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1999 | 26 | 2 | 121-131
Tytuł artykułu

Least-squares trigonometric regression estimation

Treść / Zawartość
Warianty tytułu
Języki publikacji
EN
Abstrakty
EN
The problem of nonparametric function fitting using the complete orthogonal system of trigonometric functions $e_k$, k=0,1,2,..., for the observation model $y_i = f(x_{in}) + η_i$, i=1,...,n, is considered, where $η_i$ are uncorrelated random variables with zero mean value and finite variance, and the observation points $x_{in} ∈ [0,2π]$, i=1,...,n, are equidistant. Conditions for convergence of the mean-square prediction error $(1/n)\sum_{i=1}^n E(f(x_{in})-\widehat f_{N(n)}(x_{in}))^2$, the integrated mean-square error $E ‖f-\widehat f_{N(n)}‖^2$ and the pointwise mean-square error $E(f(x)-\widehatf_{N(n)}(x))^2$ of the estimator $\widehat f_{N(n)}(x) = \sum_{k=0}^{N(n)} \widehat c_k e_k(x)$ for f ∈ C[0,2π] and $\widehat c_0,\widehat c_1,...,\widehat c_{N(n)}$ obtained by the least squares method are studied.
Rocznik
Tom
26
Numer
2
Strony
121-131
Opis fizyczny
Daty
wydano
1999
otrzymano
1997-10-30
poprawiono
1999-01-18
Twórcy
  • Department of Standards, Central Statistical Office, Al. Niepodległości 208, 00-925 Warszawa, Poland
Bibliografia
  • [1] B. Droge, On finite-sample properties of adaptive least-squares regression estimates, Statistics 24 (1993), 181-203.
  • [2] R. L. Eubank and P. Speckman, Convergence rates for trigonometric and polynomial-trigonometric regression estimators, Statist. Probab. Lett. 11 (1991), 119-124.
  • [3] T. Gasser, L. Sroka and C. Jennen-Steinmetz, Residual variance and residual pattern in nonlinear regression, Biometrika 73 (1986), 625-633.
  • [4] P. Hall, J. W. Kay and D. M. Titterington, Asymptotically optimal difference-based estimation of variance in nonparametric regression, ibid. 77 (1990), 521-528.
  • [5] P. Hall and P. Patil, On wavelet methods for estimating smooth functions, J. Bernoulli Soc. 1 (1995), 41-58.
  • [6] G. G. Lorentz, Approximation of Functions, Holt, Reinehart & Winston, New York, 1966.
  • [7] C. L. Mallows, Some comments on $C_p$, Technometrics 15 (1973), 661-675.
  • [8] E. Nadaraya, Limit distribution of the integrated squared error of trigonometric series regression estimator, Proc. Georgian Acad. Sci. Math. 1 (1993), 221-237.
  • [9] B. T. Polyak and A. B. Tsybakov, Asymptotic optimality of the $C_p$ criterion in projection type estimation of regression functions, Teor. Veroyatnost. Primenen. 35 (1990), 305-317 (in Russian).
  • [10] E. Rafajłowicz, Nonparametric least-squares estimation of a regression function, Statistics 19 (1988), 349-358.
  • [11] A. Zygmund, Trigonometrical Series, Dover, 1955.
Typ dokumentu
Bibliografia
Identyfikatory
Identyfikator YADDA
bwmeta1.element.bwnjournal-article-zmv26i2p121bwm
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