner operation is on the nodes of a path/cut and the outer operation on all paths/cuts, e.g. shortest path of a project network, maximal flow of a flow network or lifetime of a reliability system. ForX _i random with given marginal distributions, we obtain bounds for the optimal value functions, based on common and on antithetic joint distributions.This work was carried out during a visit to RWTH Aachen, supported by DAAD.     

An approximation method to estimate the Hausdorff measure of the Sierpinski gasket

In this paper, we firstly define a decreasing sequence {P^n(S)} by the generation of the Sierpinski gasket where each P^n(S) can be obtained in finite steps. Then we prove that the Hausdorff measure H^8(S) of the Sierpinski gasket S can be approximated by {P^n(S)} with P^n(S)/(1 1/2^n-3)s ≤ H^8(S)≤ Pb(S).An algorithm is presented to get P^n(S) for n≤ 5. As an application, we obtain the best lower bound of H^8(S) till now: H^8(S) ≥ 0.5631.     

An application of the Morgenstern family with standard two‐sided power and gamma marginal distributions to the Bayes premium in the collective risk model

The Bayes premium is a quantity of interest in the actuarial collective risk model, under which certain hypotheses are assumed. The usual assumption of independence among risk profiles is very convenient from a computational point of view but is not always realistic. Recently, several authors in the field of actuarial and operational risks have examined the incorporation of some dependence in their models. In this paper, we approach this topic by using and developing a Farlie–Gumbel–Morgenstern (FGM) family of prior distributions with specified marginals given by standard two‐sided power and gamma distributions. An alternative Poisson–Lindley distribution is also used to model the count data as the number of claims. For the model considered, closed expressions of the main quantities of interest are obtained, which permit us to investigate the behavior of the Bayes premium under the dependence structure adopted (Farlie–Gumbel–Morgenstern) when the independence case is included. Copyright © 2012 John Wiley & Sons, Ltd.     

An extension of a bound for functions in Sobolev spaces,with applications to (<Emphasis Type="Italic">m</Emphasis>, <Emphasis Type="Italic">s</Emphasis>)-spline interpolation and smoothing

Given a function f on a bounded open subset Ω of with a Lipschitz-continuous boundary, we obtain a Sobolev bound involving the values of f at finitely many points of . This result improves previous ones due to Narcowich et al. (Math Comp 74, 743–763, 2005), and Wendland and Rieger (Numer Math 101, 643–662, 2005). We then apply the Sobolev bound to derive error estimates for interpolating and smoothing (m, s)-splines. In the case of smoothing, noisy data as well as exact data are considered.