Minimax weights in a~trend detection problem for a~stochastic process

I. L. Legostaeva ; A. N. Širyaev

Teoriâ veroâtnostej i ee primeneniâ, Tome 16 (1971) no. 2, pp. 339-345

Résumé

Let $F_n(M)$ be the class of real functions of the form $f(t)=a_0+a_1t+\dots+ a_nt^n+\mathrm g(t)t^{n+1}$ where $\sup\limits_t|\mathrm g(t)|\le M$, $-\infty$. The problem considered is to estimate the regression coefficient $a_0=f(0)$ from the data $\xi(t)=f(t)+\eta(t)$, $\eta(t)$ being a white noise process ($\mathbf M\eta(t)=0$, $\mathbf M\eta(s)\eta(t)=d^2\delta(t-s)$). For the class of linear estimators $\widehat f(0)=\int_{-\infty}^\infty l(t)\xi(t)\,dt$, a weight $l^*(t)$ is called minimax if $$ \sup_{f\in F_n(M)}\Delta(l^*,f)=\inf_l\sup_{f\in F_n(M)}\Delta(l,f) $$ where $\Delta(l,f)=\mathbf M[f(0)-\widehat f(0)]^2$. Theorem 1 gives necessary and sufficient conditions for a weight to be minimax. For $n=0$ and $n=1$ minimax weights are obtained in Theorem 2.