Alejandra Cabaña

Using the empirical moment generating function in testing for the Weibull and the type I extreme value distributions

We introduce two families of statistics, functionals of the empirical moment generating function process of the logarithmically transformed data, for testing goodness of fit to the two-parameter Weibull distribution or, equivalently, to the type I extreme value model. We show that when affine invariant estimators are used for the parameters of the extreme value distribution, the distributions of these statistics to not depend on the underlying parameters and one of them has a limiting chi-squared distribution. We estimate, via simulations, some finite sample quantiles for the statistics introduced and evaluate their power against a varied set of alternatives.

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.

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)).