Semiparametric analysis of long-range dependence in nonlinear regression

On spline regression under Gaussian subordination with long memory

“…Koul [41] and Koul and Baillie [42] extend results on M-estimation for long memory processes given in [2] to linear and nonlinear regression with random explanatory variables. Ivanov and Leonenko [35,36] derive asymptotic properties of least squares estimators in continuous time nonlinear regression, Ivanov and Leonenko [37] discuss estimation of the long memory parameter in a similar context. The results in [35,36] are not directly applicable to our situation, because our parameter space is unbounded and some of the regularity assumptions do not hold.…”

Asymptotic behaviour of the LS estimator in a nonlinear model with long memory

“…Examples of slowly varying functions (see Ivanov & Leonenko, 2008), for x > 0: x α j /(1+x 2 ) α j /2 for α j > 0, x α j /(1 + x α j ) for α j ∈ (0, 2], x α j /(1 + x) α j for 0 < α j < 1, x α j /(1 + x α j ) γ j for α j ∈ (0, 2], 0 < γ j , α j γ j < 1. For these functions, the autocovariances in (C1), (C2) are respectively: (1…”

“…A nonlinear regression model with long memory errors was also considered by Ivanov and Leonenko (2008), under the form: Y t = h(t, β) + η t , t = 1, . .…”

“…Koul (1996) found the convergence rate of the M-estimator when the regressors are deterministic and the errors are with long memory. Under the assumption that variable X i is deterministic, we can also recall the papers of Ivanov and Leonenko (2008) and Wang (2004) where the error is a Borel function of a stationary long memory Gaussian process. The case of stochastic regressors, has hardly been explored in nonlinear regression model with long memory errors.…”

Wilcoxon-Signed Rank Test for Long Memory Sequences

“…For the statistical estimation of D see, for example, the book of Beran (1994), Robinson (1995), or Hall et al (1997, Ivanov and Leonenko (2008), among others; a comprehensive, recent review of on semi-parametric estimators of the long-memory parameter is that of Moulines and Soulier (2003). The function L is usually equivalent to some constant related to D, therefore one usually assumes that L x → c as x → , see, for example Robinson (1994).…”