IJE TRANSACTIONS A: Basics Vol. 28, No. 7, (July 2015) 1059-1067 Please cite this article as: F. Vakilian, A. Amiri, F. Sogandi, Isotonic Change Point Estimation in the AR(1) Autocorrelated Simple Linear Profiles, International Journal of Engineering (IJE), TRANSACTIONS A: Basics Vol. 28, No. 7, (July 2015) 1059-1067 International Journal of Engineering Journal Homepage: www.ije.ir Isotonic Change Point Estimation in the AR(1) Autocorrelated Simple Linear Profiles F. Vakilian, A. Amiri*, F. Sogandi Industrial Engineering Department, Shahed University, Tehran, Iran PAPER INFO Paper history: Received 12 April 2015 Received in revised form 02 June 2015 Accepted 11 June 2015 Keywords: Simple Linear Profile Isotonic Shift Change Point Estimation AR(1) Models Phase II Hotelling T 2 Control Chart ABSTRACT Sometimes the relationship between dependent and explanatory variable(s) known as profile is monitored. Simple linear profiles among the other types of profiles have been more considered due to their applications especially in calibration. There are some studies on the monitoring them when the observations within each profile are autocorrelated. On the other hand, estimating the change point leads to meet great saving time and costs. Hence, in this paper, a maximum likelihood estimator is derived for simple linear profiles with first order autoregressive autocorrelation structure within each profile to estimate isotonic change point. The performance of the proposed estimator is appraised and compared to estimators that derived under step change and drift and a confidence set estimator presented. The results demonstrate that the proposed estimator has better performance in small and medium shifts whereas the performance of their corresponding estimators becomes better than the proposed estimator in large shifts. It is worth mentioning that knowing type of the change is not important in the proposed estimator and its only assumption is belonging of the change type to a family of isotonic shifts. Finally, the performance of the estimator is illustrated through a real case. doi: 10.5829/idosi.ije.2015.28.07a.12 1. INTRODUCTION 1 In some statistical process control applications, the relationship between dependent and explanatory variable (s) is monitored instead of monitoring a univariate or multivariate quality characteristics. This relationship which can be linear or nonlinear is known as a profile. According to the relationship, there are various types of profiles including simple linear profile, multiple linear profile, polynomial profile, nonlinear profile, waveform profile, spline profile and profiles based on generalized linear models. Profiles have different applications in manufacturing and service. A number of researchers such as Kang and Albin [1] have discussed practical applications of profiles. In the recent years, monitoring profiles especially simple linear profiles due to their applications especially in calibration has been considered by many researchers. Studies about monitoring profiles are done in two phases. Many researchers such as Mahmoud et al. [2] have studied Phase I monitoring of simple linear 1 *Corresponding Author’s Email: amiri@shahed.ac.ir (A. Amiri) profiles. Also, there are many works on Phase II monitoring in which the parameters are assumed to be known. Researchers such as Gupta et al. [3] have Studied Phase II monitoring of simple linear profiles. Keramatpour et al. [4] proposed a remedial measure to remove the effect of autocorrelation in monitoring of autocorrelated polynomial profiles. Many studies have been done by researchers on monitoring simple linear profiles when the sampling time between observations collapses and as a result the observations are autocorrelated. Recently, Kamranrad and Amiri [5] proposed a robust holt-winter based control chart in Phase II monitoring of a simple linear profile under within profile autocorrelation and the presence of outliers. On the other hand, usually when the control chart declares warning about the out-of-control status, it is different with the real-time of the process change. Real- time of change in process is known as change point. A process may be in the out-of-control state due to different change types including single-step change, drift change, isotonic change, multiple-step changes, and sporadic changes. Estimating the change point leads to meet great saving on time and costs. Hence, many