Asymptotics of nonparametric L-1 regression models with dependent data

Abstract

We investigate asymptotic properties of least-absolute-deviation or median quantile estimates of the location and scale functions in nonparametric regression models with dependent data from multiple subjects. Under a general dependence structure that allows for longitudinal data and some spatially correlated data, we establish uniform Bahadur representations for the proposed median quantile estimates. The obtained Bahadur representations provide deep insights into the asymptotic behavior of the estimates. Our main theoretical development is based on studying the modulus of continuity of kernel weighted empirical process through a coupling argument. Progesterone data is used for an illustration.

Testing for changes in autocovariances of nonparametric time series models

“…Proposition 1 below is adopted from Zhao, Wei and Lin (2012). It shows that, if { e i } satisfies (5), then its properly transformed process also satisfies (5).…”

“…For completeness, we include the proof from Zhao, Wei and Lin (2012). Let q * = q /( ς + υ ), p 1 = υ/ς +1, and p 2 = ς/υ +1 so that ςq * p 1 = q, υq * p 2 = q , and 1/ p 1 +1/ p 2 = 1.…”

Asymptotics for $L_{1}$-wavelet method for nonparametric regression

“…The local median and M-estimators have been studied; see, for example, [7][8][9][10][11][12][13]. Also see [14,15] for more details on quantile regression and robust estimation, respectively.…”