arXiv subjects

Florian Puchhammer

Publications and source records attributed to Florian Puchhammer.

11 recordsLinked to original sources

arXiv2021

Monte Carlo and Quasi-Monte Carlo Density Estimation via Conditioning

Estimating the unknown density from which a given independent sample originates is more difficult than estimating the mean, in the sense that for the best popular non-parametric density estimators, the mean integrated square error converges more slowly than at the canonical rate of $\mathcal{O}(1/n)$. When the sample is generated from a simulation model and we have control over how this is done, we can do better. We examine an approach in which conditional Monte Carlo yields, under certain conditions, a random conditional density which is an unbiased estimator of the true density at any point. By averaging independent replications, we obtain a density estimator that converges at a faster rate than the usual ones. Moreover, combining this new type of estimator with randomized quasi-Monte Carlo to generate the samples typically brings a larger improvement on the error and convergence rate than for the usual estimators, because the new estimator is smoother as a function of the underlying uniform random numbers.

math.ST↗

arXiv2021

Density Estimation by Monte Carlo and Quasi-Monte Carlo

Estimating the density of a continuous random variable X has been studied extensively in statistics, in the setting where n independent observations of X are given a priori and one wishes to estimate the density from that. Popular methods include histograms and kernel density estimators. In this review paper, we are interested instead in the situation where the observations are generated by Monte Carlo simulation from a model. Then, one can take advantage of variance reduction methods such as stratification, conditional Monte Carlo, and randomized quasi-Monte Carlo (RQMC), and obtain a more accurate density estimator than with standard Monte Carlo for a given computing budget. We discuss several ways of doing this, proposed in recent papers, with a focus on methods that exploit RQMC. A first idea is to directly combine RQMC with a standard kernel density estimator. Another one is to adapt a simulation-based derivative estimation method such as smoothed perturbation analysis or the likelihood ratio method to obtain a continuous estimator of the cdf, whose derivative is an unbiased estimator of the density. This can then be combined with RQMC. We summarize recent theoretical results with these approaches and give numerical illustrations of how they improve the convergence of the mean square integrated error.

stat.CO↗

arXiv2021

A Tool for Custom Construction of QMC and RQMC Point Sets

We present LatNet Builder, a software tool to find good parameters for lattice rules, polynomial lattice rules, and digital nets in base 2, for quasi-Monte Carlo (QMC) and randomized quasi-Monte Carlo (RQMC) sampling over the $s$-dimensional unit hypercube. The selection criteria are figures of merit that give different weights to different subsets of coordinates. They are upper bounds on the worst-case error (for QMC) or variance (for RQMC) for integrands rescaled to have a norm of at most one in certain Hilbert spaces of functions. We summarize what are the various Hilbert spaces, discrepancies, types of weights, figures of merit, types of constructions, and search methods supported by LatNet Builder. We briefly discuss its organization and we provide simple illustrations of what it can do.

stat.CO↗

arXiv2021

Variance Reduction with Array-RQMC for Tau-Leaping Simulation of Stochastic Biological and Chemical Reaction Networks

We explore the use of Array-RQMC, a randomized quasi-Monte Carlo method designed for the simulation of Markov chains, to reduce the variance when simulating stochastic biological or chemical reaction networks with $τ$-leaping. The task is to estimate the expectation of a function of molecule copy numbers at a given future time $T$ by the sample average over $n$ sample paths, and the goal is to reduce the variance of this sample-average estimator. We find that when the method is properly applied, variance reductions by factors in the thousands can be obtained. These factors are much larger than those observed previously by other authors who tried RQMC methods for the same examples. Array-RQMC simulates an array of realizations of the Markov chain and requires a sorting function to reorder these chains according to their states, after each step. The choice of sorting function is a key ingredient for the efficiency of the method, although in our experiments, Array-RQMC was never worse than ordinary Monte Carlo, regardless of the sorting method. The expected number of reactions of each type per step also has an impact on the efficiency gain.

stat.CO↗

arXiv2020

Density estimation by Randomized Quasi-Monte Carlo

We consider the problem of estimating the density of a random variable $X$ that can be sampled exactly by Monte Carlo (MC). We investigate the effectiveness of replacing MC by randomized quasi Monte Carlo (RQMC) or by stratified sampling over the unit cube, to reduce the integrated variance (IV) and the mean integrated square error (MISE) for kernel density estimators. We show theoretically and empirically that the RQMC and stratified estimators can achieve substantial reductions of the IV and the MISE, and even faster convergence rates than MC in some situations, while leaving the bias unchanged. We also show that the variance bounds obtained via a traditional Koksma-Hlawka-type inequality for RQMC are much too loose to be useful when the dimension of the problem exceeds a few units. We describe an alternative way to estimate the IV, a good bandwidth, and the MISE, under RQMC or stratification, and we show empirically that in some situations, the MISE can be reduced significantly even in high-dimensional settings.

math.ST↗

arXiv2019

Array-RQMC for option pricing under stochastic volatility models

Array-RQMC has been proposed as a way to effectively apply randomized quasi-Monte Carlo (RQMC) when simulating a Markov chain over a large number of steps to estimate an expected cost or reward. The method can be very effective when the state of the chain has low dimension. For pricing an Asian option under an ordinary geometric Brownian motion model, for example, Array-RQMC reduces the variance by huge factors. In this paper, we show how to apply this method and we study its effectiveness in case the underlying process has stochastic volatility. We show that Array-RQMC can also work very well for these models, even if it requires RQMC points in larger dimension. We examine in particular the variance-gamma, Heston, and Ornstein-Uhlenbeck stochastic volatility models, and we provide numerical results.

math.ST↗

arXiv2017

Sharp general and metric bounds for the star discrepancy of perturbed Halton--Kronecker sequences

We consider the star discrepancy of two-dimensional sequences made up as a hybrid between a Kronecker sequence and a perturbed Halton sequence in base 2, where the perturbation is achieved by a digital-sequence construction in the sense of Niederreiter whose generating matrix contains a periodic perturbing sequence of a given period length. Under the assumption that the Kronecker sequence involves a parameter with bounded continued fraction coefficients sharp discrepancy estimates are obtained. Furthermore, we study the problem from a metric point of view as well. Finally, we also present sharp general and tight metric bounds for certain lacunary trigonometric products which appear to be strongly related to these problems.

math.NT↗

arXiv2016

On an explicit lower bound for the star discrepancy in three dimensions

Following a result of D.~Bylik and M.T.~Lacey from 2008 it is known that there exists an absolute constant $η>0$ such that the (unnormalized) $L^{\infty}$-norm of the three-dimensional discrepancy function, i.e, the (unnormalized) star discrepancy $D^{\ast}_N$, is bounded from below by $D_{N}^{\ast}\geq c (\log N)^{1+η}$, for all $N\in\mathbb{N}$ sufficiently large, where $c>0$ is some constant independent of $N$. This paper builds upon their methods to verify that the above result holds with $η<1/(32+4\sqrt{41})\approx 0.017357\ldots$

math.NT↗

arXiv2015

An improved bound for the star discrepancy of sequences in the unit interval

It is known that there is a constant $c>0$ such that for every sequence $x_1, x_2,\ldots$ in $[0,1)$ we have for the star discrepancy $D^{*}_N$ of the first $N$ elements of the sequence that $N D^{*}_N\geq c\cdot \log N$ holds for infinitely many $N$. Let $c^{*}$ be the supremum of all such $c$ with this property. We show $c^{*}>0.065664679\ldots$, thereby slightly improving the estimates known until now.

math.NT↗

arXiv2014

A Functional Model for SMP Matrices and the Jacobi Flow

This is the second part of the paper arXiv:1309.0959v2 on the theory of SMP (Strong Moment Problem) matrices and their relation to the Killip-Simon problem on two disjoint intervals. In this part we define and study the Jacobi flow on SMP matrices.

math.SP↗

arXiv2013

Killip-Simon-classes of Jacobi matrices with essential spectrum on two symmetric and of SMP matrices on two arbitrary intervals

Jacobi matrices probably are the most classical object in spectral theory, while CMV matrices are a comparably fresh one, although they are related to a very classical topic, namely to orhtogonal polynomials on the unit circle (in the same way as Jacobi matrices are related to orhogonal polynomials on the real axis). We will discuss the third member of this family. Our matrices are generated by the orthonormal systems of functions related to the so-called Strong Moment Problem. For this reason we call them SMP matrices. For instance, one can describe the spectral sets of periodic SMP matrices. Similarly to the case of their counterparts, the description is given by means of conformal mappings on hyperbolic, in this case, comb domains. One can represent functional models associated with periodic and almost periodic SMP matrices. We are especially enthusiastic about the role, which such matrices can play in the Killip-Simon-problem related to Jacobi matrices with the essential spectrum on two arbitrary intervals. The parametric description of SMP matrices of the Killip-Simon-class with their essential spectrum on two arbitrary intervals is the main result of this paper.

math.SP↗