A Comparison of Autometrics and Penalization Techniques under Various Error Distributions: Evidence from Monte Carlo Simulation

This work compares Autometrics with dual penalization techniques such as minimax concave penalty (MCP) and smoothly clipped absolute deviation (SCAD) under asymmetric error distributions such as exponential, gamma, and Frechet with varying sample sizes as well as predictors. Comprehensive simulations, based on a wide variety of scenarios, reveal that the methods considered show improved performance for increased sample size. In the case of low multicollinearity, these methods show good performance in terms of potency, but in gauge, shrinkage methods collapse, and higher gauge leads to overspecification of the models. High levels of multicollinearity adversely affect the performance of Autometrics. In contrast, shrinkage methods are robust in presence of high multicollinearity in terms of potency, but they tend to select a massive set of irrelevant variables. Moreover, we find that expanding the data mitigates the adverse impact of high multicollinearity on Autometrics rapidly and gradually corrects the gauge of shrinkage methods. For empirical application, we take the gold prices data spanning from 1981 to 2020. While comparing the forecasting performance of all selected methods, we divide the data into two parts: data over 1981–2010 are taken as training data, and those over 2011–2020 are used as testing data. All methods are trained for the training data and then are assessed for performance through the testing data. Based on a root-mean-square error and mean absolute error, Autometrics remain the best in capturing the gold prices trend and producing better forecasts than MCP and SCAD.

A multiplicity result for a (p, q)-Schrödinger–Kirchhoff type equation

In this paper, we study a class of (p, q)-Schrödinger–Kirchhoff type equations involving a continuous positive potential satisfying del Pino–Felmer type conditions and a continuous nonlinearity with subcritical growth at infinity. By applying variational methods, penalization techniques and Lusternik–Schnirelman category theory, we relate the number of positive solutions with the topology of the set where the potential attains its minimum values.

Multiplicity and concentration results for magnetic relativistic Schrödinger equations

In this paper, we consider the following magnetic pseudo-relativistic Schrödinger equation $$\begin{array}{} \displaystyle \sqrt{\left(\frac{\varepsilon}{i}\nabla-A(x)\right)^2+m^2}u+V(x)u= f(|u|)u \quad {\rm in}\,\,\mathbb{R}^N, \end{array}$$ where ε > 0 is a parameter, m > 0, N ≥ 1, V : ℝN → ℝ is a continuous scalar potential satisfies V(x) ≥ − V0 > − m for any x ∈ ℝN and f : ℝN → ℝ is a continuous function. Under a local condition imposed on the potential V, we discuss the number of nontrivial solutions with the topology of the set where the potential attains its minimum. We proof our results via variational methods, penalization techniques and Ljusternik-Schnirelmann theory.

Shrinkage priors for Bayesian penalized regression.

In linear regression problems with many predictors, penalized regression techniques are often used to guard against overfitting and to select variables relevant for predicting the outcome. Classical regression techniques find coefficients that minimize a squared residual; penalized regression adds a penalty term to this residual to limit the coefficients’ sizes, thereby preventing over- fitting. Many classical penalization techniques have a Bayesian counterpart, which result in the same solutions when a specific prior distribution is used in combination with posterior mode estimates. Compared to classical penalization techniques, the Bayesian penalization techniques perform similarly or even better, and they offer additional advantages such as readily available uncertainty estimates, automatic estimation of the penalty parameter, and more flexibility in terms of penalties that can be considered. As a result, Bayesian penalization is becoming increasingly popular. The aim of this paper is to provide a comprehensive overview of the literature on Bayesian penalization. We will compare different priors for penalization that have been proposed in the literature in terms of their characteristics, shrinkage behavior, and performance in terms of prediction and variable selection in order to aid researchers to navigate the many prior options.