Recent zbMATH articles in MSC 60E05https://zbmath.org/atom/cc/60E052024-03-13T18:33:02.981707ZWerkzeugPolynomial approximations in a generalized Nyman-Beurling criterionhttps://zbmath.org/1528.410472024-03-13T18:33:02.981707Z"Alouges, François"https://zbmath.org/authors/?q=ai:alouges.francois"Darses, Sébastien"https://zbmath.org/authors/?q=ai:darses.sebastien"Hillion, Erwan"https://zbmath.org/authors/?q=ai:hillion.erwanSummary: The Nyman-Beurling criterion, equivalent to the Riemann hypothesis (RH), is an approximation problem in the space of square integrable functions on \((0, \infty)\), involving dilations of the fractional part function by factors \(\theta_k\in(0, 1)\), \(k \geq 1\). Randomizing the \(\theta_k\) gener
By the way, I had editing rights over that page, maybe they should be
transferred to you. The person I was in touch with was '''Robert
Gieseke''' <>ates new structures and criteria. One of them is a sufficient condition for RH that splits into
\begin{itemize}
\item[(i)] showing that the indicator function can be approximated by convolution with the fractional part,
\item[(ii)] a control on the coefficients of the approximation.
\end{itemize}
This self-contained paper generalizes conditions (i) and (ii) that involve a \(\sigma_0\in(1/2, 1)\), and imply \(\zeta (\sigma +it) \neq 0\) in the strip \(1/2 < \sigma \leq \sigma_0 < 1\). We then identify functions for which (i) holds unconditionally, by means of polynomial approximations. This yields in passing a short probabilistic proof of a known consequence of Wiener's Tauberian theorem. In this context, the difficulty for proving RH is then reallocated in (ii), which heavily relies on the corresponding Gram matrices, for which two remarkable structures are obtained. We show that a particular tuning of the approximating sequence leads to a striking simplification of the second Gram matrix, then reading as a block Hankel form.The basic distributional theory for the product of zero mean correlated normal random variableshttps://zbmath.org/1528.600172024-03-13T18:33:02.981707Z"Gaunt, Robert E."https://zbmath.org/authors/?q=ai:gaunt.robert-edwardSummary: The product of two zero mean correlated normal random variables, and more generally the sum of independent copies of such random variables, has received much attention in the statistics literature and appears in many application areas. However, many important distributional properties are yet to be recorded. This review paper fills this gap by providing the basic distributional theory for the sum of independent copies of the product of two zero mean correlated normal random variables. Properties covered include probability and cumulative distribution functions, generating functions, moments and cumulants, mode and median, Stein characterisations, representations in terms of other random variables, and a list of related distributions. We also review how the product of two zero mean correlated normal random variables arises naturally as a limiting distribution, with an example given for the distributional approximation of double Wiener-Itô integrals.
{{\copyright} 2022 The Author. \textit{Statistica Neerlandica} published by John Wiley \& Sons Ltd on behalf of Netherlands Society for Statistics and Operations Research.}On quantile-based asymmetric family of distributions: properties and inferencehttps://zbmath.org/1528.600182024-03-13T18:33:02.981707Z"Gijbels, Irène"https://zbmath.org/authors/?q=ai:gijbels.irene"Karim, Rezaul"https://zbmath.org/authors/?q=ai:karim.rezaul"Verhasselt, Anneleen"https://zbmath.org/authors/?q=ai:verhasselt.anneleenSummary: In this paper, we provide a detailed study of a general family of asymmetric densities. In the general framework, we establish expressions for important characteristics of the distributions and discuss estimation of the parameters via method-of-moments as well as maximum likelihood estimation. Asymptotic normality results for the estimators are provided. The results under the general framework are then applied to some specific examples of asymmetric densities. The use of the asymmetric densities is illustrated in a real-data analysis.
{{\copyright} 2019 The Authors. International Statistical Review {\copyright} 2019 International Statistical Institute}Response to the letter to the editor on ``On quantile-based asymmetric family of distributions: properties and inference''https://zbmath.org/1528.600192024-03-13T18:33:02.981707Z"Gijbels, Irène"https://zbmath.org/authors/?q=ai:gijbels.irene"Karim, Rezaul"https://zbmath.org/authors/?q=ai:karim.rezaul"Verhasselt, Anneleen"https://zbmath.org/authors/?q=ai:verhasselt.anneleenSummary: \textit{F. J. R. Alvarez} [Int. Stat. Rev. 88, No. 3, 793--796 (2020; Zbl 1528.60021)] points out an identification problem for the four-parameter family of two-piece asymmetric densities introduced by \textit{V. Nassiri} and \textit{I. Loris} [J. Appl. Stat. 40, No. 5, 1090--1105 (2013; Zbl 1514.62770)]. This implies that statistical inference for that family is problematic. Establishing probabilistic properties for this four-parameter family however still makes sense. For the three-parameter family, there is no identification problem. The main contribution in [the authors, ibid. 87, No. 3, 471--504 (2019; Zbl 1528.60018)] is to provide asymptotic results for maximum likelihood and method-of-moments estimators for members of the three-parameter quantile-based asymmetric family of distributions.
{{\copyright} 2020 International Statistical Institute}On recursions for moments of a compound random variable: an approach using an auxiliary counting random variablehttps://zbmath.org/1528.600202024-03-13T18:33:02.981707Z"Kim, Yoora"https://zbmath.org/authors/?q=ai:kim.yooraSummary: We present an identity on moments of a compound random variable by using an auxiliary counting random variable. Based on this identity, we develop a new recurrence formula for obtaining the raw and central moments of any order for a given compound random variable.Letter to the editor: ``On quantile-based asymmetric family of distributions: properties and inference''https://zbmath.org/1528.600212024-03-13T18:33:02.981707Z"Rubio Alvarez, Francisco J."https://zbmath.org/authors/?q=ai:alvarez.francisco-j-rubioSummary: We show that the family of asymmetric distributions studied in a recent publication in the International Statistical Review is equivalent to the family of two-piece distributions. Moreover, we show that the location-scale asymmetric family proposed in that publication is non-identifiable (overparameterised), and it coincides with the family of two-piece distributions after removing the redundant parameters.
{{\copyright} 2020 International Statistical Institute}
Comment to the paper [\textit{I. Gijbels} et al., Int. Stat. Rev. 87, No. 3, 471--504 (2019; Zbl 1528.60018)].