Abstract




 
   

IJE TRANSACTIONS B: Applications Vol. 28, No. 2 (February 2015) 224-233   

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  ESTIMATING THE TIME OF A STEP CHANGE IN GAMMA REGRESSION PROFILES USING MLE APPROACH
 
F. Sogandi and A. Amiri
 
( Received: May 04, 2014 – Accepted: August 14, 2014 )
 
 

Abstract    Sometimes the quality of a process or product is described by a functional relationship between a response variable and one or more explanatory variables referred to as profile. In most researches in this area the response variable is assumed to be normally distributed; however, occasionally in certain applications, the normality assumption is violated. In these cases the Generalized Linear Models (GLM) such as Gamma regression models are used to characterize the profile. Also, in statistical process control finding the real time of change in process, called as change point, is necessary because it leads to saving time and cost in finding assignable cause(s). Therefore, in this paper we consider Gamma regression profile and use maximum likelihood to estimate the real time of a step change in Phase II. We evaluate accuracy and precision of the proposed change point estimator by simulation. The results show the proposed change point estimator is effective in estimating the real time of step shifts in the process parameters of Gamma regression profiles. Also, a confidence set for the process change point based on the logarithm of the likelihood function is presented. Finally, the performance of the estimator is illustrated through a numerical example.

 

Keywords    Gamma Regression Profile, Change Point Estimation, Maximum Likelihood Estimator (MLE), Statistical Process Control (SPC), Phase II.

 

چکیده    گاهی اوقات کیفیت یک محصول یا فرآیند بوسیله یک رابطه تابعی بین یک متغیر پاسخ و یک یا چند متغیر مستقل تحت عنوان پروفایل توصیف می شود. در اکثر تحقیقات در این حوزه توزیع متغیر پاسخ نرمال فرض می شود، در حالیکه گاهی در کاربردهای خاصی فرض نرمال بودن نقض می شود. در این موارد از مدل های خطی تعمیم یافته مثل مدل رگرسیونی گاما برای توصیف پروفایل استفاده می شود. همچنین در کنترل فرآیند آماری پیدا کردن زمان واقعی تغییردر فرآیند که نقطه تغییر نام دارد ضروری است زیرا منجر به صرفه جویی زمان وهزینه در شناسایی دلایل خاص می شود. بنابراین در این مقاله پروفایل رگرسیونی گاما را در نظر گرفته می شود و از برآوردکننده حداکثر درست نمایی برای تخمین زمان واقعی یک تغییرپله ای در پارامترهای گاما در فاز 2 استفاده می شود. به علاوه دقت وصحت برآورد کننده نقطه تغییر ارائه شده بوسیله شبیه سازی مونت کارلو ارزیابی می شود. نتایج شبیه سازی نشان می دهد که برآورد کننده نقطه تغییر ارائه شده در تخمین نقطه واقعی تغییرات پله ای در پارامترهای پروفایل رگرسیونی از عملکرد مناسبی برخوردار است. همچنین يك مجموعه اطمینان برای نقطه تغییر فرآیند براساس لگاريتم تابع درست نمایی ارائه شده است. نهایتا عملکرد برآورد کننده با یک مثال عددی نشان داده شده است.

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