Search

In this paper we study two important representations for extreme value distributions and their max‐domains of attraction (MDA), namely von Mises representation (vMR) and variation representation (VR), which are convenient ways to gain limit results. Both VR and vMR are defined via so‐called auxiliary functions ψ. Up to now, however, the set of valid auxiliary functions for vMR has neither been characterized completely nor separated from those for VR. We contribute to the current literature by introducing "universal" auxiliary functions which are valid for both VR and vMR representations for the entire MDA distribution families. Then we identify exactly the sets of valid auxiliary functions for both VR and vMR. Moreover, we propose a method for finding appropriate auxiliary functions with analytically simple structure and provide them for several important distributions. [ABSTRACT FROM AUTHOR]

Well‐recommended methods of forming "confidence intervals" for a binomial proportion give interval estimates that do not actually meet the definition of a confidence interval, in that their coverages are sometimes lower than the nominal confidence level. The methods are favoured because their intervals have a shorter average length than the Clopper–Pearson (gold‐standard) method, whose intervals really are confidence intervals. As the definition of a confidence interval is not being adhered to, another criterion for forming interval estimates for a binomial proportion is needed. In this paper, we suggest a new criterion for forming one‐sided intervals and equal‐tail two‐sided intervals. Methods which meet the criterion are said to yield locally correct confidence intervals. We propose a method that yields such intervals and prove that its intervals have a shorter average length than those of any other method that meets the criterion. Compared with the Clopper–Pearson method, the proposed method gives intervals with an appreciably smaller average length. For confidence levels of practical interest, the mid‐p method also satisfies the new criterion and has its own optimality property. It gives locally correct confidence intervals that are only slightly wider than those of the new method. [ABSTRACT FROM AUTHOR]

This paper deals with the specification of probability distributions expressing ignorance concerning annual or otherwise discretized failure or mortality rates, when these rates can safely be assumed to be increasing and convex, but are completely unknown otherwise. Such distributions can be used as noninformative priors for Bayesian analysis of failure data. We demonstrate why a uniform distribution used in earlier work is unsatisfactory, especially from the point of view of insensitivity with respect to the time scale that is chosen for the problem at hand. We suggest alternative distributions based on Dirichlet distributed weights for the extreme points of relevant convex sets, and discuss which consequences a requirement for scale neutrality has for the choice of Dirichlet parameters. [ABSTRACT FROM AUTHOR]

In the literature, quite a few measures have been proposed for quantifying the deviation of a probability distribution from symmetry. The most popular of these skewness measures are based on the third centralized moment and on quantiles. However, there are major drawbacks in using these quantities. These include a strong emphasis on the distributional tails and a poor asymptotic behavior for the (empirical) moment‐based measure as well as difficult statistical inference and strange behaviour for discrete distributions for quantile‐based measures. Therefore, in this paper, we introduce skewness measures based on or connected with expectiles. Since expectiles can be seen as smoothed versions of quantiles, they preserve the advantages over the moment‐based measure while not exhibiting most of the disadvantages of quantile‐based measures. We introduce corresponding empirical counterparts and derive asymptotic properties. Finally, we conduct a simulation study, comparing the newly introduced measures with established ones, and evaluating the performance of the respective estimators. [ABSTRACT FROM AUTHOR]

The class of inhomogeneous phase‐type distributions (IPH) was recently introduced in Albrecher & Bladt (2019) as an extension of the classical phase‐type (PH) distributions. Like PH distributions, the class of IPH is dense in the class of distributions on the positive halfline, but leads to more parsimonious models in the presence of heavy tails. In this paper we propose a fitting procedure for this class to given data. We furthermore consider an analogous extension of Kulkarni's multivariate PH class (Kulkarni, 1989) to the inhomogeneous framework and study parameter estimation for the resulting new and flexible class of multivariate distributions. As a by‐product, we amend a previously suggested fitting procedure for the homogeneous multivariate PH case and provide appropriate adaptations for censored data. The performance of the algorithms is illustrated in several numerical examples, both for simulated and real‐life insurance data. [ABSTRACT FROM AUTHOR]