Asymptotic properties of wavelet estimators in heteroscedastic semiparametric model based on negatively associated innovations

Consider the heteroscedastic semiparametric regression model $y_{i}=x_{i}\beta+g(t_{i})+\varepsilon_{i}$yi=xiβ+g(ti)+εi, $i=1, 2, \ldots, n$i=1,2,…,n, where β is an unknown slope parameter, $\varepsilon_{i}=\sigma_{i}e_{i}$εi=σiei, $\sigma^{2}_{i}=f(u_{i})$σi2=f(ui), $(x_{i},t_{i},u_{i})$(xi,ti,ui) are nonrandom design points, $y_{i}$yi are the response variables, f and g are unknown functions defined on the closed interval $[0,1]$[0,1], random errors $\{e_{i} \}${ei} are negatively associated (NA) random variables with zero means. Whereas kernel estimators of β, g, and f have attracted a lot of attention in the literature, in this paper, we investigate their wavelet estimators and derive the strong consistency of these estimators under NA error assumption. At the same time, we also obtain the Berry–Esséen type bounds of the wavelet estimators of β and g.

On the Bahadur representation of sample quantiles for widely orthant dependent sequences

It can be found that widely orthant dependent (WOD) random variables are weaker than extended negatively orthant dependent (END) random variables, while END random variables are weaker than negatively orthant dependent (NOD) and negatively associated (NA) random variables. In this paper, we investigate the Bahadur representation of sample quantiles based on WOD sequences. Our results extend the corresponding ones of Ling [N.X. Ling, The Bahadur representation for sample quantiles under negatively associated sequence, Statistics and Probability Letters 78(16) (2008), 2660-2663], Xu et al. [S.F. Xu, L. Ge, Y. Miao, On the Bahadur representation of sample quantiles and order statistics for NA sequences, Journal of the Korean Statistical Society 42(1) (2013), 1-7] and Li et al. [X.Q. Li, W.Z. Yang, S.H. Hu, X.J. Wang, The Bahadur representation for sample quantile under NOD sequence, Journal of Nonparametric Statistics 23(1) (2011), 59-65] for the case of NA sequences or NOD sequences.

The Hoffmann-Jørgensen inequality of NA random variables and its applications to the logarithm law

In this paper, the Hoffmann-Jørgensen inequality for negatively associated (NA) random variables is derived. As an important tool, it will be applied to the establishing for the logarithm law of NA arrays, and the results of Su, Hu and Liang {xc[15]} are extended.