Inverse Of Cumulative Distribution Function Research Articles

A shape-preserving spline interpolation for sampling designs from inverse distributions

Given a strictly monotone cumulative function F:[a,b]⟶[c,d], with a,b,c,d∈R, a<b and c<d, such that F[a,b]=[c,d]. The possibility of using the spline interpolation procedure, to approximate the inverse of F, is a natural way in some applications, and it leads to schemes that are stable and feasible to implement. We prove that, with a shape-preserving quadratic Hermite interpolation scheme based on quadratic B-splines, we can preserve the monotonicity of the inverse of F while maintaining a third-order estimate of the interpolation error. To illustrate the effectiveness of this approach, we provide some examples of application, such as the inversion of cumulative distribution functions (arc-length and normal cumulative distributions) and the computation of the Lambert W-function.

The Schwarzian-Newton method for solving nonlinear equations, with applications

The Schwarzian-Newton method can be defined as the minimal method for solving nonlinear equations f(x) = 0 which is exact for any function f with constant Schwarzian derivative; exactness means that the method gives the exact root in one iteration for any starting value in a neighborhood of the root. This is a fourth order method which has Halley’s method as limit when the Schwarzian derivative tends to zero. We obtain conditions for the convergence of the SNM in an interval and show how this method can be applied for a reliable and fast solution of some problems, like the inversion of cumulative distribution functions (gamma and beta distributions) and the inversion of elliptic integrals. 2000MSC: Primary 65H05; Secondary 33B20, 33E05.

Estimation of quantile mixtures via L-moments and trimmed L-moments

Moments or cumulants have been traditionally used to characterize a probability distribution or an observed data set. Recently, L-moments and trimmed L-moments have been noticed as appealing alternatives to the conventional moments. This paper promotes the use of L-moments proposing new parametric families of distributions that can be estimated by the method of L-moments. The theoretical L-moments are defined by the quantile function i.e. the inverse of cumulative distribution function. An approach for constructing parametric families from quantile functions is presented. Because of the analogy to mixtures of densities, this class of parametric families is called quantile mixtures. The method of L-moments is a natural way to estimate the parameters of quantile mixtures. As an example, two parametric families are introduced: the normal-polynomial quantile mixture and the Cauchy-polynomial quantile mixture. The proposed quantile mixtures are applied to model monthly, weekly and daily returns of some major stock indexes.