The signal extraction approach to nonlinear regression and spline smoothing.

*(English)*Zbl 0536.62071This paper is concerned with an approach of G. Wahba [J. R. Stat. Soc., Ser. B 40, 364-372 (1978; Zbl 0407.62048)] to non-parametric regression, where data values \((y_ i,x_ i)\), \(i=1,...,n\), are observed and should be described by a regression
\[
y_ i=f(x_ i)+e_ i,\quad where\quad f(x)=\sum^{m-1}_{k=0}\alpha_ k\frac{(x-a)^ k}{k!}+\lambda^{1/2}\sigma \int^{x}_{a}\frac{(x-h)^{m-1}}{(m- 1)!}dW(h),
\]
W(h) a Wiener process with unit dispersion parameter. Problems of estimating the parameters \(\lambda\), \(\sigma^ 2\) and \(\alpha_ 0,...,\alpha_{m-1}\), and of constructing an estimate for f(x) together with confidence bounds, are considered. An exact maximum likelihood estimator for the parameters which can be computed in Q(n) arithmetic operations is given.

After some remarks on smoothing polynomials and stochastic processes, Markov structures and the state space formulation is demonstrated. The general state space form can be represented in the matrix form \(y=X\alpha +v\), which is treated as GLS estimation problem in order to obtain ML- estimators of \(\lambda\) and \(\sigma^ 2\). ML-estimators of all parameters can be obtained by repeated applications of GLS at alternative values of \(\lambda\). The special structure of the model admits an efficient computation of the ML-estimator using the Kalman filter approach. Because of the recursive structure of the Kalman filter, the estimates of the state vectors are based on the data only.

Some limits on the overall check of model adequacy and for the polynomial model are given, and the estimation of the regression function \(f(x)=x^ t\alpha +Z^{(m)}(x)\) and its standard error at the only value of x is discussed. The estimated signal \(\hat Z^{(m)}(x)\) is a smoothing polynomial spline. It can be shown that \(var(Z^{(m)}(x)-\hat Z^{(m)}(x))\) is a continuous piecewise polynomial of degree 4m-2 with knots \(x_ 1,...,x_ n\), and with the first 2m-2 derivatives continuous over the knots. This variance is used to construct confidence bounds for the estimate of f(x).

Finally, after some remarks on the relationship between alternative recursive algorithms for polynomial splines, an example for the method presented is given.

After some remarks on smoothing polynomials and stochastic processes, Markov structures and the state space formulation is demonstrated. The general state space form can be represented in the matrix form \(y=X\alpha +v\), which is treated as GLS estimation problem in order to obtain ML- estimators of \(\lambda\) and \(\sigma^ 2\). ML-estimators of all parameters can be obtained by repeated applications of GLS at alternative values of \(\lambda\). The special structure of the model admits an efficient computation of the ML-estimator using the Kalman filter approach. Because of the recursive structure of the Kalman filter, the estimates of the state vectors are based on the data only.

Some limits on the overall check of model adequacy and for the polynomial model are given, and the estimation of the regression function \(f(x)=x^ t\alpha +Z^{(m)}(x)\) and its standard error at the only value of x is discussed. The estimated signal \(\hat Z^{(m)}(x)\) is a smoothing polynomial spline. It can be shown that \(var(Z^{(m)}(x)-\hat Z^{(m)}(x))\) is a continuous piecewise polynomial of degree 4m-2 with knots \(x_ 1,...,x_ n\), and with the first 2m-2 derivatives continuous over the knots. This variance is used to construct confidence bounds for the estimate of f(x).

Finally, after some remarks on the relationship between alternative recursive algorithms for polynomial splines, an example for the method presented is given.

Reviewer: R.Fahrion

##### MSC:

62M10 | Time series, auto-correlation, regression, etc. in statistics (GARCH) |

41A15 | Spline approximation |

62G99 | Nonparametric inference |

62M20 | Inference from stochastic processes and prediction |