Adjusted Pearson residuals in exponential family nonlinear models
In this paper, we give matrix formulae of order O(n-1), where n is the sample size, for the first two moments of Pearson residuals in exponential family nonlinear regression models [G.M. Cordeiro and G.A. Paula, Improved likelihood ratio statistic for exponential family nonlinear models, Biometrika 76 (1989), pp. 93-100.]. The formulae are applicable to many regression models in common use and generalize the results by Cordeiro [G.M. Cordeiro, On Pearson's residuals in generalized linear models, Statist. Prob. Lett. 66 (2004), pp. 213-219.] and Cook and Tsai [R.D. Cook and C.L. Tsai, Residuals in nonlinear regression, Biometrika 72(1985), pp. 23-29.]. We suggest adjusted Pearson residuals for these models having, to this order, the expected value zero and variance one. We show that the adjusted Pearson residuals can be easily computed by weighted linear regressions. Some numerical results from simulations indicate that the adjusted Pearson residuals are better approximated by the standard normal distribution than the Pearson residuals.
pearson residual exponential family nonlinear model bias correction