### Abstract

The nonparametric component in a partially linear model is estimated by a linear combination of fixed-knot cubic B-splines with a second-order difference penalty on the adjacent B-spline coefficients. The resulting penalized least-squares estimator is used to construct two Wald-type spline-based test statistics for the null hypothesis of the linearity of the nonparametric function. When the number of knots is fixed, the first test statistic asymptotically has the distribution of a linear combination of independent chi-squared random variables, each with one degree of freedom, under the null hypothesis. The smoothing parameter is determined by specifying a value for the asymptotically expected value of the test statistic under the null hypothesis. When the number of knots is fixed and under the null hypothesis, the second test statistic asymptotically has a chi-squared distribution with K = q + 2 degrees of freedom, where q is the number of knots used for estimation. The power performances of the two proposed tests are investigated via simulation experiments, and the practicality of the proposed methodology is illustrated using a real-life data set.

Original language | English (US) |
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Pages (from-to) | 542-552 |

Number of pages | 11 |

Journal | Statistical Methodology |

Volume | 6 |

Issue number | 5 |

DOIs | |

State | Published - Sep 2009 |

### Fingerprint

### Keywords

- B-splines
- P-splines
- Penalized least-squares
- Wald-type spline-based test

### ASJC Scopus subject areas

- Statistics and Probability

### Cite this

**Using P-splines to test the linearity of partially linear models.** / Li, Chin-Shang.

Research output: Contribution to journal › Article

*Statistical Methodology*, vol. 6, no. 5, pp. 542-552. https://doi.org/10.1016/j.stamet.2009.06.001

}

TY - JOUR

T1 - Using P-splines to test the linearity of partially linear models

AU - Li, Chin-Shang

PY - 2009/9

Y1 - 2009/9

N2 - The nonparametric component in a partially linear model is estimated by a linear combination of fixed-knot cubic B-splines with a second-order difference penalty on the adjacent B-spline coefficients. The resulting penalized least-squares estimator is used to construct two Wald-type spline-based test statistics for the null hypothesis of the linearity of the nonparametric function. When the number of knots is fixed, the first test statistic asymptotically has the distribution of a linear combination of independent chi-squared random variables, each with one degree of freedom, under the null hypothesis. The smoothing parameter is determined by specifying a value for the asymptotically expected value of the test statistic under the null hypothesis. When the number of knots is fixed and under the null hypothesis, the second test statistic asymptotically has a chi-squared distribution with K = q + 2 degrees of freedom, where q is the number of knots used for estimation. The power performances of the two proposed tests are investigated via simulation experiments, and the practicality of the proposed methodology is illustrated using a real-life data set.

AB - The nonparametric component in a partially linear model is estimated by a linear combination of fixed-knot cubic B-splines with a second-order difference penalty on the adjacent B-spline coefficients. The resulting penalized least-squares estimator is used to construct two Wald-type spline-based test statistics for the null hypothesis of the linearity of the nonparametric function. When the number of knots is fixed, the first test statistic asymptotically has the distribution of a linear combination of independent chi-squared random variables, each with one degree of freedom, under the null hypothesis. The smoothing parameter is determined by specifying a value for the asymptotically expected value of the test statistic under the null hypothesis. When the number of knots is fixed and under the null hypothesis, the second test statistic asymptotically has a chi-squared distribution with K = q + 2 degrees of freedom, where q is the number of knots used for estimation. The power performances of the two proposed tests are investigated via simulation experiments, and the practicality of the proposed methodology is illustrated using a real-life data set.

KW - B-splines

KW - P-splines

KW - Penalized least-squares

KW - Wald-type spline-based test

UR - http://www.scopus.com/inward/record.url?scp=69649093729&partnerID=8YFLogxK

UR - http://www.scopus.com/inward/citedby.url?scp=69649093729&partnerID=8YFLogxK

U2 - 10.1016/j.stamet.2009.06.001

DO - 10.1016/j.stamet.2009.06.001

M3 - Article

AN - SCOPUS:69649093729

VL - 6

SP - 542

EP - 552

JO - Statistical Methodology

JF - Statistical Methodology

SN - 1572-3127

IS - 5

ER -