Computational Statistics and Data Analysis 55 (2011) 2856–2870 Contents lists available at ScienceDirect Computational Statistics and Data Analysis journal homepage: www.elsevier.com/locate/csda Full and conditional likelihood approaches for hazard change-point estimation with truncated and censored data Ülkü Gürler a , C. Deniz Yenigün b,∗ a Bilkent University, Department of Industrial Engineering, 06800 Ankara, Turkey b Bilkent University, Faculty of Business Administration, 06800 Ankara, Turkey article info Article history: Received 31 March 2010 Received in revised form 15 February 2011 Accepted 19 April 2011 Available online 27 April 2011 Keywords: Hazard function Change-point Conditional likelihood Left truncated right censored data abstract Hazard function plays an important role in reliability and survival analysis. In some real life applications, abrupt changes in the hazard function may be observed and it is of interest to detect the location and the size of the change. Hazard models with a change- point are considered when the observations are subject to random left truncation and right censoring. For a piecewise constant hazard function with a single change-point, two estimation methods based on the maximum likelihood ideas are considered. The first method assumes parametric families of distributions for the censoring and truncation variables, whereas the second one is based on conditional likelihood approaches. A simulation study is carried out to illustrate the performances of the proposed estimators. The results indicate that the fully parametric method performs better especially for estimating the size of the change, however the difference between the two methods vanish as the sample size increases. It is also observed that the full likelihood approach is not robust to model misspecification. © 2011 Elsevier B.V. All rights reserved. 1. Introduction Hazard function plays an important role in reliability and survival studies since it quantifies the instantaneous risk of failure of an item at a given time point. In the majority of the studies existing in the literature, either a smooth, continuous hazard function is assumed when the objective is the estimation of this function itself, or as in the Cox model of proportional hazards, the emphasis is more on the estimation of the effects of the covariates, rather than the hazard function itself. Estimation of the hazard function presents a more interesting and a challenging task when this function displays abrupt changes in time which may correspond to significant improvements in the health conditions of a patient due to a particular treatment, or an alarming deterioration in the physical conditions of an equipment due to fatigue. As discussed by Frobish and Ebrahimi (2009), patients may experience events according to a common hazard rate function and they may receive treatments. It is commonly observed that the treatment takes its full effect only after a time lag. The curing effect of a medication or a treatment may as well deteriorate or dampen steeply after a certain period of time. In such cases it is of interest to detect both the time when such a change occurs and the size of the change. One of the earliest works that consider changes in the hazard function is by Matthews and Farewell (1982) which studied a piecewise constant hazard model with a single change-point given by λ(t ) =  β 0 ≤ t <τ β + θ t ≥ τ, (1) where β and β + θ> 0. Here β represents the initial constant hazard rate, θ represents the size of the change in the hazard rate, and τ is the location of the change-point, all of which are unknown. Matthews and Farewell (1982) applied this ∗ Corresponding author. Tel.: +90 312 290 2701; fax: +90 312 266 4958. E-mail addresses: ulku@bilkent.edu.tr (Ü. Gürler), yenigun@bilkent.edu.tr (C. Deniz Yenigün). 0167-9473/$ – see front matter © 2011 Elsevier B.V. All rights reserved. doi:10.1016/j.csda.2011.04.014