On the Inversion of High Energy Proton

Inversion of the K-fold stochastic autoconvolution integral equation is an elementary nonlinear problem, yet there are no de facto methods to solve it with finite statistics. To fix this problem, we introduce a novel inverse algorithm based on a combination of minimization of relative entropy, the Fast Fourier Transform and a recursive version of Efron's bootstrap. This gives us power to obtain new perspectives on non-perturbative high energy QCD, such as probing the ab initio principles underlying the approximately negative binomial distributions of observed charged particle final state multiplicities, related to multiparton interactions, the fluctuating structure and profile of proton and diffraction. As a proof-of-concept, we apply the algorithm to ALICE proton-proton charged particle multiplicity measurements done at different center-of-mass energies and fiducial pseudorapidity intervals at the LHC, available on HEPData. A strong double peak structure emerges from the inversion, barely visible without it.Comment: 29 pages, 10 figures, v2: extended analysis (re-projection ratios, 2D

The Likelihood of Mixed Hitting Times

We present a method for computing the likelihood of a mixed hitting-time model that specifies durations as the first time a latent L\'evy process crosses a heterogeneous threshold. This likelihood is not generally known in closed form, but its Laplace transform is. Our approach to its computation relies on numerical methods for inverting Laplace transforms that exploit special properties of the first passage times of L\'evy processes. We use our method to implement a maximum likelihood estimator of the mixed hitting-time model in MATLAB. We illustrate the application of this estimator with an analysis of Kennan's (1985) strike data.Comment: 35 page

Computing Tails of Compound Distributions Using Direct Numerical Integration

An efficient adaptive direct numerical integration (DNI) algorithm is developed for computing high quantiles and conditional Value at Risk (CVaR) of compound distributions using characteristic functions. A key innovation of the numerical scheme is an effective tail integration approximation that reduces the truncation errors significantly with little extra effort. High precision results of the 0.999 quantile and CVaR were obtained for compound losses with heavy tails and a very wide range of loss frequencies using the DNI, Fast Fourier Transform (FFT) and Monte Carlo (MC) methods. These results, particularly relevant to operational risk modelling, can serve as benchmarks for comparing different numerical methods. We found that the adaptive DNI can achieve high accuracy with relatively coarse grids. It is much faster than MC and competitive with FFT in computing high quantiles and CVaR of compound distributions in the case of moderate to high frequencies and heavy tails

On a flexible construction of a negative binomial model

This work presents a construction of stationary Markov models with negative-binomial marginal distributions. A simple closed form expression for the corresponding transition probabilities is given, linking the proposal to well-known classes of birth and death processes and thus revealing interesting characterizations. The advantage of having such closed form expressions is tested on simulated and real data.Comment: Forthcoming in "Statistics & Probability Letters

Calculation of aggregate loss distributions

Estimation of the operational risk capital under the Loss Distribution Approach requires evaluation of aggregate (compound) loss distributions which is one of the classic problems in risk theory. Closed-form solutions are not available for the distributions typically used in operational risk. However with modern computer processing power, these distributions can be calculated virtually exactly using numerical methods. This paper reviews numerical algorithms that can be successfully used to calculate the aggregate loss distributions. In particular Monte Carlo, Panjer recursion and Fourier transformation methods are presented and compared. Also, several closed-form approximations based on moment matching and asymptotic result for heavy-tailed distributions are reviewed

Poisson inverse problems

In this paper we focus on nonparametric estimators in inverse problems for Poisson processes involving the use of wavelet decompositions. Adopting an adaptive wavelet Galerkin discretization, we find that our method combines the well-known theoretical advantages of wavelet--vaguelette decompositions for inverse problems in terms of optimally adapting to the unknown smoothness of the solution, together with the remarkably simple closed-form expressions of Galerkin inversion methods. Adapting the results of Barron and Sheu [Ann. Statist. 19 (1991) 1347--1369] to the context of log-intensity functions approximated by wavelet series with the use of the Kullback--Leibler distance between two point processes, we also present an asymptotic analysis of convergence rates that justifies our approach. In order to shed some light on the theoretical results obtained and to examine the accuracy of our estimates in finite samples, we illustrate our method by the analysis of some simulated examples.Comment: Published at http://dx.doi.org/10.1214/009053606000000687 in the Annals of Statistics (http://www.imstat.org/aos/) by the Institute of Mathematical Statistics (http://www.imstat.org