Christian H. Hesse
University of Stuttgart
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Journal of Nonparametric Statistics | 1999
Christian H. Hesse
In this paper we study an automatic empirical procedure for density deconvolution based on observations that are contaminated by additive measurement errors from a known distribution. The assumptions placed on the density to be estimated are mild and apart from continuity do not include additional smoothness conditions. The procedure uses a class of deconvoluting kernel estimates and selects the smoothing parameter so as to minimize an estimate of integrated squared error over a discret set. The resulting estimator is shown to be asymptotically optimal both in the integrated squared error and mean integrated squared error sense. A simulation study is performed to examine the practical merit of the procedure.
Journal of Nonparametric Statistics | 2004
Christian H. Hesse; Alexander Meister
Assume that n independent copies of Y = X + ϵ are observed where ϵ is an unobservable measurement error with a known distribution. We consider the problem of estimating the unknown density of X when this density is known to lie in a given smoothness class. An iterative procedure for estimating the unknown density is introduced. Rates of convergence for mean integrated squared error are studied for smoothness classes arising from Fourier conditions. Minimax rates are derived for these classes. The sequence of estimators resulting from the iterative procedure is shown to attain the optimal rates both for smooth and for supersmooth error densities. The iterative scheme allows one to perform density estimation from contaminated observations by simple additive corrections to an appropriate ordinary kernel density estimator. In this way, the effect of the perturbation due to contamination by ϵ may be quantified. In addition, we demonstrate that the sequence of estimators converges exponentially fast to a specific estimator within the class of deconvoluting kernel density estimators. We also address the subtle theoretical issues that arise when the error density is not in L 2(ℝ) leading to a modification of the iterative procedure. Finally, a comparative simulation study is performed which shows the merit of the procedure. Email: meistear@mathmatik.uni-stuttgart.de
Annals of the Institute of Statistical Mathematics | 1995
Christian H. Hesse
AbstractThe paper studies the performance of deconvoluting kernel density estimators for estimating the marginal density of a linear process. The data stem from the linear process and are partially, respectively fully contaminated by iid errors with a known distribution. If 1−p denotes the proportion of contaminated observations (and it is, of course, unknown which observations are contaminated and which are not) then for 1−p ∈ (0, 1) and under mild conditions almost sure deconvolution rates of orderO(n−2/5(logn)9/10) can be achieved for convergence in
Archive | 2017
Christian H. Hesse
Archive | 2017
Christian H. Hesse
\mathcal{L}_\infty
Archive | 2017
Christian H. Hesse
Archive | 2017
Christian H. Hesse
. This rate compares well with the existing rates foriid uncontaminated observations. Forp=0 and exponentially decreasing error characteristic function the corresponding rates are of merely logarithmic order. As a by-product the paper also gives a rate of convergence result for the empirical characteristic function in the linear process context and utilizes this to demonstrate that deconvoluting kernel density estimators attain the optimal rate in the dependence case with exponentially decreasing error characteristic function.
Archive | 2017
Christian H. Hesse
Wie geht’s eigentlich Little K? Lange nichts mehr von ihm gehort. Sein Leben platschert so vor sich hin. Ohne besondere Vorkommnisse. Eine Kleinigkeit gibt’s allerdings doch von ihm zu berichten: Little K hat seinem Schulkameraden Ali Gator vor Kurzem 12 Euro geliehen. Heute will Ali ihm das Geld zuruckzahlen. Eigentlich. Aber irgendwie will Ali das auch nicht. Er ist namlich ein elender Zocker, und erst gestern zum Beispiel hat er Little K eine Wahnsinnswette angeboten, bei der man um ein paar Ecken denken muss, um zu sehen, was da los ist. Das Bild sagt, worum es dabei ging.
Archive | 2017
Christian H. Hesse
Okay, es ist schon lange her. Rund 20 Jahre. Einige von euch, die das jetzt hier lesen, waren damals noch nicht geboren. Haben aber vielleicht spater davon gehort. Die Sache ist namlich immer noch ziemlich bekannt. Ihr werdet sehen, auch dieses Kapitel bringt gleich einen richtigen Paukenschlag als Themenspender. Nach all den Neulichkeiten fruherer Kapitel – neulich beim Sudoku, neulich beim Klassentreffen, neulich beim Schach – spulen wir hier aber in die Vergangenheit zuruck.
Archive | 2017
Christian H. Hesse
Okay, Geometrie. Das heist fur die meisten zum Beispiel Pythagoras. Der Name steht in jeder Mathe-People-Bibel. Pythagoras und sein fesches Posting uber rechtwinklige Dreiecke. Ihr wisst schon: Das Quadrat uber der Hypotenuse ist …