Abstract
This article is concerned with inference about link function in generalized linear models. A parametric and yet robust likelihood approach is introduced to accomplish the intended goal. More specifically, it is demonstrated that one can convert normal and gamma likelihoods into robust likelihood functions for the link function. The asymptotic validity of the robust likelihood requires only the existence of the second moments of the underlying distributions. The application of this novel robust likelihood method is demonstrated on the Box–Cox transformation. Simulation studies and real data analysis are provided to demonstrate the efficacy of the new parametric robust procedures.
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References
Box GEP, Cox DR (1964) An analysis of transformations. J R Stat Soc Ser B 26: 211–252
Cheng KF, Wu JW (1994) Testing goodness of fit for a parametric family of link functions. J Am Stat Assoc 89: 657–664
Cox DR, Hinkley DV (1986) Theoretical statistics. Chapman and Hall, New York
Cox D, Koh E, Wahba G, Yandell BS (1988) Testing the (parametric) null model hypothesis in (semiparametric) partial and generalized spline models. Ann Stat 16: 113–119
Draper NR, Smith H (1998) Applied regression analysis, 3rd edn. John Wiley, New York
Fienberg SE, Gong GD (1984) Comment on “Graphical methods for assessing logistic regression models,” by JM Landwehr, D Pregibon and AC Shoemaker. J Am Stat Assoc 79: 72–77
Joglekar G, Schuenemeyer JH, LaRiccia V (1989) Lack of fit testing when replicates are not available. Am Stat 43: 135–143
Landwehr JM, Pregibon D, Shoemaker AC (1984) Graphical methods for assessing logistic regression models (with discussion). J Am Stat Assoc 79: 61–83
McCullagh P (1983) Quasi-likelihood functions. Ann Stat 11: 59–67
McCullagh P, Nelder JA (1989) Generalized linear models, 2nd edn. Chapman and Hall, New York
Pregibon D (1980) Goodness of link tests for generalized linear models. Appl Stat 29: 15–24
Royall RM (2000) On the probability of observing misleading statistical evidence (with discussion). J Am Stat Assoc 95: 760–767
Royall RM, Tsou TS (2003) Interpreting statistical evidence using imperfect models: robust adjusted likelihood functions. J R Stat Soc Ser B 65: 391–404
Su JQ, Wei LJ (1991) A lack of fit test for the mean function in a generalized linear model. J Am Stat Assoc 86: 420–426
Tsou TS (2005) Inferences of variance functions—a parametric robust way. J Appl Stat 32: 785–796
Tsou TS (2007) A simple and exploratory way to determine the mean-variance relationship in generalized linear models. Stat Med 26: 1623–1631
Tsou TS (2009) Performing legitimate parametric regression analysis without knowing the true underlying random mechanisms. Comm Stat Theor Methods 38: 1680–1689
Tsou TS, Shen CW (2008) Parametric robust inferences for correlated ordinal data. Stat Med 27: 3550–3562
White H (1982) Maximum likelihood estimation of misspecified models. Econometrica 82: 1–25
Wu JW, Lee WC (2001) The quasi-score statistic in quasi-likelihood model. Statistics 35: 523–535
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Tsou, TS. Likelihood inferences for the link function without knowing the true underlying distributions. Comput Stat 26, 507–519 (2011). https://doi.org/10.1007/s00180-010-0222-4
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DOI: https://doi.org/10.1007/s00180-010-0222-4