Statistical Analysis of Non Linear Least Squares Estimation for Harmonic Signals in Multiplicative and Additive Noise

Abstract

In this paper we consider the problem of parameter estimation for the multicomponent harmonic signals in multiplicative and additive noise. The nonlinear least squares (NLLS) estimators, NLLS₁ and NLLS₂ proposed by Ghogho et al. (1999b) to estimate the coherent model parameters for single-component harmonic signal, are generalized to the multicomponent harmonic signals for the cases of nonzero- and zero-mean multiplicative noise, respectively. By statistical analysis, some asymptotic results of the NLLS estimators are derived, including the strong consistency, the strong convergence rate and the asymptotic normality. Furthermore, the NLLS₁- and NLLS₂- based estimators are proposed to estimate the noncoherent model parameters for the cases of nonzero- and zero-mean multiplicative noise, respectively, meanwhile the strong consistency and the asymptotic normality of the NLLS-based estimators are also derived. Finally some numerical experiments are performed to see how the asymptotic results work for finite sample sizes.

Keywords:

• Asymptotic normality
• Harmonic signal
• Multiplicative noise
• Non-linear least squares
• Strong consistency
• Strong convergence rate

Mathematics Subject Clssification:

• Primary 62F12
• Secondary 62E20