Papers by Enrique M. Cabaña
Transformed empirical processes (TEPs) have been used by the authors in a previous paper to const... more Transformed empirical processes (TEPs) have been used by the authors in a previous paper to construct consistent and selectively efficient goodness-of-fit tests of the Kolmogorov-Smirnov type. A straightforward application of the same ideas to the construction of tests of the Cramér-von Mises type with the same properties leads to cumbersome computations. This short note exhibits some of the inconvenients encountered, and introduces a new family of quadratic statistics of the Cramér-von Mises type, in order to circumvent the difficulties.
arXiv (Cornell University), 2012
Technically our composition of the OU operator is easy to manipulate and its parameters can be co... more Technically our composition of the OU operator is easy to manipulate and its parameters can be computed efficiently because, as we show, the iteration of OU operators leads to a process that can be expressed as a linear combination of basic OU processes. Using this expression we obtain a closed formula for the covariance of the iterated OU process, and consequently estimate the parameters of an OU$(p)$ process by maximum likelihood or, as an alternative, by matching correlations, the latter being a procedure resembling the method of moments.
Trends in mathematics, 2017
We present a new construction of continuous ARMA processes based on iterating an Ornstein-Uhlenbe... more We present a new construction of continuous ARMA processes based on iterating an Ornstein-Uhlenbeck operator OU Ä that maps a random variable y.t/ onto OU Ä y.t/ D R t 1 e Ä.t s/ dy.s/. This construction resembles the procedure to build an AR(p) from an AR(1) and derives in a parsimonious model for continuous autoregression, with fewer parameters to compute than the known CARMA obtained as a solution of a system of stochastic differential equations. We show properties of this operator, give state space representation of the iterated Ornstein-Uhlenbeck process and show how to estimate the parameters of the model.
Stochastic Processes and their Applications, Sep 1, 1984
Integrals of the form 5 f(s) d Y (I, wl xl) are defined for nonanticipating processes f with resp... more Integrals of the form 5 f(s) d Y (I, wl xl) are defined for nonanticipating processes f with respect to the composition of regular functions p and a Wiener process with parameter in (R')'I. Using such integrals, a forward diferential equation (23) is established for the density CJ a Wiener process killed when it reaches one or two constant barriers. stochastic integrals * multiparameter Wiener process * diffusion equariony
Journal of Statistical Planning and Inference, Dec 1, 2018
Given any stationary time series {X n : n ∈ Z} satisfying an ARMA(p, q) model for arbitrary p and... more Given any stationary time series {X n : n ∈ Z} satisfying an ARMA(p, q) model for arbitrary p and q with infinitely divisible innovations, we construct a continuous time stationary process {x t : t ∈ R} such that the distribution of {x n : n ∈ Z}, the process sampled at discrete time, coincides with the distribution of {X n }. In particular the autocovariance function of {x t } interpolates that of {X n }.
Sort-statistics and Operations Research Transactions, Dec 19, 2016
We present a construction of a family of continuous-time ARMA processes based on p iterations of ... more We present a construction of a family of continuous-time ARMA processes based on p iterations of the linear operator that maps a Lévy process onto an Ornstein-Uhlenbeck process. The construction resembles the procedure to build an AR(p) from an AR(1). We show that this family is in fact a subfamily of the well-known CARMA(p, q) processes, with several interesting advantages, including a smaller number of parameters. The resulting processes are linear combinations of Ornstein-Uhlenbeck processes all driven by the same Lévy process. This provides a straightforward computation of covariances, a state-space model representation and methods for estimating parameters. Furthermore, the discrete and equally spaced sampling of the process turns to be an ARMA(p, p − 1) process. We propose methods for estimating the parameters of the iterated Ornstein-Uhlenbeck process when the noise is either driven by a Wiener or a more general Lévy process, and show simulations and applications to real data.
Communications in Statistics - Simulation and Computation, Jul 1, 2005
... on Weibull Alternatives. Authors: Cabaña, Alejandra 1 ; Cabaña, Enrique 2. Source: Communicat... more ... on Weibull Alternatives. Authors: Cabaña, Alejandra 1 ; Cabaña, Enrique 2. Source: Communications in Statistics: Simulation and Computation, Volume 34, Number 3, 2005 , pp. 711-723(13). Publisher: Taylor and Francis Ltd. ...
Zeitschrift für Wahrscheinlichkeitstheorie und Verwandte Gebiete, 1972
Barrier problems for the vibrating string forced by plane white noise have lead us to derive a ge... more Barrier problems for the vibrating string forced by plane white noise have lead us to derive a generalization of D. Andr6 Reflection Principle, and to apply it to solve some particular barrier problems for sums of independent symmetric random variables, and for processes with independent symmetric increments, The main results, concerning the solution of the problems about the vibrating string, are stated and proved in w 3. The generalization of the Reflection Principle is Theorem 1.1, and w 2 is devoted to obtain some steps for the proofs in w 3, by applying Theorem 1.1. No attempt is made in this paper to describe further applications of the generalized Reflection Principle.
Zeitschrift für Wahrscheinlichkeitstheorie und Verwandte Gebiete, 1970
The equation of the vibrating string forced by white noise is formally solved, using stochastic i... more The equation of the vibrating string forced by white noise is formally solved, using stochastic integrals with respect to a plane Brownian motion, and it is proved that a certain process associated to the energy is a martingale. Then Doob's martingale inequality is used to furnish some probability bounds for the energy. Such bounds provide a solution for the double barrier problem for the class of Gaussian stationary processes which can be represented as linear functionals of the positions and the velocities of the string.
Journal of Applied Probability, Jun 1, 1981
Using Slepian processes as a standard of comparison, estimates are given for the probability that... more Using Slepian processes as a standard of comparison, estimates are given for the probability that a centered multiparameter stationary Gaussian process reaches a constant barrier u on a subset of the parameter domain.
Statistics & Probability Letters, Aug 1, 1990
... Letters 10 (1990) 263270 August 1990 NorthHolland ON A MARTINGALE CHARACTERIZATION OF TWOPARA... more ... Letters 10 (1990) 263270 August 1990 NorthHolland ON A MARTINGALE CHARACTERIZATION OF TWOPARAMETER WIENER PROCESS Enrique M. CABADA ... formal equation ur1 = u11 + W * (1.6) with zero initial conditions (see Cabana, 1970; and also Walsh, 1986, where ...
Proceedings of the American Mathematical Society, 1969
Siam Journal on Applied Mathematics, Aug 1, 1987
The class of affine processes is defined as a reasonable alternative to test the null hypothesis ... more The class of affine processes is defined as a reasonable alternative to test the null hypothesis that a given process is isotropic. Estimators for the affinity parameters based only on one level set of the observed process are provided, and they are shown to be consistent under partial independence assumptions. Then asymptotic procedures are proposed to test the isotropy of a stationary process with parameter in ${\bf R}^2 $.
Actas del XXX Congreso Nacional de Estadística e Investigación Operativa y de las IV Jornadas de Estadística Pública, 2007, ISBN 978-84-690-7249-3, 2007
En la solucion de muchos problemas de inferencia aparecen procesos aleatorios que describen la i... more En la solucion de muchos problemas de inferencia aparecen procesos aleatorios que describen la informacion procedente de los datos. En algunos casos, cuando el tama~no n de los conjuntos de datos tiende a in nito, estos procesos tienen lmites gaussianos y eso permite estudiar el comportamiento asintotico de los procesos bajo alternativas contiguas. Vamos a estudiar transformaciones aplicables a esos procesos y a sus lmites gaussianos, para obtener tests consistentes frente a cualquier alternativa y que resulten especialmente sensibles a ciertas alternativas de interes. Los procedimientos que vamos a describir pueden expresarse en terminos de pruebas de pruebas de bondad de ajuste. Mostraremos un procedimiento comun para tratar todos los ejemplos mencionados.
Boletín de la Asociación Matemática Venezolana, 2002
El proceso de Wiener permite dar una demostración del Teorema del Límite Central mucho más probab... more El proceso de Wiener permite dar una demostración del Teorema del Límite Central mucho más probabilística que las habituales, que se basan fuertemente en la utilización de la transformada de Fourier o en otros argumentos de carácter analítico. Con el mismo esfuerzo, se obtiene como resultado un TLC funcional, y con poco esfuerzo más, la convergencia en distribución del proceso empírico al puente browniano. Estas notas describen el contenido de una charla que tuvo lugar en la Universidad de Oriente, en el marco de los Terceros Talleres de Formación Matemática que se desarrollaron en Cumaná, Venezuela, entre el 29 de julio y el 2 de agosto de 2002. En ella hemos intentado mostrar esta aplicación del proceso de Wiener, sin detallar los aspectos de carácter técnico, a un público con cierta familiaridad con el cálculo de probabilidades, como la que se alcanzaría luego de, al menos, un curso en el que se estudien el cálculo de esperanzas y la convergencia en probabilidad.
The statistical bibliography frequently refers to omnibus tests intended to be sensitive to all o... more The statistical bibliography frequently refers to omnibus tests intended to be sensitive to all or at least a wide variety of alternatives, and focused or directional tests directed to detect efficiently some specific alternatives. In fact, the apparent opposition between omnibus and focused is artificial, and, for instance, K-S test is focused on changes in position of Double Exponential distribution, as well as Cramér-von Mises is focused on changes in position of the distribution with density f (t) = 1/(2 cosh(πt/2)). We provide in this article a simple proof of this latter fact.
Zeitschrift für Wahrscheinlichkeitstheorie und Verwandte Gebiete, 1979
The probability density p of a plane Brownian motion stopped by a two-sided constant barrier is s... more The probability density p of a plane Brownian motion stopped by a two-sided constant barrier is shown to be a solution of a Kolmogorov forward equation of the form 5~*p=0. The operator 5('* is the product of two second order differential operators, each of them corresponding to a related one-dimensional Brownian motion.
Zeitschrift für Wahrscheinlichkeitstheorie und Verwandte Gebiete, 1972
ABSTRACT The barrier problem consisting in estimating the probability (1) is solved when the proc... more ABSTRACT The barrier problem consisting in estimating the probability (1) is solved when the process u(z) belongs to a class of sine series with independent coefficients. The solution is obtained by identifying the process u with the positions of a vibrating string forced by white noise, for which the same barrier problem has been solved in a previous paper ([2]).
Uploads
Papers by Enrique M. Cabaña