SummaryFor any probability model M ≡ {p(x | θ, ω), θ ∈ Θ, ω ∈ Ω} assumed to describe the probabilistic behaviour of data x ∈ X, it is argued that testing whether or not the available data are compatible with the hypothesis H 0 ≡ {θ = θ 0 } is best considered as a formal decision problem on whether to use (a 0 ), or not to use (a 1 ), the simpler probability model (or null model) The BRC criterion provides a general reference Bayesian solution to hypothesis testing which does not assume a probability mass concentrated on M 0 and, hence, it is immune to Lindley's paradox. The theory is illustrated within the context of multivariate normal data, where it is shown to avoid Rao's paradox on the inconsistency between univariate and multivariate frequentist hypothesis testing.

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.

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Sufficient statistics, Goodness of fit, Conditional distributions,

The use of the probability generating function in testing the fit of discrete distributions was proposed by Kocherlakota & Kocherlakota (1986), and further studied by Mbques and P6rez-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 Cram& von Mises test of fit described in Spinelli (1994) and Spinelli and Stephens (1997), thus providing additional insight into these procedures. Also, a cor-