Parameter estimation and outlier detection in simultaneous linear functional relationship model for linear variables

This study addresses the necessity of extending the bivariate linear functional relationship model (LFRM) to simultaneous LFRM to investigate relationships between multiple linear variables concurrently while considering errors in each variable. The existing bivariate LFRM only enables the study of the relationship between two linear variables at a time with error consideration. This study aims to bridge this gap by extending the parameter estimation from bivariate LFRM to simultaneous LFRM using the maximum likelihood estimation (MLE) method. This facilitates examining relationships among multiple linear variables with error consideration in each variable. The covariance matrix of the parameter estimates is derived using the Fisher information matrix. Simulation results demonstrate parameter estimation accuracy across various sample sizes, assessed through measures such as mean, estimated bias, and mean absolute percentage error (MAPE) of parameter estimates in simultaneous LFRM. Subsequently, a COVRATIO statistic is derived from the model’s covariance matrix to detect the outliers in simultaneous LFRM. The cut-off points equation at the 1%, 5%, and 10% upper percentiles of the maximum value of the COVRATIO statistic is obtained through a Monte Carlo simulation study. Outliers are identified when the COVRATIO statistic exceeds these cut-off points. Simulation results of power performance indicate that the COVRATIO statistic effectively detects outliers for simultaneous LFRM. In the final segment of the study, the applicability of simultaneous LFRM is illustrated using synthetic data and Malaysia environmental datasets involving linear variables such as wind speed, humidity, and temperature. The Kolmogorov-Smirnov test and Q-Q plots support the normality of the data. The study contributes to the body of knowledge itself and offers applications for environmental analysis. Theoretical implications of this research include providing a more comprehensive approach to examining relationships between multiple linear variables while accounting for errors in each variable. This advancement enhances our understanding of multivariate relationships and error handling, which is critical in fields such as environmental science, economics, and engineering, where multiple interacting variables are common. Practical implications highlight that this enhancement improves predictive capabilities and facilitates informed decision-making across diverse domains, extending beyond outdoor activities. For instance, the ability to accurately model and predict the interplay between multiple climatic factors can lead to better resource management and policy-making in environmental analysis. Further research endeavours can explore applications of simultaneous LFRM in various fields, extending the scope of its utility.