zbMATH — the first resource for mathematics

Geometry Search for the term Geometry in any field. Queries are case-independent.
Funct* Wildcard queries are specified by * (e.g. functions, functorial, etc.). Otherwise the search is exact.
"Topological group" Phrases (multi-words) should be set in "straight quotation marks".
au: Bourbaki & ti: Algebra Search for author and title. The and-operator & is default and can be omitted.
Chebyshev | Tschebyscheff The or-operator | allows to search for Chebyshev or Tschebyscheff.
"Quasi* map*" py: 1989 The resulting documents have publication year 1989.
so: Eur* J* Mat* Soc* cc: 14 Search for publications in a particular source with a Mathematics Subject Classification code (cc) in 14.
"Partial diff* eq*" ! elliptic The not-operator ! eliminates all results containing the word elliptic.
dt: b & au: Hilbert The document type is set to books; alternatively: j for journal articles, a for book articles.
py: 2000-2015 cc: (94A | 11T) Number ranges are accepted. Terms can be grouped within (parentheses).
la: chinese Find documents in a given language. ISO 639-1 language codes can also be used.

a & b logic and
a | b logic or
!ab logic not
abc* right wildcard
"ab c" phrase
(ab c) parentheses
any anywhere an internal document identifier
au author, editor ai internal author identifier
ti title la language
so source ab review, abstract
py publication year rv reviewer
cc MSC code ut uncontrolled term
dt document type (j: journal article; b: book; a: book article)
Nonparametric estimates of regression quantiles and their local Bahadur representation. (English) Zbl 0728.62042
Summary: Let (X,Y) be a random vector such that X is d-dimensional, Y is real valued and $Y=\theta (X)+\epsilon$, where X and $\epsilon$ are independent and the $\alpha$ th quantile of $\epsilon$ is 0 ($\alpha$ is fixed such that $0<\alpha <1)$. Assume that $\theta$ is a smooth function with order of smoothness $p>0$, and set $$ r=(p-m)/(2p+d), $$ where m is a nonnegative integer smaller than p. Let T($\theta$) denote a derivative of $\theta$ of order m. It is proved that there exists a pointwise estimate $\hat T\sb n$ of T($\theta$), based on a set of i.i.d. observations $(X\sb 1,Y\sb 1),...,(X\sb n,Y\sb n)$, that achieves the optimal nonparametric rate of convergence $n\sp{-r}$ under appropriate regularity conditions. Further, a local Bahadur type representation is shown to hold for the estimate $\hat T\sb n$ and this is used to obtain some useful asymptotic results.

62G07Density estimation
62G20Nonparametric asymptotic efficiency
62G35Nonparametric robustness
62E20Asymptotic distribution theory in statistics
Full Text: DOI