### Abstract

This article proposes some simplifications of the residual variance estimator of Gasser, Sroka, and jennen-Steinmetz (GSJ, 1986) which is often used in conjunction with nonparametric regression. The GSJ estimator is a quadratic form of the data, which depends on the relative spacings of the design points. When the errors are independent, identically distributed Gaussian variables, and the true regression curve is flat, theestimate is distributed as a weighted sum of x^{2} variables. By matching the first two moments, the distribution can be approximated by a x^{2}^{w}ith degrees of freedom determined by the coefficients of the quadratic for. Computation ofthe estimated degrees of freedom requires computing the trace of the square of an n x n matrix, where n is the number of design points. In this article, (n-2)/3 is shown to be a conservative estimate of the approximate degrees of freedom, and (n-2)/2 is shown to be conservative for many designs. In addition, a simplified version of the estimator is shown to be asymptotically equivalent, under many conditions.

Original language | English (US) |
---|---|

Pages (from-to) | 1045-1051 |

Number of pages | 7 |

Journal | Communications in Statistics - Theory and Methods |

Volume | 22 |

Issue number | 4 |

DOIs | |

State | Published - Jan 1 1993 |

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### All Science Journal Classification (ASJC) codes

- Statistics and Probability

### Cite this

*Communications in Statistics - Theory and Methods*,

*22*(4), 1045-1051. https://doi.org/10.1080/03610928308831072

}

*Communications in Statistics - Theory and Methods*, vol. 22, no. 4, pp. 1045-1051. https://doi.org/10.1080/03610928308831072

**Some remarks about the gasser-sroka-jennen-steinnietz variance estimator.** / Altman, Naomi S.; Paulson, C. P.

Research output: Contribution to journal › Article

TY - JOUR

T1 - Some remarks about the gasser-sroka-jennen-steinnietz variance estimator

AU - Altman, Naomi S.

AU - Paulson, C. P.

PY - 1993/1/1

Y1 - 1993/1/1

N2 - This article proposes some simplifications of the residual variance estimator of Gasser, Sroka, and jennen-Steinmetz (GSJ, 1986) which is often used in conjunction with nonparametric regression. The GSJ estimator is a quadratic form of the data, which depends on the relative spacings of the design points. When the errors are independent, identically distributed Gaussian variables, and the true regression curve is flat, theestimate is distributed as a weighted sum of x2 variables. By matching the first two moments, the distribution can be approximated by a x2with degrees of freedom determined by the coefficients of the quadratic for. Computation ofthe estimated degrees of freedom requires computing the trace of the square of an n x n matrix, where n is the number of design points. In this article, (n-2)/3 is shown to be a conservative estimate of the approximate degrees of freedom, and (n-2)/2 is shown to be conservative for many designs. In addition, a simplified version of the estimator is shown to be asymptotically equivalent, under many conditions.

AB - This article proposes some simplifications of the residual variance estimator of Gasser, Sroka, and jennen-Steinmetz (GSJ, 1986) which is often used in conjunction with nonparametric regression. The GSJ estimator is a quadratic form of the data, which depends on the relative spacings of the design points. When the errors are independent, identically distributed Gaussian variables, and the true regression curve is flat, theestimate is distributed as a weighted sum of x2 variables. By matching the first two moments, the distribution can be approximated by a x2with degrees of freedom determined by the coefficients of the quadratic for. Computation ofthe estimated degrees of freedom requires computing the trace of the square of an n x n matrix, where n is the number of design points. In this article, (n-2)/3 is shown to be a conservative estimate of the approximate degrees of freedom, and (n-2)/2 is shown to be conservative for many designs. In addition, a simplified version of the estimator is shown to be asymptotically equivalent, under many conditions.

UR - http://www.scopus.com/inward/record.url?scp=84947611135&partnerID=8YFLogxK

UR - http://www.scopus.com/inward/citedby.url?scp=84947611135&partnerID=8YFLogxK

U2 - 10.1080/03610928308831072

DO - 10.1080/03610928308831072

M3 - Article

AN - SCOPUS:84947611135

VL - 22

SP - 1045

EP - 1051

JO - Communications in Statistics - Theory and Methods

JF - Communications in Statistics - Theory and Methods

SN - 0361-0926

IS - 4

ER -