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José A. Adell

Publications and source records attributed to José A. Adell.

15 recordsLinked to original sources
arXiv2026

Explicit confidence bands and intervals for distribution functions and their derivatives via random Weierstrass-type operators

Classical kernel estimators of second order are interpreted in terms of random Weierstrass-type operators, particularly random Steklov operators. This leads us to obtain explicit nonasymptotic confidence bands and intervals for distribution functions $F$ and their derivatives $F^{(k)}$. Under the only assumption that $F^{(k)}$ is uniformly continuous, confidence bands for $F^{(k)}$ are established by using the Dvoretzky-Kiefer-Wolfowitz inequality. To give confidence intervals, we allow $F^{(k)}$ to have isolated discontinuities of the first kind, so that we really estimate the midpoint function $(F^{(k)})_{\star}(x)$. The proofs are based either on concentration inequalities for subordinated stochastic processes or accurate estimates of the MSE of the corresponding estimators. The length of the confidence bands and intervals depends on the degree of smoothness of $F^{(k)}$ measured in terms of the second modulus of continuity. Both lengths are of order $n^{-1 / 2}$ if $F$ is locally a polynomial of degree $k+1$ at most.

math.ST↗
arXiv2025

Tight bounds on Poisson tails with applications to Szász-Mirakyan operators

We obtain tight bounds on Poisson tails which are easy to handle. A short proof based on the median of the gamma distribution is given. Numerical comparisons with other known estimates are made. As an application, we consider the rates of pointwise convergence for $S_nf(x)-f(x)$, as $n\rightarrow \infty$, where $S_n$ is the Szász-Mirakyan operator and $x$ belongs to an open interval $I$. If the function $f$ is affine on $I$, the rate of convergence is exponential. This property is no longer true at the boundary of $I$

math.PR↗
arXiv2025

Random positive linear operators and their applications to nonparametric statistics

We outline a general procedure on how to apply random positive linear operators in nonparametric estimation. As a consequence, we give explicit confidence bands and intervals for a distribution function $F$ concentrated on $[0,1]$ by means of random Bernstein polynomials, and for the derivatives of $F$ by using random Bernstein-Kantorovich type operators. In each case, the lengths of such bands and intervals depend upon the degree of smoothness of $F$ or its corresponding derivatives, measured in terms of appropriate moduli of smoothness. In particular, we estimate the uniform distribution function by means of a random polynomial of second order. This estimator is much simpler and performs better than the classical uniform empirical process used in the celebrated Dvoretzky-Kiefer-Wolfowitz inequality.

math.ST↗
arXiv2024

On the rates of pointwise convergence for Bernstein polynomials

Let $f$ be a real function defined on the interval $[0,1]$ which is constant on $(a,b)\subset [0,1]$, and let $B_nf$ be its associated $n$th Bernstein polynomial. We prove that, for any $x\in (a,b)$, $|B_nf(x)-f(x)|$ converges to $0$ as $n\rightarrow \infty $ at an exponential rate of decay. Moreover, we show that this property is no longer true at the boundary of $(a,b)$. Finally, an extension to Bernstein-Kantorovich type operators is also provided

math.CA↗
arXiv2024

Approximation of discontinuous functions by positive linear operators. A probabilistic approach

We obtain approximation results for general positive linear operators satisfying mild conditions, when acting on discontinuous functions and absolutely continuous functions having discontinuous derivatives. The upper bounds, given in terms of a local first modulus of continuity, are best possible, in the sense that we can construct particular sequences of operators attaining them. When applied to functions of bounded variation or absolutely continuous functions having derivatives of bounded variation, these upper bounds are better and simpler to compute than the usual total variation bounds. The particular case of the Bernstein polynomials is thoroughly discussed. We use a probabilistic approach based on representations of such operators in terms of expectations of random variables.

math.PR↗
arXiv2024

$B$-Stirling numbers associated to potential polynomials

We introduce the $B$-Stirling numbers of the first and second kind, which are the coefficients of the potential polynomials when we express them in terms of the monomials and the falling factorials, respectively. These numbers include, as particular cases, the partial and complete Bell polynomials, the degenerate and probabilistic Stirling numbers, and the $S$-restricted Stirling numbers, among others. Special attention is devoted to the computation of such numbers. On the one hand, a recursive formula is provided. On the other, we can compute Stirling numbers of one kind in terms of the other, with the help of the classical Stirling numbers.

math.CO↗
arXiv2024

Strong Converse Inequalities for Bernstein Polynomials with Explicit Asymptotic Constants

We obtain strong converse inequalities for the Bernstein polynomials with explicit asymptotic constants. We give different estimation procedures in the central and non-central regions of [0,1]. The main ingredients in our approach are the following: representation of the derivatives of the Bernstein polynomials in terms of the Krawtchouk polynomials, estimates of different inverse moments of various random variables, sharp estimates of both absolute central moments of Bernstein polynomials and the total variation distance between binomial and Poisson distributions, and iterates of the Bernstein polynomials, together with their probabilistic representations.

math.CA↗
arXiv2024

Explicit estimates for the Stirling numbers of the second kind

We give explicit estimates for the Stirling numbers of the second kind $S(n,m)$. With a few exceptions, such estimates are asymptotically sharp. The form of these estimates varies according to $m$ lying in the central or non-central regions of $\{1,\ldots ,n\}$. In each case, we use a different probabilistic representation of $S(n,m)$ in terms of well known random variables to show the corresponding results.

math.CO↗
arXiv2023

Strong Converse Inequalities for Bernstein Operators via Krawtchouk Polynomials

We obtain strong converse inequalities for the Bernstein operators with explicit constants. One of the main ingredients in our approach is the representation of the derivatives of the Bernstein operators in terms of the orthogonal polynomials with respect to the binomial distribution, namely, the Krawtchouk polynomials

math.CA↗
arXiv2020

Probabilistic Stirling numbers of the second kind and applications

Associated to each complex-valued random variable satisfying appropriate integrability conditions, we introduce a different generalization of the Stirling numbers of the second kind. Various equivalent definitions are provided. Attention, however, is focused on applications. Indeed, such numbers describe the moments of sums of i.i.d. random variables, determining their precise asymptotic behavior without making use of the central limit theorem. Such numbers also allow us to obtain explicit and simple Edgeworth expansions. Applications to Lévy processes and cumulants are discussed, as well.

math.PR↗
arXiv2018

Convolution identities for scale transformations of Appell polynomials

We obtain closed form expressions for convolutions of scale transformations within a certain subset of Appell polynomials. This subset contains the Bernoulli, Apostol-Euler, and Cauchy polynomials, as well as various kinds of their generalizations, among others. We give a unified approach mainly based on a probabilistic generalization of the Stirling numbers of the second kind. Different illustrative examples, including reformulations of convolution identities already known in the literature, are discussed in detail.

math.NT↗
arXiv2018

Closed form expressions for Appell polynomials

We show that any Appell sequence can be written in closed form as a forward difference transformation of the identity. Such transformations are actually multipliers in the abelian group of the Appell polynomials endowed with the operation of binomial convolution. As a consequence, we obtain explicit expressions for higher order convolution identities referring to various kinds of Appell polynomials in terms of the Stirling numbers. Applications of the preceding results to generalized Bernoulli and Apostol-Euler polynomials of real order are discussed in detail.

math.NT↗
arXiv2018

A probabilistic generalization of the Stirling numbers of the second kind

Associated to each random variable $Y$ having a finite moment generating function, we introduce a different generalization of the Stirling numbers of the second kind. Some characterizations and specific examples of such generalized numbers are provided. As far as their applications are concerned, attention is focused in extending in various ways the classical formula for sums of powers on arithmetic progressions. Illustrations involving rising factorials, Bell polynomials, polylogarithms, and a certain class of Appell polynomials, in connection with appropriate random variables $Y$ in each case, are discussed in detail.

math.NT↗
arXiv2017

Explicit expressions for a certain class of Appell polynomials. A probabilistic approach

We consider the class $\mathcal{E}_t(Y)$ of Appell polynomials whose generating function is given by means of a real power $t$ of the moment generating function of a certain random variable $Y$. For such polynomials, we obtain explicit expressions depending on the moments of $Y$. It turns out that various kinds of generalizations of Bernoulli and Apostol-Euler polynomials belong to $\mathcal{E}_t(Y)$ and can be written and investigated in a unified way. In particular, explicit expression for such polynomials can be given in terms of suitable probabilistic generalizations of the Stirling numbers of the second kind.

math.NT↗