Goodness-of-fit testing and quadratic functional estimation from indirect observations
By Cristina Butucea. 2006
58 citations recorded by the source corpus.
We consider the convolution model where i.i.d. random variables X i having unknown density f are observed with additive i.i.d. noise, independent of the X's. We assume that the density f belongs to either a Sobolev class or a class of supersmooth functions. The noise distribution is known and its characteristic function decays either polynomially or exponentially asymptotically. We consider the problem of goodness-of-fit testing in the convolution model. We prove upper bounds for the risk of a test statistic derived from a kernel estimator of the quadratic functional f f 2 based on indirect observations. When the unknown density is smoother enough than the noise density, we prove that this estimator is n -1/2 consistent, asymptotically normal and efficient (for the variance we compute). Otherwise, we give nonparametric upper bounds for the risk of the same estimator. We give an approach unifying the proof of nonparametric minimax lower bounds for both problems. We establish them for Sobolev densities and for supersmooth densities less smooth than exponential noise. In the two setups we obtain exact testing constants associated with the asymptotic minimax rates.
Subfields: Management Science and Operations Research, Statistics and Probability
View Paper: https://www.semanticscholar.org/paper/d59d0c1f2205b262216085d414ae8f46d2593f83
How researchers see this paper
The researchers whose trusted networks rank it highest, strongest advocate first. Every researcher gets a different order.
In Management Science and Operations Research
In Statistics and Probability
- Richard Nickl
- Evarist Gin'e
- Kolyan Ray
- I. Castillo
- D. Chetverikov
- Bharath K. Sriperumbudur
- Kengo Kato
- Victor Chernozhukov
Citations in the corpus
- Minimax Testing of a Composite null Hypothesis Defined via a Quadratic Functional in the Model of regression (2012, unknown): “In the case where the covariates t i are not observable and only x i ’s are available, our model coincides with the convolution model, for which the minimax rates of testing were obtained by Butucea (2007) in the one-dimensional case with simple null hypothesis.”
- A simple adaptive estimator of the integrated square of a density (2008, methodology): “Butucea (2007) also obtains such a bound, but only for the special kernel K(x) = sin(x)/πx, and it is the use of Lemma 1 that allows us to consider the case of general kernels.”