On a Birnbaum–Saunders distribution arising from a non-homogeneous Poisson process

A New Inference Approach for Type-II Generalized Birnbaum-Saunders Distribution

The Birnbaum-Saunders (BS) distribution, with its generalizations, has been successfully applied in a wide variety of fields. One generalization, type-II generalized BS (denoted as GBS-II), has been developed and attracted considerable attention in recent years. In this article, we propose a new simple and convenient procedure of inference approach for GBS-II distribution. An extensive simulation study is carried out to assess performance of the methods under various settings of parameter values with different sample sizes. Real data are analyzed for illustrative purposes to display the efficiency of the proposed method.

Statistical Inference for Progressive Stress Accelerated Life Testing with Birnbaum-Saunders Distribution

As a result of the two-parameter Birnbaum–Saunders (BS) distribution being successful in modelling fatigue failure times, several extensions of this model have been explored from different aspects. In this article, we consider a progressive stress accelerated life testing for the BS model to introduce a generalized Birnbaum–Saunders (we call it Type-II GBS) distribution on the lifetime of products in the test. We outline some interesting properties of this highly flexible distribution, present the Fisher’s information in the maximum likelihood estimation method, and propose a new Bayesian approach for inference. Simulation studies are carried out to assess the performance of the methods under various settings of parameter values and sample sizes. Real data are analyzed for illustrative purposes to demonstrate the efficiency and accuracy of the proposed Bayesian method over the likelihood-based procedure.

The Hawkes Process with Different Exciting Functions and its Asymptotic Behavior

The standard Hawkes process is constructed from a homogeneous Poisson process and uses the same exciting function for different generations of offspring. We propose an extension of this process by considering different exciting functions. This consideration may be important in a number of fields; e.g. in seismology, where main shocks produce aftershocks with possibly different intensities. The main results are devoted to the asymptotic behavior of this extension of the Hawkes process. Indeed, a law of large numbers and a central limit theorem are stated. These results allow us to analyze the asymptotic behavior of the process when unpredictable marks are considered.

Fatigue-life distributions for reaction time data

The family of fatigue-life distributions is introduced as an alternative model of reaction time data. This family includes the shifted Wald distribution and a shifted version of the Birnbaum-Saunders distribution. Although the former has been proposed as a way to model reaction time data, the latter has not. Hence, we provide theoretical, mathematical and practical arguments in support of the shifted Birnbaum-Saunders as a suitable model of simple reaction times and associated cognitive mechanisms.

The membrane potential process of a single neuron seen as a cumulative damage process

A simple integrate-and-fire mechanism of a single neuron can be compared with a cumulative damage process, where the spiking process is analogous to rupture sequences of a material under cycles of stress. Although in some cases lognormal-like patterns can be recognized in the inter-spike times under a simple integrate-and-fire mechanism, fatigue life models as the inverse Gaussian distribution and the Birnbaum-Saunders distribution (which was recently introduced in the neural activity framework) provide theoretical arguments that make them more suitable for the modeling of the resulting inter-spike times.

The Hawkes Process with Different Exciting Functions and its Asymptotic Behavior

The standard Hawkes process is constructed from a homogeneous Poisson process and uses the same exciting function for different generations of offspring. We propose an extension of this process by considering different exciting functions. This consideration may be important in a number of fields; e.g. in seismology, where main shocks produce aftershocks with possibly different intensities. The main results are devoted to the asymptotic behavior of this extension of the Hawkes process. Indeed, a law of large numbers and a central limit theorem are stated. These results allow us to analyze the asymptotic behavior of the process when unpredictable marks are considered.

Modeling neural activity with cumulative damage distributions

Neurons transmit information as action potentials or spikes. Due to the inherent randomness of the inter-spike intervals (ISIs), probabilistic models are often used for their description. Cumulative damage (CD) distributions are a family of probabilistic models that has been widely considered for describing time-related cumulative processes. This family allows us to consider certain deterministic principles for modeling ISIs from a probabilistic viewpoint and to link its parameters to values with biological interpretation. The CD family includes the Birnbaum-Saunders and inverse Gaussian distributions, which possess distinctive properties and theoretical arguments useful for ISI description. We expand the use of CD distributions to the modeling of neural spiking behavior, mainly by testing the suitability of the Birnbaum-Saunders distribution, which has not been studied in the setting of neural activity. We validate this expansion with original experimental and simulated electrophysiological data.

Birnbaum–Saunders statistical modelling: a new approach

Modelling based on the Birnbaum–Saunders distribution has received considerable attention in recent years. In this article, we introduce a new approach for Birnbaum–Saunders regression models, which allows us to analyze data in their original scale and to model non-constant variance. In addition, we propose four types of residuals for these models and conduct a simulation study to establish which of them has a better performance. Moreover, we develop methods of local influence by calculating the normal curvatures under different perturbation schemes. Finally, we perform a statistical analysis with real data by using the approach proposed in the article. This analysis shows the potentiality of our proposal.