Asymptotic Normality for Wavelet Estimators in Heteroscedastic Semiparametric Model with Random Errors
Tóm tắt
For the heteroscedastic regression model Yi = xiβ + g(ti) + σiei, 1 ≤ i ≤ n, where σ
2
= f (ui), the design points (xi, ti, ui) are known and nonrandom, g(·) and f(·) are defined on the closed interval [0,1]. When f(·) is known, we investigate the asymptotic normality for wavelet estimators of β and g(·) under {ei, 1 ≤ i ≤ n} is a sequence of identically distributed a-mixing errors; when f(·) is unknown, the asymptotic normality for wavelet estimators of β, g(·) and f(·) are established under independent errors. A simulation study is provided to illustrate the feasibility of the theoretical result that the authors derived.
Tài liệu tham khảo
Xu D K, Statistical inference for heteroscedastic models, Doctoral dissertation, Beijing University of Technology, Beijing, 2013.
Ruppert D, Wand M P, and Carroll R J, Semiparametric Regression, Cambridge University Press, Cambridge, New York, 2003.
You J H and Chen G M, Testing heteroscedasticity in partially linear regression models, Statistics and Probability Letters, 2005, 73: 61–70.
Fan G L, Liang H Y, and Xu H X, Empirical likelihood for a heteroscedastic partial linear model, Communications in Statistics — Theory and Methods, 2011, 40(8): 1396–1417.
Zhang J J and Liang H Y, Berry-Esseen type bounds in heteroscedastic semi-parametric model, Journal of Statistical Planning and Inference, 2011, 141(11): 3447–3462.
Zhang J J and Liang H Y, Asymptotic normality of estimators in heteroscedastic semi-parametric model with strong mixing errors, Communications in Statistics — Theory and Methods, 2012, 41(12): 2172–2201.
Engle R, Granger C, Rice J, et al., Nonparametric estimates of the relation between weather and electricity sales, Journal of the American Statistics Association, 1986, 81: 310–320.
Gao J T and Zhao L C, Adaptive estimation in partly linear regression models, Science in China (Series A), 1993, 36(1): 14–27.
Qian W M and Chai G X, Strong approximation of wavelet estimate in semiparametric regression model, Science in China, 1999, 29A(3): 233–240.
Xue L G, Rates of random weighting approximation of wavelet estimates in semiparametric regression model, Acta Mathematicae Applicatae Sinica, 2003, 26: 11–25.
Liang H Y and Fan G L, Berry-Esseen type bounds of estimators in a semiparametric model with linear process errors, Journal of Multivariate Analysis, 2009, 100(1): 1–15.
Zhou X C and Lin J G, Asymptotic properties of wavelet estimators in semiparametric regression models under dependent errors, Journal of Multivariate Analysis, 2013, 122: 251–270.
Wang X J, Deng X, Xia F X, et al., The consistency for the estimators of semiparametric regression model based on weakly dependent errors, Statistical Papers, 2017, 58: 303–318.
Liang H Y, Asymptotic normality of wavelet estimator in heteroscedastic model with α-mixing errors, Journal of Systems Science and Complexity, 2011, 24(4): 725–737.
Liang H Y and Liu Y M, Asymptotic normality of variance estimator in a heteroscedastic model with dependent errors, Journal of Nonparametric Statistics, 2011, 23(2): 351–365.
Georgiev A A, Consistent nonparametric multiple regression: The fixed design case, Journal of Multivariate Analysis, 1988, 25(1): 100–110.
Yang S C, Moment bounds for strong mixing sequences and their application, Journal of Mathematical Research and Exposition, 2000, 20(3): 349–359.
Yang S C, Maximal moment inequality for partial sums of strong mixing sequences and application, Acta Mathematicae Sinica, English Series, 2007, 26B(3): 1013–1024.
Li Y M and Guo J H, Asymptotic normality of wavelet estimator for strong mixing errors, Journal of the Korean Statistical Society, 2009, 38(4): 383–390.
Wang X J, Xu C, Hu T C, et al., On complete convergence for widely orthant-dependent random variables and its applications in nonparametric regression models, Test, 2014, 23: 607–629.
Shen A T, Zhang Y, and Volodin A, Applications of the Rosenthal-type inequality for negatively superadditive dependent random variables, Metrika, 2015, 78(3): 295–311.
Yang W Z, Xu H Y, Chen L, et al., Complete consistency of estimators for regression models based on extended negatively dependent errors, Statistical Papers, 2018, 59(2): 449–465.
Roussas G G, Tran L T, and Ioannides D A, Fixed design regression for time series: Asymptotic normality, Journal of Multivariate Analysis, 1992, 40: 262–291.
You J H, Chen M, and Chen G M, Asymptotic normality of some estimators in a fixed-design semipararmetric regression model with linear time series errors, Journal of Systems Science and Complexity, 2004, 17(4): 511–522.
Antoniadis A, Grégoire G, and Mckeague I M, Wavelet methods for curve estimation, Journal of the American Statistics Association, 1994, 89: 1340–1352.
Hall P and Heyde C C, Martingale Limit Theory and Its Application, Academic Press, New York, 1980.
Volkonskii V and Razanov Y, Some limit theorems for random functions, Theory of Probability and Its Applications, 1959, 4(2): 178–197.