A new statistical distribution via the Phi-4 equation with its wide-ranging applications

Area of Science:

• Mathematical Physics
• Statistical Modeling
• Nonlinear Dynamics

Background:

• Nonlinear partial differential equations (PDEs) are crucial for modeling complex phenomena.
• Statistical distributions are essential for data analysis and interpretation.
• The hyperbolic secant (HS) distribution offers unique properties for modeling.

Purpose of the Study:

• To develop a novel statistical framework integrating nonlinear PDEs and statistical distributions.
• To derive and analyze the probability density function (PDF) of the HS distribution for the nonlinear Phi-4 equation.
• To explore the applicability of the proposed HS distribution model to real-world datasets.

Main Methods:

• Formulation of a new framework based on nonlinear partial differential equations.
• Derivation of the probability density function (PDF) for the hyperbolic secant (HS) distribution applied to the nonlinear Phi-4 equation.
• Employment of eight distinct parameter estimation techniques.
• Investigation of HS model parameter behavior using simulated datasets.
• Application of the HS distribution model to real-world data.

Main Results:

• The derived HS distribution exhibits versatile density shapes (left-skewed, symmetric, right-skewed).
• Eight estimation methods were successfully applied to estimate model parameters.
• Parameter behavior analysis confirmed the robustness of the estimation techniques.
• Demonstrated successful application of the HS distribution for modeling real data.

Conclusions:

• The proposed framework effectively integrates nonlinear PDEs with statistical modeling.
• The HS distribution provides a flexible and applicable tool for data analysis.
• This work is expected to benefit researchers exploring hyperbolic function-based distributions and their real-world applications.