A mixed-Weibull regression model for the analysis of automotive warranty data Laura Attardi a , Maurizio Guida b , Gianpaolo Pulcini c, * a Department of Aeronautical Engineering, University Federico II of Naples, Naples, Italy b Department of Information Engineering and Electrical Engineering, University of Salerno, Fisciano (SA), Italy c Department of Statistics and Reliability, Istituto Motori CNR, via Marconi 8, 80125 Naples, Italy Received 3 December 2003; accepted 12 May 2004 Abstract This paper presents a case study regarding the reliability analysis of some automotive components based on field failure warranty data. The components exhibit two different failure modes, namely early and wearout failures, and are mounted on different vehicles, which differ among themselves for car model and engine type, thus involving different operating conditions. Hence, the failure time of each component is a random variable with a bimodal pdf which also depends upon a vector of covariates that indexes the specific operating condition. Then, a mixed-Weibull distribution, where the pdf of each subpopulation (namely the ‘weak’ and ‘strong’ subpopulation) depends on the covariates through the scale parameter, is used to analyze the component lifetime. A Fortran algorithm for the maximum likelihood estimation of model parameters has been implemented and a stepwise procedure, in its backwards version, has been used to test the significance of covariates and to construct the regression model. The presence of a weak subpopulation has been verified and the fraction of weak units in the population has also been estimated. Finally, the adequacy of the proposed model to fit the observed data has been assessed. q 2004 Elsevier Ltd. All rights reserved. Keywords: Mixed-Weibull distribution; Regression model; Automotive data 1. Introduction In this paper a case study regarding automotive field failure warranty data is analyzed. The warranty period is that time and/or mileage during which the manufacturer will repair, with no charge or minimum charge to the customer, all failures which occur to the vehicle. Usually, all the repairs performed during the warranty period at authorized dealerships are recorded by manufacturers in ad hoc database. In fact, warranty data are a primary source of information on the behavior of the product in use and they can be useful for an early detection of unusual failure rates. However, in the analysis of warranty data, one is often faced by two problems. The first problem is that many electronic and electro- mechanical components exhibit more than one failure mode. For example, it is well known that some product populations contain a mixture of ‘weak’ units (i.e. units with substandard strengths due to material or manufacturing defects) and ‘strong’ units (i.e. units with strengths close to a designed value). Weak units usually have short lifetimes, since there is a high probability that they will encounter early in the use a stress greater than their strength. Instead, strong units will have much longer lifetimes since they will fail mainly due to some degradation mechanism (e.g. wear, fatigue, corrosion, etc.). In such a case two failure modes are present: namely early and wearout. Then, the lifetime pdf of these components exhibits a bimodal (or, more generally, a multimodal) shape. When the product population is a mixture of n independent subpopulations and each subpopulation has 0951-8320/$ - see front matter q 2004 Elsevier Ltd. All rights reserved. doi:10.1016/j.ress.2004.05.003 Reliability Engineering and System Safety 87 (2005) 265–273 www.elsevier.com/locate/ress * Corresponding author. Fax: C39-081-239-6097. E-mail address: g.pulcini@im.cnr.it (G. Pulcini).