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A lack-of-fit test for generalized linear models via single-index techniques

Author

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  • Chin-Shang Li

    (University of California)

  • Minggen Lu

    (University of Nevada)

Abstract

A generalized partially linear single-index model (GPLSIM) is proposed in which the unknown smooth function of single index is approximated by a spline function that can be expressed as a linear combination of B-spline basis functions. The regression coefficients and the unknown smooth function are estimated simultaneously via a modified Fisher-scoring method. It can be shown that the estimators of regression parameters are asymptotically normally distributed. The asymptotic covariance matrix of the estimators can be estimated directly and consistently by using the least-squares method. As an application, the proposed GPLSIM can be employed to assess the lack of fit of a postulated generalized linear model (GLM) based on the comparison of the goodness of fit of the GPLSIM and postulated GLM to construct a likelihood ratio test. An extensive simulation study is conducted to examine the finite-sample performance of the likelihood ratio test. The practicality of the proposed methodology is illustrated with a real-life data set from a study of nesting horseshoe crabs.

Suggested Citation

  • Chin-Shang Li & Minggen Lu, 2018. "A lack-of-fit test for generalized linear models via single-index techniques," Computational Statistics, Springer, vol. 33(2), pages 731-756, June.
    • Handle: RePEc:spr:compst:v:33:y:2018:i:2:d:10.1007_s00180-018-0802-2
    DOI: 10.1007/s00180-018-0802-2
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    1. Yu Y. & Ruppert D., 2002. "Penalized Spline Estimation for Partially Linear Single-Index Models," Journal of the American Statistical Association, American Statistical Association, vol. 97, pages 1042-1054, December.
    2. Lu, Minggen & Zhang, Ying & Huang, Jian, 2009. "Semiparametric Estimation Methods for Panel Count Data Using Monotone B-Splines," Journal of the American Statistical Association, American Statistical Association, vol. 104(487), pages 1060-1070.
    3. Minggen Lu & Ying Zhang & Jian Huang, 2007. "Estimation of the mean function with panel count data using monotone polynomial splines," Biometrika, Biometrika Trust, vol. 94(3), pages 705-718.
    4. Stoker, Thomas M, 1986. "Consistent Estimation of Scaled Coefficients," Econometrica, Econometric Society, vol. 54(6), pages 1461-1481, November.
    5. Anne B. Koehler & Emily S. Murphree, 1988. "A Comparison of the Akaike and Schwarz Criteria for Selecting Model Order," Journal of the Royal Statistical Society Series C, Royal Statistical Society, vol. 37(2), pages 187-195, June.
    6. Powell, James L & Stock, James H & Stoker, Thomas M, 1989. "Semiparametric Estimation of Index Coefficients," Econometrica, Econometric Society, vol. 57(6), pages 1403-1430, November.
    7. Yingcun Xia, 2009. "Model checking in regression via dimension reduction," Biometrika, Biometrika Trust, vol. 96(1), pages 133-148.
    8. Delecroix, Michel & Härdle, Wolfgang & Hristache, Marian, 2003. "Efficient estimation in conditional single-index regression," Journal of Multivariate Analysis, Elsevier, vol. 86(2), pages 213-226, August.
    9. Newey, Whitney K & Stoker, Thomas M, 1993. "Efficiency of Weighted Average Derivative Estimators and Index Models," Econometrica, Econometric Society, vol. 61(5), pages 1199-1223, September.
    10. Jianhua Z. Huang & Linxu Liu, 2006. "Polynomial Spline Estimation and Inference of Proportional Hazards Regression Models with Flexible Relative Risk Form," Biometrics, The International Biometric Society, vol. 62(3), pages 793-802, September.
    11. Sun, Jie & Kopciuk, Karen A. & Lu, Xuewen, 2008. "Polynomial spline estimation of partially linear single-index proportional hazards regression models," Computational Statistics & Data Analysis, Elsevier, vol. 53(1), pages 176-188, September.
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    Cited by:

    1. Mustapha Rachdi & Mohamed Alahiane & Idir Ouassou & Abdelaziz Alahiane & Lahoucine Hobbad, 2024. "Cross-Validated Functional Generalized Partially Linear Single-Functional Index Model," Mathematics, MDPI, vol. 12(17), pages 1-22, August.
    2. Mohamed Alahiane & Idir Ouassou & Mustapha Rachdi & Philippe Vieu, 2022. "High-Dimensional Statistics: Non-Parametric Generalized Functional Partially Linear Single-Index Model," Mathematics, MDPI, vol. 10(15), pages 1-21, July.
    3. Mohamed Alahiane & Idir Ouassou & Mustapha Rachdi & Philippe Vieu, 2021. "Partially Linear Generalized Single Index Models for Functional Data (PLGSIMF)," Stats, MDPI, vol. 4(4), pages 1-21, September.

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