Skew-normal Model Research Articles

Default Bayesian goodness-of-fit tests for the skew-normal model

In this paper we propose a series of goodness-of-fit tests for the family of skew-normal models when all parameters are unknown. As the null distributions of the considered test statistics depend only on asymmetry parameter, we used a default and proper prior on skewness parameter leading to the prior predictive p-value advocated by G. Box. Goodness-of-fit tests, here proposed, depend only on sample size and exhibit full agreement between nominal and actual size. They also have good power against local alternative models which also account for asymmetry in the data.

Generalized skew-normal models: properties and inference

In this article, we introduce a new family of asymmetric distributions, which depends on two parameters namely, α and β, and in the special case where β = 0, the skew-normal (SN) distribution considered by Azzallini [Azzalini, A., 1985, A class of distributions which includes the normal ones. Scandinavian Journal of Statistics, 12, 171–178.] is obtained. Basic properties such as a stochastic representation and the derivation of maximum likelihood and moment estimators are studied. The asymptotic behaviour of both types of estimators is also investigated. Results of a small-scale simulation study is provided illustrating the usefulness of the new model. An application to a real data set is reported showing that it can present better fit than the SN distribution.

A note on reference priors for the scalar skew-normal distribution

In this note, we discuss some peculiar features of default Bayes analysis of the scalar skew-normal model. In particular we show that, when we consider the simplest model with a single skewness parameter λ unknown, the reference—or Jeffreys’—prior for this parameter, whose support is unbounded, is proper. From a Bayesian perspective this is important because the likelihood function for λ happens to be monotone with positive sampling probability and the use of improper priors would be impossible. We also explore the frequentist properties of the default Bayes approach. Finally, we consider the general scalar case with unknown location and scale parameters and we derive a closed form expression for the integrated likelihood function for λ when the location and scale parameters are integrated out with respect to the usual conditional reference prior.

A class of multivariate skew-normal models

The existing model for multivariate skew normal data does not cohere with the joint distribution of a random sample from a univariate skew normal distribution. This incoherence causes awkward interpretation for data analysis in practice, especially in the development of the sampling distribution theory. In this paper, we propose a refined model that is coherent with the joint distribution of the univariate skew normal random sample, for multivariate skew normal data. The proposed model extends and strengthens the multivariate skew model described in Azzalini (1985,Scandinavian Journal of Statistics,12, 171–178). We present a stochastic representation for the newly proposed model, and discuss a bivariate setting, which confirms that the newly proposed model is more plausible than the one given by Azzalini and Dalla Valle (1996,Biometrika,83, 715–726).

Species and size selectivity in a Portuguese multispecies artisanal long-line fishery

The species and size selectivity of long-lines using small hooks were studied off the south coast of Portugal using “Mustad” brand round bent, flatted sea hooks (Quality 2316 DT) numbers 15, 13, and 11 baited with razor shell clam (Ensis siliqua). Hook numbers 13 and 11 are 49 and 109% larger respectively than number 15 hooks in terms of overall size (maximum width × maximum length). A total of 39 900 hooks were fished in 45 sets and 35 species of fish and cephalopods were caught. As a group, 13 species of sea breams (Sparidae) dominated the catch by numbers (58%) and weight (73%). Six species of sea breams, along with the greater weever fish (Trachinus draco) accounted for 81% of the total catch by weight, with the common or white sea bream (Diplodus sargus) being the most important (29%). Catch size distributions by hook size were, in general, highly overlapped for all species and hook size had little apparent effect on minimum size at capture. All hooks caught a wide range of sizes per species, but the catch rate (number of fish per 100 hooks) was significantly lower for the largest hook. Except for the black sea bream (Spondyliosoma cantharus), capture of illegally sized or immature fish was minimal. Small increases in average size with hook size were evident for four species: Diplodus sargus, D. vulgaris, Lithognathus mormyrus and Serranus cabrilla. No differences in size selectivity were detected for Boops boops, D. annularis, Spondyliosoma cantharus and Trachinus draco. A skew-normal model adequately described differences in size selectivity in five of six species.