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23
Asymptotic equivalence for nonparametric regression with multivariate and random design
, 2008
"... We show that nonparametric regression is asymptotically equivalent in Le Cam’s sense with a sequence of Gaussian white noise experiments as the number of observations tends to infinity. We propose a general constructive framework based on approximation spaces, which permits to achieve asymptotic equ ..."
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Cited by 20 (2 self)
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We show that nonparametric regression is asymptotically equivalent in Le Cam’s sense with a sequence of Gaussian white noise experiments as the number of observations tends to infinity. We propose a general constructive framework based on approximation spaces, which permits to achieve asymptotic equivalence even in the cases of multivariate and random design.
Minimax hypothesis testing for curve registration
 Electron. J. Statist
"... Abstract: This paper is concerned with the problem of goodnessoffit for curve registration, and more precisely for the shifted curve model, whose application field reaches from computer vision and road traffic prediction to medicine. We give bounds for the asymptotic minimax separation rate, when ..."
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Cited by 8 (4 self)
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Abstract: This paper is concerned with the problem of goodnessoffit for curve registration, and more precisely for the shifted curve model, whose application field reaches from computer vision and road traffic prediction to medicine. We give bounds for the asymptotic minimax separation rate, when the functions in the alternative lie in Sobolev balls and the separation from the null hypothesis is measured by the l2norm. We use the generalized likelihood ratio to build a nonadaptive procedure depending on a tuning parameter, which we choose in an optimal way according to the smoothness of the ambient space. Then, a Bonferroni procedure is applied to give an adaptive test over a range of Sobolev balls. Both achieve the asymptotic minimax separation rates, up to possible logarithmic factors.
Asymptotic equivalence of nonparametric autoregression and nonparametric
, 2006
"... It is proved that nonparametric autoregression is asymptotically equivalent in the sense of Le Cam’s deficiency distance to nonparametric regression with random design as well as with regular nonrandom design. 1. Introduction. We assume that observations X0,...,Xn from a stationary autoregressive pr ..."
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Cited by 7 (0 self)
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It is proved that nonparametric autoregression is asymptotically equivalent in the sense of Le Cam’s deficiency distance to nonparametric regression with random design as well as with regular nonrandom design. 1. Introduction. We assume that observations X0,...,Xn from a stationary autoregressive process (Xi)i=0,...,n are available which obey the model equation (1) Xi = f(Xi−1) + εi,
Calibration of selfdecomposable Lévy models
 SFB 649 Berlin, Discussion Paper No
, 2011
"... We study the nonparametric calibration of exponential Lévy models with infinite jump activity. In particular our analysis applies to selfdecomposable processes whose jump density can be characterized by the kfunction, which is typically nonsmooth at zero. On the one hand the estimation of the dri ..."
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Cited by 6 (1 self)
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We study the nonparametric calibration of exponential Lévy models with infinite jump activity. In particular our analysis applies to selfdecomposable processes whose jump density can be characterized by the kfunction, which is typically nonsmooth at zero. On the one hand the estimation of the drift, of the activity measure α: = k(0+)+k(0−) and of analogous parameters for the derivatives of the kfunction are considered and on the other hand we estimate nonparametrically the kfunction. Minimax convergence rates are derived. Since the rates depend on α, we construct estimators adapting to this unknown parameter. Our estimation method is based on spectral representations of the observed option prices and on a regularization by cutting off high frequencies. Finally, the procedure is applied to simulations and real data.
ASYMPTOTIC EQUIVALENCE AND ADAPTIVE ESTIMATION FOR ROBUST NONPARAMETRIC REGRESSION
, 2009
"... Asymptotic equivalence theory developed in the literature so far are only for bounded loss functions. This limits the potential applications of the theory because many commonly used loss functions in statistical inference are unbounded. In this paper we develop asymptotic equivalence results for rob ..."
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Cited by 6 (4 self)
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Asymptotic equivalence theory developed in the literature so far are only for bounded loss functions. This limits the potential applications of the theory because many commonly used loss functions in statistical inference are unbounded. In this paper we develop asymptotic equivalence results for robust nonparametric regression with unbounded loss functions. The results imply that all the Gaussian nonparametric regression procedures can be robustified in a unified way. A key step in our equivalence argument is to bin the data and then take the median of each bin. The asymptotic equivalence results have significant practical implications. To illustrate the general principles of the equivalence argument we consider two important nonparametric inference problems: robust estimation of the regression function and the estimation of a quadratic functional. In both cases easily implementable procedures are constructed and are shown to enjoy simultaneously a high degree of robustness and adaptivity. Other problems such as construction of confidence sets and nonparametric hypothesis testing can be handled in a similar fashion.
A note on quantile coupling inequalities and their applications
 Submitted. Available from www.stat.yale.edu/˜hz68
, 2006
"... A relationship between the large deviation and quantile coupling is studied. We apply this relationship to the coupling of the sum of n i.i.d. symmetric random variables with a normal random variable, improving the classical quantile coupling inequalities (the key part in the celebrated KMT construc ..."
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Cited by 3 (2 self)
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A relationship between the large deviation and quantile coupling is studied. We apply this relationship to the coupling of the sum of n i.i.d. symmetric random variables with a normal random variable, improving the classical quantile coupling inequalities (the key part in the celebrated KMT constructions) with a rate 1 = p n for random variables with continuous distributions, or the same rate modulo constants for the general case. Applications to the asymptotic equivalence theory and nonparametric function estimation are discussed.
ASYMPTOTIC EQUIVALENCE FOR INHOMOGENEOUS JUMP DIFFUSION PROCESSES AND WHITE NOISE.
"... Abstract. We prove the global asymptotic equivalence between the experiments generated by the discrete (high frequency) or continuous observation of a path of a time inhomogeneous jumpdiffusion process and a Gaussian white noise experiment. Here, the considered parameter is the drift function, and ..."
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Abstract. We prove the global asymptotic equivalence between the experiments generated by the discrete (high frequency) or continuous observation of a path of a time inhomogeneous jumpdiffusion process and a Gaussian white noise experiment. Here, the considered parameter is the drift function, and we suppose that the observation time T tends to ∞. The approximation is given in the sense of the Le Cam ∆distance, under smoothness conditions on the unknown drift function. These asymptotic equivalences are established by constructing explicit Markov kernels that can be used to reproduce one experiment from the other. 1.
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"... SFB 649 Discussion Paper 2011086 Spectral estimation of covolatility from noisy observations using local weights ..."
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SFB 649 Discussion Paper 2011086 Spectral estimation of covolatility from noisy observations using local weights