Statistical analysis for fractional differential equations models

Challenges related to parameter estimation in models based on fractional differential equations have garnered increased attention in recent years. Specifically, significant literature exists concerning the point estimation of parameters and the efficiency of the fractional models. While many studies have made valuable contributions to elucidating these matters, there remains a need for further investigation to broaden our comprehension. Additionally, rigorous scrutiny of the underlying assumptions is required in any statistical analysis. This paper identifies several inferential issues that require consideration, including nested models containing the classical differential equation models; the trade-off between parameter estimation accuracy and model efficiency when the fractional derivative parameter is statistically insignificant; determination of an appropriate sample size to maintain the originally planned type I error rate while enhancing the power of hypothesis tests; the application of goodness-and-fit tests; computational complexities associated with solving the fractional differential equation; and considerations related to the statistical model such as the covariance structure. To address these issues systematically, we introduce a range of statistical methods and computational tools. These include the utilization of a likelihood-based approach, statistical tests, comprehensive simulation studies, and an efficient numerical scheme for solving the fractional differential equation. We illustrate the applicability of our proposed methods with examples and real-world data extracted from the existing body of literature.