Impulsive problems on the half-line with infinite impulse moments<Superscript>*</Superscript>

We present a two-point impulsive boundary value problem on the half-line with infinite impulsive effects on the unknown function and its derivative given by generalized functions.In this way, this problem can be applied to phenomena where the occurrence of infinite jumps depends not only on the instant, but also on their amplitude and frequency. The arguments apply Green’s functions and Schauder’s fixed-point theorem. The concept of equiconvergence at +∞ and at each impulsive moment is a key point to have a compact operator.     

Exact upper bounds for the number of the polynomials with given discriminants

We generalize the result of Davenport on the sum of absolute values of discriminants of integer polynomials of degree three. For the first time, we find the exact upper bound for the number of polynomials with given discriminant in the class of cubic polynomials of bounded height for the whole interval of values of discriminants.     

Extremes of Gaussian processes with smooth random expectation and smooth random variance

Let ξ(t), t ∈ [0, T],T > 0, be a Gaussian stationary process with expectation 0 and variance 1, and let η(t) and μ(t) be other sufficiently smooth random processes independent of ξ(t). In this paper, we obtain an asymptotic exact result for P(sup _t∈[0,T](η(t)ξ(t) + μ(t)) > u) as u→∞.     

Multi-poly-Bernoulli numbers and polynomials with a <Emphasis Type="Italic">q</Emphasis> parameter

We introduce themulti-poly-Bernoulli numbers and polynomialswith a q parameter, which are generalizations of the poly-Bernoulli numbers and polynomials with a q parameter, respectively.We give several combinatorial identities and properties of these new numbers and polynomials.     

Symmetry of zeros of Lerch zeta-function for equal parameters

For most values of parameters λ and α, the zeros of the Lerch zeta-function L(λ, α, s) are distributed very chaotically. In this paper, we consider the particular case of equal parameters L(λ, λ, s) and show by calculations that the nontrivial zeros either lie extremely close to the critical line σ = 1/2 or are distributed almost symmetrically with respect to the critical line. We also investigate this phenomenon theoretically.     

Fast implementation of the Tukey depth

Tukey depth function is one of the most famous multivariate tools serving robust purposes. It is also very well known for its computability problems in dimensions \(p \ge 3\). In this paper, we address this computing issue by presenting two combinatorial algorithms. The first is naive and calculates the Tukey depth of a single point with complexity \(O\left( n^{p-1}\log (n)\right) \), while the second further utilizes the quasiconcave of the Tukey depth function and hence is more efficient than the first. Both require very minimal memory and run much faster than the existing ones. All experiments indicate that they compute the exact Tukey depth.     

Bootstrap and permutation tests in ANOVA for directional data

The problem of testing the null hypothesis of a common direction across several populations defined on the hypersphere arises frequently when we deal with directional data. We may consider the Analysis of Variance (ANOVA) for testing such hypotheses. However, for the Watson distribution, a commonly used distribution for modeling axial data, the ANOVA test is only valid for large concentrations. So we suggest to use alternative tests, such as bootstrap and permutation tests in ANOVA. Then, we investigate the performance of these tests for data from Watson populations defined on the hypersphere.     

A comparison of variational approximations for fast inference in mixed logit models

Variational Bayesian methods aim to address some of the weaknesses (computation time, storage costs and convergence monitoring) of mainstream Markov chain Monte Carlo based inference at the cost of a biased but more tractable approximation to the posterior distribution. We investigate the performance of variational approximations in the context of the mixed logit model, which is one of the most used models for discrete choice data. A typical treatment using the variational Bayesian methodology is hindered by the fact that the expectation of the so called log-sum-exponential function has no explicit expression. Therefore additional approximations are required to maintain tractability. In this paper we compare seven different possible bounds or approximations. We found that quadratic bounds are not sufficiently accurate. A recently proposed non-quadratic bound did perform well. We also found that the Taylor series approximation used in a previous study of variational Bayes for mixed logit models is only accurate for specific settings. Our proposed approximation based on quasi Monte Carlo sampling performed consistently well across all simulation settings while remaining computationally tractable.     

Bayesian inference using a noninformative prior for linear Gaussian random coefficient regression with inhomogeneous within-class variances

A Bayesian inference for a linear Gaussian random coefficient regression model with inhomogeneous within-class variances is presented. The model is motivated by an application in metrology, but it may well find interest in other fields. We consider the selection of a noninformative prior for the Bayesian inference to address applications where the available prior knowledge is either vague or shall be ignored. The noninformative prior is derived by applying the Berger and Bernardo reference prior principle with the means of the random coefficients forming the parameters of interest. We show that the resulting posterior is proper and specify conditions for the existence of first and second moments of the marginal posterior. Simulation results are presented which suggest good frequentist properties of the proposed inference. The calibration of sonic nozzle data is considered as an application from metrology. The proposed inference is applied to these data and the results are compared to those obtained by alternative approaches.