On the asymptotic normality of sequences of weak dependent random variables

Peligrad, Magda

doi:10.1007/bf02214083

Cited by 68 publications

(64 citation statements)

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“…We refer the reader to Bergstr om (1972), Krieger (1984) and Samur (1984) for the central limit theorem for -mixing triangular arrays with stationary rows and to Tikhomirov (1980) andG otze andHipp (1983) for rates of convergence and asymptotic expansions in the central limit theorem for mixing sequences. Let us also mention the recent w orks of Peligrad and Utev (1994) and Peligrad (1995), which improve the previous results for triangular arrays.…”

Section: Introductionsupporting

confidence: 57%

About the Lindeberg method for strongly mixing sequences

Rio

1997

ESAIM: PS

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Section: Introductionsupporting

confidence: 57%

About the Lindeberg method for strongly mixing sequences

Rio

1997

ESAIM: PS

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“…For simplicity, we discuss only iid random errors although the asymptotic results in Sections 1 and 2 are valid also for strongly mixing random errors: the asymptotic normality of GM estimator may be established by using a Central Limit Theorem for nonstationary weakly dependent triangular arrays of random variables [14] and the uniform strong consistency rate, needed for establishing the asymptotic normality of empirical zero, may be derived by applying a Hoeffding type exponential inequality for strongly mixing sequences [17] and by proceeding similarly as in [9], [11].…”

Section: Nonparametric Estimation Of Zerosmentioning

confidence: 99%

Optimal designs for nonparametric estimation of zeros of regression functions

Hlávka¹

2012

Tatra Mountains Mathematical Publications

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ABSTRACT. We investigate nonparametric estimators of zeros of a regression function and its derivatives and we derive the distribution of design points minimizing the expected width of a confidence interval and the expected variance of the proposed estimator.The main goal of this contribution is to derive the optimal distribution of design points for a nonparametric regression estimator of the so-called zero of a regression function, i.e., the location in which the unknown regression function intersects the horizontal axis.A survey of literature concerning optimal design for nonparametric regression models may be found in [20]. The immediately following discussion [7] suggests that there seem to exist two different general approaches. One approach uses the classical optimal design theory in order to find a finite set of support points with associated weights minimizing, e.g., the sum of variances of the nonparametric regression estimator in some pre-defined points [5] , we aim to find the design minimizing the variability of the empirical zero, i.e., the location at which the nonparametric regression estimator intersects the horizontal axis, instead of minimizing the variability of the nonparametric regression estimator itself. Similarly as in [10], the optimal design is obtained by applying standard calculus of variations. The main advantage of our approach is that prior

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“…It follows from a standard result for a weakly dependent sequence of random variables (Peligrad, 1996, Theorem 2.1) that the asymptotic distribution of elements in S can be assumed to be multivariate Gaussian as the number of scan volumes n → ∞,…”

Section: Measures Of Reliabilitymentioning

confidence: 99%