# 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)
Fixed-design regression for linear time series. (English) Zbl 0862.62069

Summary: This investigation is concerned with recovering a regression function $g\left({x}_{i}\right)$ on the basis of noisy observations taken at uniformly spaced design points ${x}_{i}$. It is presumed that the corresponding observations are corrupted by additive dependent noise, and that the noise is, in fact, induced by a general linear process in which the summand law can be discrete, as well as continuously distributed. Discreteness induces a complication because such noise is not known to be strong mixing, the postulate by which regression estimates are often shown to be asymptotically normal. In fact, as cited, there are processes of this character which have been proven not to be strong mixing.

The main analytic result of this study is that, in general circumstances which include the non-strong mixing example, the smoothers we propose are asymptotically normal. Some motivation is offered, and a simple illustrative example calculation concludes this investigation. The innovative elements of this work, mainly, consist of encompassing models with discrete noise, important in practical applications, and in dispensing with mixing assumptions. The ensuing mathematical difficulties are overcome by sharpening standard arguments.

##### MSC:
 62M10 Time series, auto-correlation, regression, etc. (statistics) 62G07 Density estimation 62J02 General nonlinear regression