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In the presence of fixed threshold effects, the least squares (LS) estimator of the threshold parameter poses challenges for statistical inference due to its nonstandard limiting distribution, which also presents challenges for bootstrap methods. To address this issue, we propose a novel estimator: a two-step smoothed gradient least squares (SGLS) estimator. Our proposed method achieves a normal limiting distribution for the threshold parameter with minimal efficiency loss compared to the LS estimator. Furthermore, our modified bootstrap method significantly enhances computational efficiency, leading to improved bootstrap confidence intervals (CIs) for the threshold parameter compared to asymptotic CIs. Our method is validated through a small Monte Carlo study and demonstrated with an empirical application. [ABSTRACT FROM AUTHOR]

In order to solve partial differential equations (PDEs) numerically, one first needs to approximate the field functions (such as the displacement functions), and then obtain the derivatives of the field functions (such as the strains), by directly differentiating the field functions. Using such direct-derivatives in formulating a numerical method is common and is used in the standard finite element method (FEM), but such models are often found to be 'stiff'. In the weakened weak (W2) formulations, it is found that the use of properly re-constructed derivatives can be beneficial in ways because the model can become 'softer'. This paper presents a novel 'pick-out' theory and technique for re-constructing the derivatives (such as the strains) of functions defined in a local domain, using smoothing operations. The local domain can be a smoothing domain used in the smoothed finite element methods (S-FEMs), smoothed point interpolation methods (S-PIMs), and smoothed particle hydrodynamics (SPH). It is discovered that through the use of a set of linearly independent smoothing functions that are continuous in the local domain, one can simply pick out various orders of smoothed derivatives (at the center of a domain) from any given function that may discontinuous (strictly) inside the local domain. As long as the smoothing function is continuous in the smoothing domain, the picked out 'smoothed derivatives' are equivalent (in a local integral sense) to the compatible direct-derivatives, which ensures the convergence of the smoothed model (such as the S-FEM) when the smoothing domains shrinking to zero. The pick-out technique can be used in strong, weak, local weak, weak-strong, or weakened weak formulations to create stable and convergent numerical models. It may offer a new window of opportunity to develop new effective numerical models using smoothed derivatives (strains) that are 'softer' and can produce accurate solutions also in the derivatives (strains and stresses) of the field functions (displacements). [ABSTRACT FROM AUTHOR]

This paper presents a study of meshless Point Interpolation Methods based on weakened-weak forms. The mathematical formulations of the methods are presented as well as the procedures for the support nodes selection called T-schemes. The numerical results are shown for four different types of electromagnetic static problems in order to ponctuate the characteristics of the approximation generated by these new methods. [ABSTRACT FROM AUTHOR]

This paper presents a study of meshless Point Interpolation Methods based on weakened-weak forms. The mathematical formulations of the methods are presented as well as the procedures for the support nodes selection called T-schemes. The numerical results are shown for four different types of electromagnetic static problems in order to ponctuate the characteristics of the approximation generated by these new methods.