Comparison of five methods for parameter estimation under Taylor's power law

Taylor’s power law (TPL) can be applied to the mean-variance relationship for various quantities, e.g., population densities over space and time, biomass of plants in the growth processes, developmental rates and growth rates of arthropods at different temperatures, etc. When TPL holds in the grouped data (e.g., the biomass of plants at different investigation times with limited replicates at each investigation), we must consider the heterocedasticity when carrying out the parameter estimation for a given mathematical model. We propose two new target functions based on TPL and attempt to more accurately estimate the model parameter(s). By capturing the mean-variance relationship, the accuracy of parameter estimation and the power of hypothesis testing are improved compared to the traditional methods which assume homogeneous variance. Five approaches, including ordinary least squares (OLS), chi-squared (χ²), weighted least squares (WLS), psi-squared (ψ²) and maximum likelihood estimation (MLE), are compared under both linear and nonlinear scenarios regarding the prediction accuracy using both computer simulations and experimental data. We further simulate irregular measurements on the predictors to examine the robustness of parameter estimation. In computer simulations, psi-squared, WLS, and MLE outperform OLS and chi-squared with respect to parameter estimation. In the simulation study of irregular measurements, psi-squared and MLE outperform WLS when the sample mean is smaller than the population mean. On the other hand, WLS outperforms psi-squared and MLE when the sample mean is larger than the population mean. Psi-squared has an advantage over MLE when large irregular deviations are present. This study strongly suggests using psi-squared and WLS in place of OLS or chi-squared for both linear and nonlinear regressions, the choice of method depending on the observations.