Optimal statistical estimates of a periodic function observed in random noise (Q1079310)

Consider the following class \(\chi\) of observed processes \[ X(t)=f(t)+\xi (t)+\eta (t); \] the signal \(f(t)=\sum^{\infty}_{- \infty}c_ ke^{ikt}\) is a periodic function with \(| c_ k| \leq d_ k\), \(d_ k=d_{-k}\geq 0\), \(\sum^{\infty}_{-\infty}d_ k<\infty\); the process \(\xi\) (t) is stationary in the wide sense with E \(\xi\) (t)\(=0\) and spectral density \(f_{\xi}(\lambda)\) satisfying \(f_{\xi}(\lambda)\leq g(\lambda)\), \(g(\lambda)=g(-\lambda)\geq 0\), \(\int^{\infty}_{-\infty}g(\lambda)d\lambda <\infty\); and \(\eta\) (t) is i.i.d. with E \(\eta\) (t)\(=0\), E \(\eta\) \({}^ 2(t)\leq \sigma^ 2<\infty\), where \((d_ k)\), g and \(\sigma^ 2\) are given. \(N=nm\) observations \(X(t_ j)\), \(j=0,...,N-1\), are taken periodically: \(t_ j=2\pi j/n\), \(n=2\nu +1\). Consider the class \({\mathcal K}\) of estimates \(\hat f\) of f having the following form: \[ \hat f(t)=\sum^{\nu}_{k=-\nu}w_ kc_ ke^{ikt},\quad c_ k=(1/N)\sum^{N-1}_{j=0}X(t_ j)e^{-ikt_ j} \] where \(w_ k\) are complex numbers and \(\bar w_ k=w_{-k}\). It is shown that there exists an optimal estimate \(f^*\) in the following sense \[ \sup_{X\in \chi}E\int^{2\pi}_{0}| f^*(t)-f(t)|^ 2dt=\inf_{\hat f\in {\mathcal K}\quad}\sup_{X\in \chi}E\int^{2\pi}_{0}| \hat f(t)- f(t)|^ 2dt \] and as m,n\(\to \infty\), \[ \sup_{X\in \chi}E\int^{2\pi}_{0}| f^*(t)-f(t)|^ 2dt\to 0. \] The rate of convergence is also investigated in certain specific cases.