Statistics and Probability Letters 78 (2008) 3269–3273 Contents lists available at ScienceDirect Statistics and Probability Letters journal homepage: www.elsevier.com/locate/stapro Residuals and their statistical properties in symmetrical nonlinear models Francisco José A. Cysneiros ∗ , Luis Hernando Vanegas Departamento de Estatística, CCEN-UFPE-Cidade Universitária-Recife, PE-Brazil 50740-540, Brazil article info Article history: Received 11 July 2005 Received in revised form 24 April 2008 Accepted 2 June 2008 Available online 17 June 2008 abstract In this work we present theoretical details of a general residual for symmetric nonlinear regression models. This class of models includes all symmetric continuous distributions such as normal, Student-t , Pearson VII, power exponential and logistic. Such regression models are used for the analysis of data sets containing influential or outlying observations, that can significantly influence inferential conclusions. On the basis of expansions of Cox and Snell [Cox, D.R., Snell, E.J., 1968. A general definition of residuals. Journal of the Royal Statistical Society B 30, 248–275], we calculate first and second moments of a general definition of a residual for symmetrical nonlinear regression models. Also, the statistical properties of some proposed residuals are studied using Monte Carlo simulations for the Michaelis–Menten model, frequently used in chemical and biological experiments. © 2008 Elsevier B.V. All rights reserved. 1. Introduction Statistical modelling is one of the tools most used by researchers in several areas of the knowledge whose main interest is in answering research questions through the statistical analysis of a data set. Classical analysis based on models with normal errors is one of the most popular methods used when the response is continuous, because it is easily applicable and has a great number of theories developed for it. However, it is very well known that modelling under the assumption of normally distributed errors can be highly influenced by extreme observations (see, e.g., Cook and Weisberg (1982)). As an alternative to the classical analysis for data sets with extreme observations we have models with error distributions belonging to the class of symmetrical distributions (see Fang et al. (1990)). In this article we present theoretical details of a general residual for the symmetrical nonlinear regression models where the standardized residual proposed by Galea et al. (2005) is a particular case. Also, we derived the deviance and quantal residual and, following the methodology proposed by Cox and Snell (1968), we obtained approximate expressions for the expectation and the variance of these residuals. The work is organized as follows. In Section 2 we present the symmetrical nonlinear regression model. In Section 3 we develop the first and second moments for a general residual and the standardization of the residual is presented. In Section 4 we present the results of a simulation experiment using the Michaelis–Menten model, in which statistical properties of the residuals are studied empirically. In the last section, we present some conclusions. 2. Symmetrical nonlinear regression model Let random variables Y 1 ,..., Y n be independent and let each Y i have a density of the symmetrical class of distributions with location parameter µ i ∈ R and of scale φ> 0. The density function of Y i , denoted by Y i ∼ S (µ i ,φ), is defined as ∗ Corresponding author. E-mail addresses: cysneiros@de.ufpe.br (F.J.A. Cysneiros), hvanegasp@gmail.com (L.H. Vanegas). 0167-7152/$ – see front matter © 2008 Elsevier B.V. All rights reserved. doi:10.1016/j.spl.2008.06.011