On the average sensitivity of the weighted sum function

Journal of Nonparametric Statistics, 2013

In this paper we address the problem of efficient estimation of Sobol sensitivy indices. First, we focus on general functional integrals of conditional moments of the form E(ψ(E(ϕ(Y )|X))) where (X, Y ) is a random vector with joint density f and ψ and ϕ are functions that are differentiable enough. In particular, we show that asymptotical efficient estimation of this functional boils down to the estimation of crossed quadratic functionals. An efficient estimate of first-order sensitivity indices is then derived as a special case. We investigate its properties on several analytical functions and illustrate its interest on a reservoir engineering case.

Bulletin of the Malaysian Mathematical Sciences Society, 2021

In this paper, we obtain error bound for pseudo-binomial and negative binomial approximations to weighted sums of locally dependent random variables, using Stein's method. We also discuss approximation results for weighted sums of independent random variables. We demonstrate our results through some applications in finance and runs in statistics.

Bulletin of the American Mathematical Society, 2020

The sensitivity conjecture is a long-standing problem in theoretical computer science that seeks to fit the sensitivity of a Boolean function into a unified framework formed by the other complexity measures of Boolean functions, such as block sensitivity and certificate complexity. After more than thirty years of attacks on this conjecture, Hao Huang (2019) gave a very succinct proof of the conjecture. In this survey, we explore the ideas that inspired the proof of this conjecture by an exposition of four papers that had the most impact on the conjecture. We also discuss progress on further research directions that the conjecture leads us to.

We prove the convergence of weighted sums of associated random variables normalized by n 1/p , p ∈ (1, 2), assuming the existence of moments somewhat larger than p, depending on the behaviour of the weights, improving on previous results by getting closer to the moment assumption used for the case of constant weights. Besides moment conditions, we assume a convenient behaviour either on truncated covariances or on joint tail probabilities. Our results extend analogous characterizations known for sums of independent or negatively dependent random variables.

2003

Let Y n,k , k = 0, 1, 2, ..., n ≥ 1, be a collection of random variables, where for each n, Y n,k , k = 0, 1, 2 ..., are independent. Let A = [p n,k ] be a regular summability method. We provide some rates of convergence (Berry-Esseen type bounds) for the weak convergence of summability transform (AY). We show that when A = [p n,k ] is the classical Cesaro summability method, the rate of convergence of the resulting central limit theorem is best possible among all regular triangular summability methods with rows adding up to one. We further provide some summability results concerning l 2 -negligibility. An application of these results characterizes the rate of convergence of Schnabl operators while approximating Lipschitz continuous functions.

Fibonacci Quarterly, 1991

RePEc: Research Papers in Economics, 2007

A sensitivity index quantifies the degree of smoothness with which it responds to fluctuations in the wishes of the members of a voting body. This paper characterizes the Banzhaf-Coleman-Dubey-Shapley sensitivity index using a set of independent axioms. Bounds on the index for a very general class of games are also derived.

LATIN 2018: Theoretical Informatics, 2018

The Fourier-Entropy Influence (FEI) Conjecture states that for any Boolean function f : {+1, −1} n → {+1, −1}, the Fourier entropy of f is at most its influence up to a universal constant factor. While the FEI conjecture has been proved for many classes of Boolean functions, it is still not known whether it holds for the class of Linear Threshold Functions. A natural question is: Does the FEI conjecture hold for a "random" linear threshold function? In this paper, we answer this question in the affirmative. We consider two natural distributions on the weights defining a linear threshold function, namely uniform distribution on [−1, 1] and Normal distribution.

Journal of Statistical Planning and Inference, 2003

For one-parameter natural exponential and power series families we give some general characterizations from an a ne relation connecting the mean of member distributions and their length biased versions. Examples subsume many known cases. Other characterizations are explored using random variable relations involving the length biased version of partial sums. Finally, characterizations of the law of X admitting more general weights are obtained by equating the laws of a function H (X ) and the weighted version of X .

2003

We prove an analog of the Kolmogorov-Rogozin inequality for the value concentration of completely additive functions defined on random permutations.