Finite-sample and asymptotic sign-based tests for parameters of non-linear quantile regression with Markov noise

Sign-based statistical procedures are known to be more robust to outliers than least squares and ability to control precise significance level for finite samples when testing simple hypothesis. In this paper, the sign-based approach is extended to the case of non-linear model with dependent noise. The model of multi-quantile regression is considering, which allows to test hypotheses both on the parameters of the regression y _t = g _t (0) + e _t, t = 1, n , and the parameters ofthe noise e _t, which forms a stationary Markov process of order (r-1). According to that, the signs of observations are calculated with respect to a set of quantiles (at specified levels) of one-dimensional distribution et. The quantiles depend on the unknown parameter ц. For example, in a symmetric two-quantile regression, the quantiles (-ц) and ц correspond to levels ofp and 1-p, where p is given, and ц is unknown. For the three-quantile regression, the quantiles (-ц), 0, ц correspond to levels of p, 1/2 and 1-p. In both cases, the parameter ц is a scale parameter of noise. According to the sign-based approach, the residuals are substituting with indicators of their belonging to the interquantile intervals s = (si,...,s _n), where s _t takes a finite number of values. The unknown parameters in this scheme are U = (0',ц',Q )', where the vector Q contains linearly independent r-dimensional joint probabilities of the states of generated by process of indicators {s _t} . Since the problem is considering in a nonparametric setting, then each fixed value of parameters ц and Q corresponds to a class of finite-dimensional distributions of the initial process e _t. However, we can show that for any continuous parameterization of finite- dimensional distributions, all the derivatives of the likelihood of indicators P(s | u) are expressed in the same way. In the problem of testing a simple hypothesis H ₀ : U = U ₀ it gives the opportunity to build a test based on the principle of maximal likelihood ratio. In this paper, we consider the problem of calculating the critical values to provide the desired significance level with any accuracy for finite samples, as well as the critical values based on the asymptotic distribution of the test statistic. The obtained tests can be used as a basis for estimating the parameters и by the principle of maximal p-values, as well as for the development of tests for linear hypothesis.

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