Exponential-type inequalities for martingale difference sequences. Application to nonparametric regression estimation

Laïb, Naâmane

doi:10.1080/03610929908832373

Cited by 19 publications

(13 citation statements)

References 7 publications

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“…) is a bounded triangular array of martingale differences with respect to F i . Behavior of this sequence may be studied using the following lemma due to Laïb [22]. Lemma 6.5 Let {(X i , S i ) : i ≥ 1} be a sequence of martingale difference such that |X i | ≤ B a.s. for 1 ≤ i ≤ n. For all > 0, one has P max…”

Section: Lemma 63 Under (A0) We Havementioning

confidence: 99%

Generalised kernel smoothing for non-negative stationary ergodic processes

Chaubey

Laïb

Sen

2010

Journal of Nonparametric Statistics

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In this paper, we consider a generalized kernel smoothing estimator of the regression function with non-negative support, using gamma probability densities as kernels, which are nonnegative and have naturally varying shapes. It is based on a generalization of Hille's lemma and a perturbation idea that allows us to deal with the problem at the boundary. Its uniform consistency and asymptotic normality are obtained at interior and boundary points, under a stationary ergodic process assumption, without using traditional mixing conditions. The asymptotic mean squared error of the estimator is derived and the optimal value of smoothing parameter is also discussed. Graphical illustrations of the proposed estimator are provided for simulated as well as for real data. A simulation study is also carried out to compare our method with the competing local linear method.

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Section: Lemma 63 Under (A0) We Havementioning

confidence: 99%

Generalised kernel smoothing for non-negative stationary ergodic processes

Chaubey

Laïb

Sen

2010

Journal of Nonparametric Statistics

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“…For improvements of the Hoeffding inequalities and related results see, for example, Talagrand 1995, McDiarmid 1989, Godbole and Hitczenko 1998, Pinelis 1998, Laib 1999, B 2001, van de Geer 2002, Perron 2003, BGZ 2006-2006, BGPZ 2006, BKZ 2006, BZ 2003 Up to certain constant factors, some of these improvements are close to the final optimal inequalities, see B 2004B , BKZ 2006. However so far no bounds taking into account information related to skewness and/or kurtosis are known, not to mention certain results related to symmetric random variables, see BGZ 2006, BGPZ 2006.…”

Section: Introduction and Resultsmentioning

confidence: 99%

Bounds for tail probabilities of martingales using skewness and kurtosis

Bentkus¹,

Juškevičius²

2008

Lith Math J

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Abstract. Let M n = X 1 + · · · + X n be a sum of independent random variables such that X k ≤ 1, E X k = 0 and E X 2 k = σ 2 k for all k. Hoeffding 1963, Theorem 3, proved thatBentkus 2004 improved Hoeffding's inequalities using binomial tails as upper bounds. Letk stand for the skewness and kurtosis of X k . In this paper we prove (improved) counterparts of the Hoeffding inequality replacing σ 2 by certain functions of γ 1 , . . . , γ n respectively κ 1 , . . . , κ n . Our bounds extend to a general setting where X k are martingale differences, and they can combine the knowledge of skewness and/or kurtosis and/or variances of X k . Up to factors bounded by e 2 /2 the bounds are final. All our results are new since no inequalities incorporating skewness or kurtosis control so far are known.

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“…Tran et al (1996) obtained the asymptotic normality of g n (x) assuming that the errors form a linear time series, more precisely, a weakly stationary linear process based on a martingale difference sequence, and Hu et al (2003) generalized the main results of Tran et al (1996). Laib (1999) studied the rate of convergence in the law of large numbers and the consistency of nonparametric regression models when the errors are martingale differences are given. Liang and Jing (2005) presented some asymptotic properties for estimates of nonparametric regression models base on negatively associated sequences.…”

Section: Introductionmentioning

confidence: 98%

The consistency for the estimator of nonparametric regression model based on martingale difference errors

Chen

Wang

2015

Stat Papers

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In this paper, by using the inequalities for martingale difference sequence, we investigate the consistency for the estimator of nonparametric regression model based on martingale difference errors. Some results on consistency for the estimator of g(x) are presented, including the mean consistency, complete consistency and strong consistency. As an application, the consistency for the nearest neighbor estimator is obtained.

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Exponential-type inequalities for martingale difference sequences. Application to nonparametric regression estimation

Cited by 19 publications

References 7 publications

Generalised kernel smoothing for non-negative stationary ergodic processes

Generalised kernel smoothing for non-negative stationary ergodic processes

Bounds for tail probabilities of martingales using skewness and kurtosis

The consistency for the estimator of nonparametric regression model based on martingale difference errors

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