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Frédéric Ouimet

Publications and source records attributed to Frédéric Ouimet.

At least 55 records · Page 3Linked to original sources
arXiv2023

Asymptotic comparison of negative multinomial and multivariate normal experiments

This note presents a refined local approximation for the logarithm of the ratio between the negative multinomial probability mass function and a multivariate normal density, both having the same mean-covariance structure. This approximation, which is derived using Stirling's formula and a meticulous treatment of Taylor expansions, yields an upper bound on the Hellinger distance between the jittered negative multinomial distribution and the corresponding multivariate normal distribution. Upper bounds on the Le Cam distance between negative multinomial and multivariate normal experiments ensue.

math.ST↗
arXiv2023

Non-asymptotic approximations for Pearson's chi-square statistic and its application to confidence intervals for strictly convex functions of the probability weights of discrete distributions

In this paper, we develop a non-asymptotic local normal approximation for multinomial probabilities. First, we use it to find non-asymptotic total variation bounds between the measures induced by uniformly jittered multinomials and the multivariate normals with the same means and covariances. From the total variation bounds, we also derive a comparison of the cumulative distribution functions and quantile coupling inequalities between Pearson's chi-square statistic (written as the normalized quadratic form of a multinomial vector) and its multivariate normal analogue. We apply our results to find confidence intervals for the negative entropy of discrete distributions. Our method can be applied more generally to find confidence intervals for strictly convex functions of the weights of discrete distributions.

math.ST↗
arXiv2023

The Gaussian product inequality conjecture for multinomial covariances

In this short note, we find an equivalent combinatorial condition only involving finite sums under which a centered Gaussian random vector with multinomial covariance matrix satisfies the Gaussian product inequality (GPI) conjecture. These covariance matrices are relevant since their off-diagonal elements are negative, which is the hardest case to cover for the GPI conjecture, as mentioned by [Russell, O., \& Sun, W. 2022. Some new {G}aussian product inequalities. {\em J. Math. Anal. Appl.}, {\bf 515}(2), Paper No. 126439, 21 pp.].

math.PR↗
arXiv2023

Moments of the negative multinomial distribution

The negative multinomial distribution appears in many areas of applications such as polarimetric image processing and the analysis of longitudinal count data. In previous studies, Mosimann (1963) derived general formulas for the falling factorial moments of the negative multinomial distribution, while Withers & Nadarajah (2014) obtained expressions for the cumulants. Despite the availability of the moment generating function, no comprehensive formulas for the moments have been calculated thus far. This paper addresses this gap by presenting general formulas for both central and non-central moments of the negative multinomial distribution. These formulas are expressed in terms of binomial coefficients and Stirling numbers of the second kind. Utilizing these formulas, we provide explicit expressions for all central moments up to the 4th order and all non-central moments up to the 8th order.

math.PR↗
arXiv2023

A refined continuity correction for the negative binomial distribution and asymptotics of the median

In this paper, we prove a local limit theorem and a refined continuity correction for the negative binomial distribution. We present two applications of the results. First, we find the asymptotics of the median for a $\mathrm{Negative\hspace{0.5mm}Binomial}\hspace{0.2mm}(r,p)$ random variable jittered by a $\mathrm{Uniform}\hspace{0.2mm}(0,1)$, which answers a problem left open in Coeurjolly & Trépanier (2020). This is used to construct a simple, robust and consistent estimator of the parameter $p$, when $r > 0$ is known. The case where $r$ is unknown is also briefly covered. Second, we find an upper bound on the Le Cam distance between negative binomial and normal experiments.

math.ST↗
arXiv2023

Minimax properties of Dirichlet kernel density estimators

This paper considers the asymptotic behavior in $β$-Hölder spaces, and under $L^p$ losses, of a Dirichlet kernel density estimator proposed by Aitchison and Lauder (1985) for the analysis of compositional data. In recent work, Ouimet and Tolosana-Delgado (2022) established the uniform strong consistency and asymptotic normality of this estimator. As a complement, it is shown here that the Aitchison-Lauder estimator can achieve the minimax rate asymptotically for a suitable choice of bandwidth whenever $(p,β) \in [1, 3) \times (0, 2]$ or $(p, β) \in \mathcal{A}_d$, where $\mathcal{A}_d$ is a specific subset of $[3, 4) \times (0, 2]$ that depends on the dimension $d$ of the Dirichlet kernel. It is also shown that this estimator cannot be minimax when either $p \in [4, \infty)$ or $β\in (2, \infty)$. These results extend to the multivariate case, and also rectify in a minor way, earlier findings of Bertin and Klutchnikoff (2011) concerning the minimax properties of Beta kernel estimators.

math.ST↗
arXiv2023

Miscellaneous results related to the Gaussian product inequality conjecture for the joint distribution of traces of Wishart matrices

This note reports partial results related to the Gaussian product inequality (GPI) conjecture for the joint distribution of traces of Wishart matrices. In particular, several GPI-related results from Wei (2014) and Liu et al. (2015) are extended in two ways: by replacing the power functions with more general classes of functions, and by replacing the usual Gaussian and multivariate gamma distributional assumptions by the more general trace-Wishart distribution assumption. These findings suggest that a Kronecker product form of the GPI holds for diagonal blocks of any Wishart distribution.

math.PR↗
arXiv2022

On the boundary properties of Bernstein estimators on the simplex

In this paper, we study the asymptotic properties (bias, variance, mean squared error) of Bernstein estimators for cumulative distribution functions and density functions near and on the boundary of the $d$-dimensional simplex. Our results generalize those found by Leblanc (2012), who treated the case $d=1$, and complement the results from Ouimet (2021) in the interior of the simplex. Since the "edges" of the $d$-dimensional simplex have dimensions going from $0$ (vertices) up to $d - 1$ (facets) and our kernel function is multinomial, the asymptotic expressions for the bias, variance and mean squared error are not straightforward extensions of one-dimensional asymptotics as they would be for product-type estimators studied by almost all past authors in the context of Bernstein estimators or asymmetric kernel estimators. This point makes the mathematical analysis much more interesting.

math.ST↗
arXiv2022

Goodness-of-fit tests for Laplace, Gaussian and exponential power distributions based on $λ$-th power skewness and kurtosis

Temperature data, like many other measurements in quantitative fields, are usually modeled using a normal distribution. However, some distributions can offer a better fit while avoiding underestimation of tail event probabilities. To this point, we extend Pearson's notions of skewness and kurtosis to build a powerful family of goodness-of-fit tests based on Rao's score for the exponential power distribution $\mathrm{EPD}_λ(μ,σ)$, including tests for normality and Laplacity when $λ$ is set to 1 or 2. We find the asymptotic distribution of our test statistic, which is the sum of the squares of two $Z$-scores, under the null and under local alternatives. We also develop an innovative regression strategy to obtain $Z$-scores that are nearly independent and distributed as standard Gaussians, resulting in a $χ_2^2$ distribution valid for any sample size (up to very high precision for $n\geq 20$). The case $λ=1$ leads to a powerful test of fit for the Laplace($μ,σ$) distribution, whose empirical power is superior to all $39$ competitors in the literature, over a wide range of $400$ alternatives. Theoretical proofs in this case are particularly challenging and substantial. We applied our tests to three temperature datasets. The new tests are implemented in the R package PoweR.

math.ST↗
arXiv2022

A combinatorial proof of the Gaussian product inequality beyond the MTP${}_2$ case

A combinatorial proof of the Gaussian product inequality (GPI) is given under the assumption that each component of a centered Gaussian random vector $\boldsymbol{X} = (X_1, \ldots, X_d)$ of arbitrary length can be written as a linear combination, with coefficients of identical sign, of the components of a standard Gaussian random vector. This condition on $\boldsymbol{X}$ is shown to be strictly weaker than the assumption that the density of the random vector $(|X_1|, \ldots, |X_d|)$ is multivariate totally positive of order $2$, abbreviated MTP${}_2$, for which the GPI is already known to hold. Under this condition, the paper highlights a new link between the GPI and the monotonicity of a certain ratio of gamma functions.

math.PR↗
arXiv2022

Local normal approximations and probability metric bounds for the matrix-variate $T$ distribution and its application to Hotelling's $T$ statistic

In this paper, we develop local expansions for the ratio of the centered matrix-variate $T$ density to the centered matrix-variate normal density with the same covariances. The approximations are used to derive upper bounds on several probability metrics (such as the total variation and Hellinger distance) between the corresponding induced measures. This work extends some of the results of Shafiei & Saberali (2015) and Ouimet (2022) for the univariate Student distribution to the matrix-variate setting.

math.ST↗
arXiv2022

A comprehensive empirical power comparison of univariate goodness-of-fit tests for the Laplace distribution

In this paper we present the results from an empirical power comparison of 40 goodness-of-fit tests for the univariate Laplace distribution, carried out using Monte Carlo simulations with sample sizes $n = 20, 50, 100, 200$, significance levels $α= 0.01, 0.05, 0.10$, and 400 alternatives consisting of asymmetric and symmetric light/heavy-tailed distributions taken as special cases from 11 models. In addition to the unmatched scope of our study, an interesting contribution is the proposal of an innovative design for the selection of alternatives. The 400 alternatives consist of 20 specific cases of 20 submodels drawn from the main 11 models. For each submodel, the 20 specific cases corresponded to parameter values chosen to cover the full power range. An analysis of the results leads to a recommendation of the best tests for five different groupings of the alternative distributions. A real-data example is also presented, where an appropriate test for the goodness-of-fit of the univariate Laplace distribution is applied to weekly log-returns of Amazon stock over a recent four-year period.

stat.ME↗
arXiv2022

Refined normal approximations for the central and noncentral chi-square distributions and some applications

In this paper, we prove a local limit theorem for the chi-square distribution with $r > 0$ degrees of freedom and noncentrality parameter $λ\geq 0$. We use it to develop refined normal approximations for the survival function. Our maximal errors go down to an order of $r^{-2}$, which is significantly smaller than the maximal error bounds of order $r^{-1/2}$ recently found by Horgan & Murphy (2013) and Seri (2015). Our results allow us to drastically reduce the number of observations required to obtain negligible errors in the energy detection problem, from $250$, as recommended in the seminal work of Urkowitz (1967), to only $8$ here with our new approximations. We also obtain an upper bound on several probability metrics between the central and noncentral chi-square distributions and the standard normal distribution, and we obtain an approximation for the median that improves the lower bound previously obtained by Robert (1990).

math.ST↗
arXiv2022

Refined normal approximations for the Student distribution

In this paper, we develop a local limit theorem for the Student distribution. We use it to improve the normal approximation of the Student survival function given in Shafiei & Saberali (2015) and to derive asymptotic bounds for the corresponding maximal errors at four levels of approximation. As a corollary, approximations for the percentage points (or quantiles) of the Student distribution are obtained in terms of the percentage points of the standard normal distribution.

math.ST↗
arXiv2022

A multivariate normal approximation for the Dirichlet density and some applications

In this short note, we prove an asymptotic expansion for the ratio of the Dirichlet density to the multivariate normal density with the same mean and covariance matrix. The expansion is then used to derive an upper bound on the total variation between the corresponding probability measures and rederive the asymptotic variance of the Dirichlet kernel estimators introduced by Aitchison & Lauder (1985) and studied theoretically in Ouimet (2020). Another potential application related to the asymptotic equivalence between the Gaussian variance regression problem and the Gaussian white noise problem is briefly mentioned but left open for future research.

math.ST↗
arXiv2021

On the Le Cam distance between multivariate hypergeometric and multivariate normal experiments

In this short note, we develop a local approximation for the log-ratio of the multivariate hypergeometric probability mass function over the corresponding multinomial probability mass function. In conjunction with the bounds from Carter (2002) and Ouimet (2021) on the total variation between the law of a multinomial vector jittered by a uniform on $(-1/2,1/2)^d$ and the law of the corresponding multivariate normal distribution, the local expansion for the log-ratio is then used to obtain a total variation bound between the law of a multivariate hypergeometric random vector jittered by a uniform on $(-1/2,1/2)^d$ and the law of the corresponding multivariate normal distribution. As a corollary, we find an upper bound on the Le Cam distance between multivariate hypergeometric and multivariate normal experiments.

math.ST↗
arXiv2021

A symmetric matrix-variate normal local approximation for the Wishart distribution and some applications

The noncentral Wishart distribution has become more mainstream in statistics as the prevalence of applications involving sample covariances with underlying multivariate Gaussian populations as dramatically increased since the advent of computers. Multiple sources in the literature deal with local approximations of the noncentral Wishart distribution with respect to its central counterpart. However, no source has yet developed explicit local approximations for the (central) Wishart distribution in terms of a normal analogue, which is important since Gaussian distributions are at the heart of the asymptotic theory for many statistical methods. In this paper, we prove a precise asymptotic expansion for the ratio of the Wishart density to the symmetric matrix-variate normal density with the same mean and covariances. The result is then used to derive an upper bound on the total variation between the corresponding probability measures and to find the pointwise variance of a new density estimator on the space of positive definite matrices with a Wishart asymmetric kernel. For the sake of completeness, we also find expressions for the pointwise bias of our new estimator, the pointwise variance as we move towards the boundary of its support, the mean squared error, the mean integrated squared error away from the boundary, and we prove its asymptotic normality.

math.ST↗