Raúl Rueda

TESTS OF FIT FOR DISCRETE DISTRIBUTIONS BASED ON THE PROBABILITY GENERATING FUNCTION

The use of the probability generating function in testing the fit of discrete distributions was proposed by Kocherlakota & Kocherlakota (1986), and further studied by Marques and Perez-Abreu (1989). In Rueda et al. (1991), a quadratic statistic to test the fit of a discrete distribution was proposed using the probability generating function and its empirical counterpart. This was illustrated for the Poisson case with known parameter. Here, we deal with some extensions: the Poisson case with unknown parameter and the negative Binomial distribution with known or unknown parameter p. We find the asymptotic distribution of the test statistic in each case, and show with the aid of some Monte Carlo studies the closeness of these asymptotic distributions. A connection is established between this quadratic test and the Cramer von Mises test of fit described in Spinelli (1994) and Spinelli and Stephens (1997), thus providing additional insight into these procedures. Also, a correction is made on the expression of ...

Goodness of fit for the inverse Gaussian distribution

For testing the fit of the inverse Gaussian distribution with unknown parameters, the empirical distribution-function statistic A2 is studied. Two procedures are followed in constructing the test statistic; they yield the same asymptotic distribution. In the first procedure the parameters in the distribution function are directly estimated, and in the second the distribution function is estimated by its Rao-Blackwell distribution estimator. A table is given for the asymptotic critical points of A2 These are shown to depend only on the ratio of the unknown parameters. An analysis is provided of the effect of estimating the ratio to enter the table for A2. This analysis enables the proposal of the complete operating procedure, which is sustained by a Monte Carlo study.

Reference priors for exponential families

Abstract Reference analysis, introduced by Bernardo (J. Roy. Statist. Soc. 41 (1979) 113) and further developed by Berger and Bernardo (On the development of reference priors (with discussion). In: J.M. Bernardo, J.O. Berger, A.P. Dawid, A.F.M. Smith (Eds.), Bayesian Statistics, Vol. 4, Clarendon Press, Oxford, pp. 35–60), has proved to be one of the most successful general methods to derive noninformative prior distributions. In practice, however, reference priors are typically difficult to obtain. In this paper we show how to find reference priors for a wide class of exponential family likelihoods.

A Bayesian alternative to parametric hypothesis testing

SummaryA unified approach to parametric hypothesis testing from a decision-theoretical viewpoint is proposed. A measure of the discrepancy between two models is incorporated as part of the utility function. Specifically, the Kullback logarithmic divergence is considered as such a measure. Under certain conditions, when using this measure of discrepancy there is correspondence with the classical solutions.

On the Selection of Ground‐Motion Prediction Equations for Probabilistic Seismic‐Hazard Analysis

Abstract In current practice of probabilistic seismic‐hazard analysis (PSHA), the difference between ground‐motion prediction equations (GMPEs), which are in principle equally valid related to their quality and applicability, is attributed to epistemic uncertainty. The standard practice is to include this uncertainty through logic trees. Based on probability concepts, we present a method to assist during the selection and weighting of GMPEs to be included in different branches of a logic tree. We find that in regions with abundant recorded data, only those models with large likelihood should be considered. Although the presented method is not the only option to define the weighting of GMPEs for PSHA, it offers an ordered way to combine different sources of knowledge, such as recorded data and prior information.

A note on the fit for the levy distribution

The problem of testing the fit of the Levy distribution with unknown scale parameter is addressed. The corresponding empirical process is analysed, and the Cramer-von Mises W2 and the Anderson-Darlings A2 statistics are used. Well known results regarding the relationship between the Levy distribution and the gamma and inverse Gaussian are exploited. Some remarks are made regarding the use of the Rao-Blackwell estimator of the distribution function in the empirical process.

ON CONVERGENCE THEOREMS FOR QUANTILES

Given a real random variable Xits quantiles are always well defined, unlike its expectation. Based on a general definition of population quantiles we generalize known results for the case in which the quantiles are not unique. We propose estimators for the non unique case that also work in the unique case, giving a general approach for real random variables. Some simulations and applications of these results are also included.

Fiducial Inferences for the Truncated Exponential Distribution

Lindqvist and Taraldsen (2005) introduced an interesting parametric family of distributions in the unit interval. In this note, inference procedures are given, both from the classical and the Bayesian view point. It is shown numerically through various examples that the posterior distribution for the parameter and the induced fiducial distribution are almost equivalent. The parametric family under study is a regular member of the Natural Exponential Family and so use of this fact permits induction of a unique fiducial in terms of the minimal sufficient statistic.

How Important Is the Effect of Rounding in Goodness-of-Fit

Abstract In the area of goodness-of-fit there is a clear distinction between the problem of testing the fit of a continuous distribution and that of testing a discrete distribution. In all continuous problems the data is recorded with a limited number of decimals, so in theory one could say that the problem is always of a discrete nature, but it is a common practice to ignore discretization and proceed as if the data is continuous. It is therefore an interesting question whether in a given problem of test of fit, the “limited resolution” in the observed recorded values may be or may be not of concern, if the analysis done ignores this implied discretization. In this article, we address the problem of testing the fit of a continuous distribution with data recorded with a limited resolution. A measure for the degree of discretization is proposed which involves the size of the rounding interval, the dispersion in the underlying distribution and the sample size. This measure is shown to be a key characteristic which allows comparison, in different problems, of the amount of discretization involved. Some asymptotic results are given for the distribution of the EDF (empirical distribution function) statistics that explicitly depend on the above mentioned measure of degree of discretization. The results obtained are illustrated with some simulations for testing normality when the parameters are known and also when they are unknown. The asymptotic distributions are shown to be an accurate approximation for the true finite n distribution obtained by Monte Carlo. A real example from image analysis is also discussed. The conclusion drawn is that in the cases where the value of the measure for the degree of discretization is not “large”, the practice of ignoring discreteness is of no concern. However, when this value is “large”, the effect of ignoring discreteness leads to an exceded number of rejections of the distribution tested, as compared to what would be the number of rejections if no rounding is taking into account. The error made in the number of rejections might be huge.