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)
A study of logspline density estimation. (English) Zbl 0825.62442
Summary: A method of estimating an unknown density function $f$ based on sample data is studied. Our approach is to use maximum likelihood estimation to estimate log($f$) by a function $s$ from a space of cubic splines that have a finite number of prespecified knots and are linear in the tails. The knots are placed at selected order statistics of the sample data. The number of knots can be determined either by a simple rule or by minimizing a variant of $AIC$. Examples using both simulated and real data show that the method works well both in obtaining smooth estimates and in picking up small details. The method is fully automatic and can easily be extended to yield estimates and confidence bounds for quantiles.

62G07Density estimation
Full Text: DOI
[1] De Boor, C.: A practical guide to splines. (1978) · Zbl 0406.41003
[2] Box, G. E. P.; Cox, D. R.: An analysis of transformations (with discussion). J. roy. Statist. soc. 26, 211-252 (1964) · Zbl 0156.40104
[3] Breiman, L.; Stone, C. J.; Kooperberg, C.: Robust confidence bounds for extreme upper quantiles. J. statist. Comp. simul. 37, 127-149 (1990) · Zbl 0775.62117
[4] Copas, J. B.; Freyer, M. J.: Density estimation and suicide risk in psychiatric treatment. J. roy. Statist. soc. 143, 167-176 (1980) · Zbl 0457.62087
[5] . Annual basetapes and reports (1968--1983) (1983)
[6] Kooperberg, C.: Smoothing images, splines and densities, ph.d. Thesis. (1990)
[7] O’sullivan, F.: Fast computation of fully automated log-density and log-hazard estimators. SIAM J. Sci. stat. Comput. 9, 363-379 (1988) · Zbl 0688.65083
[8] Schwarz, G.: Estimating the dimension of a model. Annals of statistics 6, 461-464 (1978) · Zbl 0379.62005
[9] Silverman, B. W.: Density estimation for statistics and data analysis. (1986) · Zbl 0617.62042
[10] Smith, P. L.: Curve Fitting and modeling with splines using statistical variable selection methods. NASA report 166034 (1982)
[11] Stone, C. J.: Large sample inference for logspline model. Annals of statistics 18, 717-741 (1990) · Zbl 0712.62036
[12] Stone, C. J.; Koo, C. -Y.: Logspline density estimation. Contemporary mathematics 59, 1-15 (1986) · Zbl 0618.62038
[13] Wand, M. P.; Marron, S. J.; Ruppert, D.: Transformation in density estimation (with discussion). J.a.s.a. (1991) · Zbl 0742.62046