The asymptotic normality of Baxter-type estimates for parameters of nonstationary processes. (English. Ukrainian original) Zbl 0934.62084

Maiboroda, R. E.

Theory Probab. Math. Stat. 53, 107-112 (1996); translation from Teor. Jmovirn. Mat. Stat. 53, 97-102 (1995).

Let \(x(t)\) be a diffusion process that satisfies the equation \[ dx(t)=f(x(t),t)dt + g(t,\vartheta)dw(t), \] where \(f(x,t)\) is an unknown function, \(g(t,\vartheta)\) is a function that is known up to the parameter \(\vartheta\in\Theta\subset R^d\). Observations of the diffusion process \(x(t)\) at the points \(t_j^N=j/N\) are used to construct the Baxter-type sums \[ B_N(x,a)=\sum_{j=1}^N a(t_{j-1}^N)(x(t_j^N)-x(t_{j-1}^N))^2 \] with some weight function \(a(t)\) and a criterion \[ R_N(\alpha)=\int_0^1 g^4(\cdot,\alpha)dt - 2B_N(x,g^2(\cdot,\alpha)). \] The estimate for \(\vartheta\) is defined by the relation \(\hat\vartheta_N=\text{arg min}_{\alpha\in\Theta} R_N(\alpha).\) Conditions for asymptotic normality of \[ \sqrt{N}(B_N(x,a)- E B_N(x,a))\;\text{and} \sqrt{N}(\hat\vartheta_N-\vartheta) \] are obtained.

Reviewer: A.Ya.Olenko (Kyïv)

MSC:

62M05	Markov processes: estimation; hidden Markov models
62G20	Asymptotic properties of nonparametric inference
60F05	Central limit and other weak theorems
60G35	Signal detection and filtering (aspects of stochastic processes)

Keywords:

diffusion processes; Baxter-type theorems; asymptotic normality of estimates