Computing tails of compound distributions using direct numerical integration

Efficient Numerical Inversion of Characteristic Functions for Computing Tails of Compound Distributions

An adaptive direct numerical integration (DNI) algorithm is developed for inverting characteristic functions of compound distributions, enabling efficient computations of high quantiles and conditional Value at Risk (CVaR). A key innovation of the numerical scheme is an effective tail integration approximation that reduces the truncation errors significantly. High precision results of the 0.999...

Numerical integration using spline quasi-interpolants

In this paper, quadratic rules for obtaining approximate solution of definite integrals as well as single and double integrals using spline quasi-interpolants will be illustrated. The method is applied to a few test examples to illustrate the accuracy and the implementation of the method.

A Numerical Integration Method by Using Generalized Series of Functions

Tails of Condition Number Distributions

Let κ be the condition number of an m-by-n matrix with independent standard Gaussian entries, either real (β = 1) or complex (β = 2). The major result is the existence of a constant C (depending on m, n, β) such that P [κ > x] < C x−β for all x. As x→∞, the bound is asymptotically tight. An analytic expression is given for the constant C, and simple estimates are given, one involving a Tracy-Wi...

Degree and Sensitivity: Tails of Two Distributions

The sensitivity of a Boolean function f is the maximum, over all inputs x, of the number of sensitive coordinates of x (namely the number of Hamming neighbors of x with different f -value). The well-known sensitivity conjecture of Nisan (see also Nisan and Szegedy) states that every sensitivity-s Boolean function can be computed by a polynomial over the reals of degree poly(s). The best known u...