# Efficient computation of the cumulative distribution function of a linear mixture of independent random variables

## TL;DR

This paper proves the correctness of an improved algorithm for efficiently computing the cumulative distribution function of a linear mixture of independent random variables, with applications in bootstrap methods and hypothesis testing.

## Contribution

It provides a proof of convergence for a variant of an existing algorithm, ensuring accurate approximation of the distribution function from finite samples.

## Key findings

- The algorithm converges to the true distribution as resolution increases.
- It efficiently computes the complete distribution function from sample data.
- Applications include bootstrap estimates and hypothesis testing in linear models.

## Abstract

For a variant of the algorithm in [Pit19] (arXiv:1903.10816) to compute the approximate density or distribution function of a linear mixture of independent random variables known by a finite sample, it is presented a proof of the functional correctness, i.e. the convergence of the computed distribution function towards the true distribution function (given the observations) as the algorithm resolution is increased to infinity. The algorithm (like its predecessor version) bears elements which are closely related to early known methods for numerical inversion of the characteristic function of a probability distribution, however here efficiently computes the complete distribution function. Possible applications are in computing the distribution of the bootstrap estimate in any linear bootstrap method (e.g. in the block bootstrap for the mean as parameter of interest, or residual bootstrap in linear regression with fixed design), or in elementary analysis-of-variance hypothesis testing.