On the estimation of periodic signals in the diffusion process using a high-frequency scheme

Abstract

Abstract The estimation of the frequency component is very interesting to study, considering its unique nature when these parameters are together in their amplitude. The periodicity of the frequency components is also thought to affect the convergence of these parameters. In this paper, we consider the problem of estimating the frequency component of a periodic continuous-time sinusoidal signal. Under the high-frequency sampling setting, we provide the frequency components’ consistency and asymptotic normality. It is observed that the convergence rate of the continuous-time sinusoidal signal of the diffusion process is the same as the continuous-time sinusoidal signal of the Ornstein–Uhlenbeck process, which is mentioned in [G. Pramesti, Parameter least-squares estimation for time-inhomogeneous Ornstein–Uhlenbeck process, Monte Carlo Methods Appl. 29 (2023), 1, 1–32]. The result of this study deduces that the convergence rate of the frequency is the same as long as the signal is periodic. In this case, the existence of the rate of reversion does not affect the convergence rate of the frequency components. Further, the result of the study, that is, the convergence rate of the frequency is ( n ⁢ h ) 3 \sqrt{(nh)^{3}} , also revised the previous one in [G. Pramesti, The least-squares estimator of sinusoidal signal of diffusion process for discrete observations, J. Math. Comput. Sci. 11 (2021), 5, 6433–6443], which mentioned ( n ⁢ h ) 3 ⁢ h \sqrt{(nh)^{3}h} . The proposed approach is demonstrated with a ten-minute sampling rate of real data on the energy consumption of light fixtures in one Belgium household.