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Frédéric Ouimet

Publications and source records attributed to Frédéric Ouimet.

At least 73 records · Page 4Linked to original sources
arXiv2021

Counterexamples to the classical central limit theorem for triplewise independent random variables having a common arbitrary margin

We present a general methodology to construct triplewise independent sequences of random variables having a common but arbitrary marginal distribution $F$ (satisfying very mild conditions). For two specific sequences, we obtain in closed form the asymptotic distribution of the sample mean. It is non-Gaussian (and depends on the specific choice of $F$). This allows us to illustrate the extent of the 'failure' of the classical central limit theorem (CLT) under triplewise independence. Our methodology is simple and can also be used to create, for any integer $K$, new $K$-tuplewise independent sequences that are not mutually independent. For $K \geq 4$, it appears that the sequences created using our methodology do verify a CLT, and we explain heuristically why this is the case.

math.ST↗
arXiv2021

Bounding Means of Discrete Distributions

We introduce methods to bound the mean of a discrete distribution (or finite population) based on sample data, for random variables with a known set of possible values. In particular, the methods can be applied to categorical data with known category-based values. For small sample sizes, we show how to leverage the knowledge of the set of possible values to compute bounds that are stronger than for general random variables such as standard concentration inequalities.

math.ST↗
arXiv2021

An improvement of Tusnády's inequality in the bulk

We prove a non-asymptotic generalization of the refined continuity correction developed in Cressie (1978) for the Binomial distribution, which we then use to improve the versions of Tusnády's inequality from Massart (2002) and Carter & Pollard (2004) in the bulk.

math.ST↗
arXiv2021

Asymptotic properties of Dirichlet kernel density estimators

We study theoretically, for the first time, the Dirichlet kernel estimator introduced by Aitchison and Lauder (1985) for the estimation of multivariate densities supported on the $d$-dimensional simplex. The simplex is an important case as it is the natural domain of compositional data and has been neglected in the literature on asymmetric kernels. The Dirichlet kernel estimator, which generalizes the (non-modified) unidimensional Beta kernel estimator from Chen (1999), is free of boundary bias and non-negative everywhere on the simplex. We show that it achieves the optimal convergence rate $O(n^{-4/(d+4)})$ for the mean squared error and the mean integrated squared error, we prove its asymptotic normality and uniform strong consistency, and we also find an asymptotic expression for the mean integrated absolute error. To illustrate the Dirichlet kernel method and its favorable boundary properties, we present a case study on minerals processing.

math.ST↗
arXiv2021

Asymptotic properties of Bernstein estimators on the simplex

Bernstein estimators are well-known to avoid the boundary bias problem of traditional kernel estimators. The theoretical properties of these estimators have been studied extensively on compact intervals and hypercubes, but never on the simplex, except for the mean squared error of the density estimator in Tenbusch (1994) when $d = 2$. The simplex is an important case as it is the natural domain of compositional data. In this paper, we make an effort to prove several asymptotic results (bias, variance, mean squared error (MSE), mean integrated squared error (MISE), asymptotic normality, uniform strong consistency) for Bernstein estimators of cumulative distribution functions and density functions on the $d$-dimensional simplex. Our results generalize the ones in Leblanc (2012) and Babu et al. (2002), who treated the case $d = 1$, and significantly extend those found in Tenbusch (1994). In particular, our rates of convergence for the MSE and MISE are optimal.

math.ST↗
arXiv2021

Moments of the Riemann zeta function on short intervals of the critical line

We show that as $T\to \infty$, for all $t\in [T,2T]$ outside of a set of measure $\mathrm{o}(T)$, $$ \int_{-(\log T)^θ}^{(\log T)^θ} |ζ(\tfrac 12 + \mathrm{i} t + \mathrm{i} h)|^β \mathrm{d} h = (\log T)^{f_θ(β) + \mathrm{o}(1)}, $$ for some explicit exponent $f_θ(β)$, where $θ> -1$ and $β> 0$. This proves an extended version of a conjecture of Fyodorov and Keating (2014). In particular, it shows that, for all $θ> -1$, the moments exhibit a phase transition at a critical exponent $β_c(θ)$, below which $f_θ(β)$ is quadratic and above which $f_θ(β)$ is linear. The form of the exponent $f_θ$ also differs between mesoscopic intervals ($-1<θ<0$) and macroscopic intervals ($θ>0$), a phenomenon that stems from an approximate tree structure for the correlations of zeta. We also prove that, for all $t\in [T,2T]$ outside a set of measure $\mathrm{o}(T)$, $$ \max_{|h| \leq (\log T)^θ} |ζ(\tfrac{1}{2} + \mathrm{i} t + \mathrm{i} h)| = (\log T)^{m(θ) + \mathrm{o}(1)}, $$ for some explicit $m(θ)$. This generalizes earlier results of Najnudel (2018) and Arguin et al. (2019) for $θ= 0$. The proofs are unconditional, except for the upper bounds when $θ> 3$, where the Riemann hypothesis is assumed.

math.NT↗
arXiv2021

A precise local limit theorem for the multinomial distribution and some applications

In Siotani & Fujikoshi (1984), a precise local limit theorem for the multinomial distribution is derived by inverting the Fourier transform, where the error terms are explicit up to order $N^{-1}$. In this paper, we give an alternative (conceptually simpler) proof based on Stirling's formula and a careful handling of Taylor expansions, and we show how the result can be used to approximate multinomial probabilities on most subsets of $\mathbb{R}^d$. Furthermore, we discuss a recent application of the result to obtain asymptotic properties of Bernstein estimators on the simplex, we improve the main result in Carter (2002) on the Le Cam distance bound between multinomial and multivariate normal experiments while simultaneously simplifying the proof, and we mention another potential application related to finely tuned continuity corrections.

math.ST↗
arXiv2021

On the Le Cam distance between Poisson and Gaussian experiments and the asymptotic properties of Szasz estimators

In this paper, we prove a local limit theorem for the ratio of the Poisson distribution to the Gaussian distribution with the same mean and variance, using only elementary methods (Taylor expansions and Stirling's formula). We then apply the result to derive an upper bound on the Le Cam distance between Poisson and Gaussian experiments, which gives a complete proof of the sketch provided in the unpublished set of lecture notes by Pollard (2010), who uses a different approach. We also use the local limit theorem to derive the asymptotics of the variance for Bernstein c.d.f. and density estimators with Poisson weights on the positive half-line (also called Szasz estimators). The propagation of errors in the literature due to the incorrect estimate in Lemma 2 (iv) of Leblanc (2012) is addressed in the Appendix.

math.ST↗
arXiv2021

General formulas for the central and non-central moments of the multinomial distribution

We present the first general formulas for the central and non-central moments of the multinomial distribution, using a combinatorial argument and the factorial moments previously obtained in Mosimann (1962). We use the formulas to give explicit expressions for all the non-central moments up to order 8 and all the central moments up to order 4. These results expand significantly on those in Newcomer (2008) and Newcomer et al. (2008), where the non-central moments were calculated up to order 4.

math.PR↗
arXiv2020

A study of seven asymmetric kernels for the estimation of cumulative distribution functions

In Mombeni et al. (2019), Birnbaum-Saunders and Weibull kernel estimators were introduced for the estimation of cumulative distribution functions (c.d.f.s) supported on the half-line $[0,\infty)$. They were the first authors to use asymmetric kernels in the context of c.d.f. estimation. Their estimators were shown to perform better numerically than traditional methods such as the basic kernel method and the boundary modified version from Tenreiro (2013). In the present paper, we complement their study by introducing five new asymmetric kernel c.d.f. estimators, namely the Gamma, inverse Gamma, lognormal, inverse Gaussian and reciprocal inverse Gaussian kernel c.d.f. estimators. For these five new estimators, we prove the asymptotic normality and we find asymptotic expressions for the following quantities: bias, variance, mean squared error and mean integrated squared error. A numerical study then compares the performance of the five new c.d.f. estimators against traditional methods and the Birnbaum-Saunders and Weibull kernel c.d.f. estimators from Mombeni et al. (2019). By using the same experimental design, we show that the lognormal and Birnbaum-Saunders kernel c.d.f. estimators perform the best overall, while the other asymmetric kernel estimators are sometimes better but always at least competitive against the boundary kernel method.

math.ST↗
arXiv2020

A counterexample to the central limit theorem for pairwise independent random variables having a common arbitrary margin

The Central Limit Theorem (CLT) is one of the most fundamental results in statistics. It states that the standardized sample mean of a sequence of $n$ mutually independent and identically distributed random variables with finite first and second moments converges in distribution to a standard Gaussian as $n$ goes to infinity. In particular, pairwise independence of the sequence is generally not sufficient for the theorem to hold. We construct explicitly a sequence of pairwise independent random variables having a common but arbitrary marginal distribution $F$ (satisfying very mild conditions) for which the CLT is not verified. We study the extent of this 'failure' of the CLT by obtaining, in closed form, the asymptotic distribution of the sample mean of our sequence. This is illustrated through several theoretical examples, for which we provide associated computing codes in the R language.

math.PR↗
arXiv2019

Complete monotonicity of a ratio of gamma functions and some combinatorial inequalities for multinomial coefficients

For $m,n\in \mathbb{N}$, let $0 < α_i,β_j,λ_{ij} \leq 1$ be such that $\sum_{j=1}^n λ_{ij} = α_i$, $\sum_{i=1}^m λ_{ij} = β_j$, and $\sum_{i=1}^m α_i = \sum_{j=1}^n β_j \leq 1$. We prove that the ratio of gamma functions \begin{equation*} \hspace{-15mm}t \mapsto \frac{\prod_{i=1}^m Γ(α_i t + 1) \prod_{j=1}^n Γ(β_j t + 1)}{\prod_{i=1}^m \prod_{j=1}^n Γ(λ_{ij} t + 1)} \end{equation*} is logarithmically completely monotonic on $(0,\infty)$. This result complements the logarithmically complete monotonicity of multinomial probabilities shown in Ouimet (2018), Qi et al (2018), and the recent survey of Qi & Argawal (2019) on the complete monotonicity of functions related to ratios of gamma functions. As a consequence of the log-convexity, we obtain new combinatorial inequalities for multinomial coefficients.

math.CA↗
arXiv2019

Discretization of the maximum for the derivatives of a random model of $\log|ζ|$ on the critical line

In this short note, we study the derivatives of all orders for the random field $$ X_T(h) = \sum_{p \leq T} \frac{\text{Re}(U_p \, p^{-i h})}{p^{1/2}}, \quad h\in [0,1], $$ where $(U_p, \, p ~\text{primes})$ is an i.i.d. sequence of uniform random variables on the unit circle in $\mathbb{C}$. We show that the maximum of $X_{T}$, and more generally the maximum of its $j$-th derivative, varies on a $(\log T)^{-\frac{1}{2}(j+2)}$ scale, which improves and extends the main result in Arguin & Ouimet (2019) and makes further progress towards the open problem of the tightness of the recentered maximum of $X_{T}$. Our proof is also much simpler and shorter.

math.PR↗
arXiv2019

Large deviations and continuity estimates for the derivative of a random model of $\log |ζ|$ on the critical line

In this paper, we study the random field \begin{equation*} X(h) \circeq \sum_{p \leq T} \frac{\text{Re}(U_p \, p^{-i h})}{p^{1/2}}, \quad h\in [0,1], \end{equation*} where $(U_p, \, p ~\text{primes})$ is an i.i.d. sequence of uniform random variables on the unit circle in $\mathbb{C}$. Harper (2013) showed that $(X(h), \, h\in (0,1))$ is a good model for the large values of $(\log |ζ(\frac{1}{2} + i (T + h))|, \, h\in [0,1])$ when $T$ is large, if we assume the Riemann hypothesis. The asymptotics of the maximum were found in Arguin, Belius & Harper (2017) up to the second order, but the tightness of the recentered maximum is still an open problem. As a first step, we provide large deviation estimates and continuity estimates for the field's derivative $X'(h)$. The main result shows that, with probability arbitrarily close to $1$, \begin{equation*} \max_{h\in [0,1]} X(h) - \max_{h\in \mathcal{S}} X(h) = O(1), \end{equation*} where $\mathcal{S}$ a discrete set containing $O(\log T \sqrt{\log \log T})$ points.

math.PR↗
arXiv2018

Maxima of branching random walks with piecewise constant variance

This article extends the results of Fang & Zeitouni (2012a) on branching random walks (BRWs) with Gaussian increments in time inhomogeneous environments. We treat the case where the variance of the increments changes a finite number of times at different scales in [0,1] under a slight restriction. We find the asymptotics of the maximum up to an OP(1) error and show how the profile of the variance influences the leading order and the logarithmic correction term. A more general result was independently obtained by Mallein (2015b) when the law of the increments is not necessarily Gaussian. However, the proof we present here generalizes the approach of Fang & Zeitouni (2012a) instead of using the spinal decomposition of the BRW. As such, the proof is easier to understand and more robust in the presence of an approximate branching structure.

math.PR↗
arXiv2018

A uniform $L^1$ law of large numbers for functions of i.i.d. random variables that are translated by a consistent estimator

We develop a new $L^1$ law of large numbers where the $i$-th summand is given by a function $h(\cdot)$ evaluated at $X_i - θ_n$, and where $θ_n \circeq θ_n(X_1,X_2,\ldots,X_n)$ is an estimator converging in probability to some parameter $θ\in \mathbb{R}$. Under broad technical conditions, the convergence is shown to hold uniformly in the set of estimators interpolating between $θ$ and another consistent estimator $θ_n^{\star}$. Our main contribution is the treatment of the case where $|h|$ blows up at $0$, which is not covered by standard uniform laws of large numbers.

math.PR↗
arXiv2018

Poisson-Dirichlet statistics for the extremes of a randomized Riemann zeta function

In Arguin & Tai (2018), the authors prove the convergence of the two-overlap distribution at low temperature for a randomized Riemann zeta function on the critical line. We extend their results to prove the Ghirlanda-Guerra identities. As a consequence, we find the joint law of the overlaps under the limiting mean Gibbs measure in terms of Poisson-Dirichlet variables. It is expected that we can adapt the approach to prove the same result for the Riemann zeta function itself.

math.PR↗