Multiscale inference about a density

In the setting of high-dimensional linear models with Gaussian noise, we investigate the possibility of confidence statements connected to model selection. Although there exist numerous procedures for adaptive (point) estimation, the construction of adaptive confidence regions is severely limited (cf. Li, 1989). The present paper sheds new light on this gap. We develop exact and adaptive confidence regions for the best approximating model in terms of risk. One of our constructions is based on a multiscale procedure and a particular coupling argument. Utilizing exponential inequalities for noncentral χ 2 -distributions, we show that the risk and quadratic loss of all models within our confidence region are uniformly bounded by the minimal risk times a factor close to one.AMS 2000 subject classifications: 62G15, 62G20.

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“…Hence we may utilize (15)(16), replacing (n, δ 2 , u) with (|D|, 0, α /(2µ n )), to complement (31) with the following observation:…”

Section: Thus Maxmentioning

confidence: 99%

Statistical inference for the optimal approximating model

Rohde

Dümbgen

2012

Probab. Theory Relat. Fields

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“…The above statistic was introduced in [17] where its large deviations were studied. Similar but different statistics were considered in [5,6]. The next theorem describes the almost-sure limiting behavior of the statistic (2.1).…”

Section: Multiscale Signal Detectionmentioning

confidence: 99%

Shao's theorem on the maximum of standardized random walk increments for multidimensional arrays

Kabluchko¹,

Munk²

2009

ESAIM: PS

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Abstract.We generalize a theorem of Shao [Proc. Amer. Math. Soc. 123 (1995) 575-582] on the almost-sure limiting behavior of the maximum of standardized random walk increments to multidimensional arrays of i.i.d. random variables. The main difficulty is the absence of an appropriate strong approximation result in the multidimensional setting. The multiscale statistic under consideration was used recently for the selection of the regularization parameter in a number of statistical algorithms as well as for the multiscale signal detection.Mathematics Subject Classification. 60F15.

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“…one simultaneously examines local regions of various sizes and locations. This approach was used by Marron (1999,2000) and Dümbgen and Spokoiny (2001) in conjunction with kernel estimates with varying bandwidths, by Dümbgen (2002) and Rohde (2006) with local rank tests, by Hall and Heckman (2000) with local linear smoothers, by Ganguli and Wand (2004) with local splines, and by Gijbels and Heckman (2004) and Dümbgen and Walther (2008) with local spacings.…”

Section: Introductionmentioning

confidence: 99%

“…For a concise exposition we will focus our theoretical investigations on the problem of inference about a density as in Dümbgen and Walther (2008). But it will become clear that the methodology introduced in this paper can be adapted to the other contexts cited above, and it may be relevant for multiscale methods beyond the area of shape-restricted inference.…”

Section: Introductionmentioning

confidence: 99%

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The Block Criterion for Multiscale Inference About a Density, With Applications to Other Multiscale Problems

Rufibach¹,

Walther²

2010

Journal of Computational and Graphical Statistics

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The use of multiscale statistics, i.e. the simultaneous inference about various stretches of data via multiple localized statistics, is a natural and popular method for inference about e.g. local qualitative characteristics of a regression function, a density, or its hazard rate. We focus on the problem of providing simultaneous confidence statements for the existence of local increases and decreases of a density and address several statistical and computational issues concerning such multiscale statistics. We first review the benefits of employing scaledependent critical values for multiscale statistics and then derive an approximation scheme that results in a fast algorithm while preserving statistical optimality properties. The main contribution is a methodology for calibrating multiscale statistics that does not require a caseby-case derivation of its specific form. We show that in the above density context the methodology possesses statistical optimality properties and allows for a fast algorithm. We illustrate the methodology with two further examples: A multiscale statistic introduced by Gijbels and Heckman for inference about a hazard rate and local rank tests introduced by D¨umbgen for inference in nonparametric regression. Code for the density application is available as the R package modehunt on CRAN. Additional code to compute critical values, reproduce the hazard rate and local rank example and the plots in the paper as well as datasets containing simulation results and an appendix with all the proofs of the theorems are available online as supplemental material. Code for the density application is available as the R package modehunt on CRAN. Additional code to compute critical values, reproduce the hazard rate and local rank example and the plots in the paper as well as datasets containing simulation results and an appendix with all the proofs of the theorems are available online as supplemental material.Keywords and phrases. Multiscale test, local increase in a density, fast algorithm.

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Multiscale inference about a density

Cited by 94 publications

References 31 publications

Statistical inference for the optimal approximating model

Statistical inference for the optimal approximating model

Shao's theorem on the maximum of standardized random walk increments for multidimensional arrays

The Block Criterion for Multiscale Inference About a Density, With Applications to Other Multiscale Problems

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