zbMATH — the first resource for mathematics

Examples
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.

Operators
a & b logic and
a | b logic or
!ab logic not
abc* right wildcard
"ab c" phrase
(ab c) parentheses
Fields
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)
Testing the (parametric) null model hypothesis in (semiparametric) partial and generalized spline models. (English) Zbl 0673.62017

Summary: D. R. Cox and E. Koh [A smoothing spline based test of model adequacy in polynomial regression. Tech. Rep. 787, Dept. Statistics, Univ. Wisconsin-Madison (1986)] considered the model y i =f(x(i))+ϵ i , ϵ i i.i.d. N(0,σ 2 ), with the (parametric) null hypothesis f(x), x[0,1], a polynomial of degree m-1 or less, versus the alternative f is “smooth”, based on the Bayesian model for f which leads to polynomial smoothing spline estimates for f. They showed that there was no uniformly most powerful test and found the locally most powerful (LMP) test. We extend their result to generalized smoothing spline models and to partial spline models.

We also show that the test statistic has an intimate relationship with the behavior of the generalized cross validation (GCV) function at λ=. If the GCV function has a minimum at λ=, then GCV has chosen the (parametric) model corresponding to the null hypothesis; we show that if the LMP test statistic is no larger than a certain multiple of the residual sum of squares after (parametric) regression, then the GCV function will have a (possibly local) minimum at λ=.


MSC:
62F03Parametric hypothesis testing
62G10Nonparametric hypothesis testing
62H15Multivariate hypothesis testing