## Efficient detection of random coefficients in autoregressive models.(English)Zbl 1039.62081

Summary: The problem of detecting randomness in the coefficients of an $$\text{AR}(p)$$ model, that is, the problem of testing ordinary $$\text{AR}(p)$$ dependence against the alternative of a random coefficient autoregressive $$\text{[RCAR}(p)]$$ model is considered. A nonstandard LAN property is established for $$\text{RCAR}(p)$$ models in the vicinity of $$\text{AR}(p)$$ ones. Two main problems arise in this context. The first problem is related to the statistical model itself: Gaussian assumptions are highly unrealistic in a nonlinear context, and innovation densities should be treated as nuisance parameters. The resulting semiparametric model however appears to be severely nonadaptive. In contrast with the linear ARMA case, pseudo-Gaussian likelihood methods here are invalid under non-Gaussian densities; even the innovation variance cannot be estimated without a strict loss of efficiency.
This problem is solved using a general result by M. Hallin and B. J. M. Werker [Bernoulli 9, 137–165 (2003; Zbl 1020.62042)], which provides semiparametrically efficient central sequences without going through explicit tangent space calculations. The second problem is related to the fact that the testing problem under study is intrinsically one-sided, while the case of multiparameter one-sided alternatives is not covered by classical asymptotic theory under LAN. A concept of locally asymptotically most stringent somewhere efficient test is proposed in order to cope with this one-sided nature of the problem.

### MSC:

 62M10 Time series, auto-correlation, regression, etc. in statistics (GARCH) 62G10 Nonparametric hypothesis testing 62G20 Asymptotic properties of nonparametric inference

Zbl 1020.62042
Full Text: