Enrique M. Cabaña

Goodness-of-Fit to the Exponential Distribution, Focused on Weibull Alternatives

ABSTRACT The goodness-of-fit technique based on the use of transformed empirical processes (TEPs) is applied to the construction of a test of exponentiality, focused on Weibull alternatives. The resulting procedure shares some desirable properties with other existing applications of the same technique: (a) the tests are consistent against fixed alternatives, (b) local alternatives belonging to the Weibull family are detected with an asymptotic power that does not differ significantly from the power of a two-sided (non consistent) likely ratio test (LRT), and (c) this asymptotic power is the same already encountered when a quadratic Cramer–von Mises–Watson type test statistic is used to test the fit to a single probability distribution, or to a parametric model with estimation of parameters–-In that sense, it is distribution free. In addition, an empirical power study shows that our test has the same level of performance than the best tests in the statistical literature.

A construction of continuous-time ARMA models by iterations of Ornstein-Uhlenbeck processes

We present a construction of a family of continuous-time ARMA processes based on p iterations of the linear operator that maps a Levy 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 Levy 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 Levy process, and show simulations and applications to real data.

A gaussian process with parabolic covariances

The centred, periodic, stationary Gaussian process X(z), 0 < z < 1 with covariances EX(y)X(z)=i(21y - z - 1)2, Iy - zl 1, appears when one studies the solutions of the vibrating string equation forced by noise, corresponding to the case of a finite string with the extremes tied together. The close relationship between this process and a Brownian bridge permits us to compute the distribution of the maximum excursion of the string at particular times.

Estimation of the spectral moment, by means of the extrema

An estimator of the standard deviation of the first derivative of a stationary Gaussian process with known variance and two continuous derivatives, based on the values of the relative maxima and minima, is proposed, and some of its properties are considered.ResumenSe propone un estimador de la desviación típica de la derivada de un proceso gaussiano estacionario de variancia conocida, basado en los valores de los extremos relativos del proceso, y se estudian algunas de sus propiedades.

Bridge-to-bridge transformations and Kolmogorov-Smirnov tests

A further step after the transformations to the empirical process introduced in Cabana (1993-a) to improve the efficiency of K-S tests provides a new class of quasi-optimum goodness-of-fit tests, and leads in particular to a constructive proof of the well-known fact that the classical K-S test has optimal ARE for shifts of double-exponential distributions (see Capon (1965)).

On a martingale characterization of two-parameter Wiener process

A generalization of martingales, named string-martingales in this paper, is introduced for two-dimensional-parameter processes. The two-parameter standard Wiener process is then characterised as a strong-martingale with continuous paths, and an additional property which states that a sort of quadratic variation of the process is deterministic, and given by the Lebesque measure.

An Alternative to CARMA Models via Iterations of Ornstein–Uhlenbeck Processes

We present a new construction of continuous ARMA processes based on iterating an Ornstein–Uhlenbeck operator \(\mathcal{O}\mathcal{U}_{\kappa }\) that maps a random variable y(t) onto \(\mathcal{O}\mathcal{U}_{\kappa }y(t) =\int _{ -\infty }^{t}\mathrm{e}^{-\kappa (t-s)}dy(s)\). 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.

Modeling Stationary Data by a Class of Generalized Ornstein-Uhlenbeck Processes: The Gaussian Case

The Ornstein-Uhlenbeck (OU) process is a well known conti- nuous-time interpolation of the discrete-time autoregressive process of order one, the AR(1). We propose a generalization of the OU process that resembles the construction of autoregressive processes of higher or- der p> 1 from the AR(1). The higher order OU processes thus obtained are called Ornstein-Uhlenbeck processes of order p (denoted OU(p)), and constitute a family of parsimonious models able to adjust slowly de- caying covariances. We show that the OU(p) processes are contained in the family of autoregressive moving averages of order (p, p − 1), the ARMA(p, p − 1), and that their parameters and covariances can be com- puted efficiently. Experiments on real data show that the empirical auto- correlation for large lags can be well modeled with OU(p) processes with approximately half the number of parameters than ARMA processes.