New error bounds for Laplace approximation via Stein's method

We use Stein's method to obtain explicit bounds on the rate of convergence for the Laplace approximation of two different sums of independent random variables; one being a random sum of mean zero random variables and the other being a deterministic sum of mean zero random variables in which the normalisation sequence is random. We make technical advances to the framework of Pike and Ren \cite {pike} for Stein's method for Laplace approximation, which allows us to give bounds in the Kolmogorov and Wasserstein metrics. Under the additional assumption of vanishing third moments, we obtain faster convergence rates in smooth test function metrics. As part of the derivation of our bounds for the Laplace approximation for the deterministic sum, we obtain new bounds for the solution, and its first two derivatives, of the Rayleigh Stein equation.

ON SMOOTH TESTS FOR THE EQUALITY OF DISTRIBUTIONS

This note re-investigates the smooth tests for the equality of distributions introduced by Bera et al. (2013, Econometric Theory 29, 419–446) and provides a modified smooth test which works for the general case with two sample sizes m and n. Asymptotic properties of the proposed test statistic under both the null and the alternative hypothesis are studied.

Pair dependent linear statistics for CβE

We study the limiting distribution of a pair counting statistics of the form [Formula: see text] for the circular [Formula: see text]-ensemble (C[Formula: see text]E) of random matrices for sufficiently smooth test function [Formula: see text] and [Formula: see text] For [Formula: see text] and [Formula: see text] our results are inspired by a classical result of Montgomery on pair correlation of zeros of Riemann zeta function.

Fully nonlinear stochastic and rough PDEs: Classical and viscosity solutions

We study fully nonlinear second-order (forward) stochastic PDEs. They can also be viewed as forward path-dependent PDEs and will be treated as rough PDEs under a unified framework. For the most general fully nonlinear case, we develop a local theory of classical solutions and then define viscosity solutions through smooth test functions. Our notion of viscosity solutions is equivalent to the alternative using semi-jets. Next, we prove basic properties such as consistency, stability, and a partial comparison principle in the general setting. If the diffusion coefficient is semilinear (i.e, linear in the gradient of the solution and nonlinear in the solution; the drift can still be fully nonlinear), we establish a complete theory, including global existence and a comparison principle.

Tensile Notch Sensitivity of Additively Manufactured IN 625 Superalloy

The notch sensitivity of additively manufactured IN 625 superalloy produces by laser powder bed fusion (LPBF) has been investigated by tensile testing of cylindrical test pieces. Smooth and V-notched test pieces with four different radii were tested both in as-built state and after a stress relief heat treatment for 1 h at 900 °C. Regardless of the notch root radius, the investigated alloy exhibits notch strength ratios higher than unity in both as-built and in stress-relieved states, showing that the additive manufactured IN 625 alloy is not prone to brittleness induced by the presence of V-notches. Higher values of notch strength ratios were recorded for the as-built material as a result of the higher internal stress level induced by the manufacturing process. Due to the higher triaxiality of stresses induced by notches, for both as-built and stress-relieved states, the proof strength of the notched test pieces is even higher than the tensile strength of the smooth test pieces tested in the same conditions. SEM fractographic analysis revealed a mixed mode of ductile and brittle fracture morphology of the V-notched specimens regardless the notch root radius. A more dominant ductile mode of fracture was encountered for stress-relieved test pieces than in the case of the as-built state. However, future research is needed to better understand the influence of notches on additive manufactured IN 625 alloy behaviour under more complex stresses.

LEROYI: A NEW TETHYAN LAGENID BENTHIC FORAMINIFERAL GENUS

Leroyi n. gen., is introduced to include the Cretaceous-Neogene (predominantly Maastrichtian-Eocene) benthic Lagenid foraminiferids from many Tethyan localities that characterized by its slightly coiled early portion of the smooth test, later slightly arcuate uniserial chambers increasing in length as added, oblique depressed sutures, aperture radial of dorsal angle. I suggest Leroyi as a new genus to accommodate foraminifera with these characters. This new genus have been previously assigned to Marginulina sp. C of LeRoy (1953), and here assigned as a genotype of the new genus. Four species were previously described from two localities in Egypt (Maqfi section, Farafra Oasis and Nekhl section, Sinai) are treated here as a new species of the new genus, and formally named as: Leroyi aegyptiaca Anan, n. sp., L. maqfiensis Anan, n. sp., L. deserti (Said & Kenawy, 1956), L. ghorabi (Said & Kenawy, 1956). One Tunisian species: Leroyi tunisiana Anan, n. sp. is added to these Egyptian species. Another European and American species: Leroyi glabra (d’Orbigny) is added to these Laginid group. These six species of the Lagenid new genus Leroyi are recorded in six localities in the Tethys (USA, France, Italy, Tunisia Egypt, UAE and India).

A Dispersion Test for the Modified Borel-Tanner Distribution

Dispersion tests based on the second order component of smooth test statistics are related to Fisher’s Index of Dispersion test, used for testing for the Poisson distribution when there are no covariates present. Such tests have been recommended in [1] to test for the Poisson distribution when covariates are present. The modified Borel-Tanner (MBT) distribution seems suited to data with extra zeroes, a monotonic decline in counts and longer tails. Here we recommend a dispersion test for the MBT distribution for both when covariates are absent and when they are present.