برآورد فاصله‌يي شاخص‌هاي Cpm و Cpmk فرايندهاي خودبرگشت به‌كمك روش‌ بوت‌استرپ بلوكي حلقوي

Interval Estimation of Cpm and Cpmk in AR(1) Process Using Circular Block Bootstrap Method

شاخص‌هاي قابليت فرايند، به‌عنوان معيارهاي كمي عملكرد فرايند در راستاي نيل به بهبود كيفيت، در صنعت كاربرد گسترده‌يي دارند. استقلال مشاهدات از متداول‌ترين مفروضات اغلب شاخص‌هاي قابليت فرايند است اما در عمل، الگوهاي همبستگي ميان اطلاعات نمونه‌يي قابل كشف است كه منجر به نقض فرض استقلال مشاهدات مي‏شود. عدم تشخيص صحيح يك فرايند خودهمبسته ممكن است باعث اشتباه در تصميم‌گيري و به بار آمدن خسارت‌هاي كيفي شود. به‌دليل توزيع نامعلوم شاخص‌هاي مذكور در حضور خودهمبستگي ميان داده‌ها، در اين نوشتار الگوريتمي براي برآوردهاي فاصله‌يي Cpm و Cpmk در فرايندهاي AR(1) با بهره‌گيري از روش ناپارامتري بوت‌استرپ بلوكي حلقوي ارايه شده است. به‌علاوه ارزيابي اين راهكار و مقايسه‌ي آن با ساير روش‌هاي موجود در ادبيات، به‌كمك مطالعات شبيه‌سازي مختلف نشان مي‌دهد كه عملكرد الگوريتم پيشنهادي با فواصلي به روش بوت‌استرپ استاندارد، نسبت به فواصل صدكي، بهتر است. همچنين روش والگرن براي برآورد فاصله‌يي شاخص‌ Cpm و روش بوت‌استرپ بلوكي حلقوي براي شاخص Cpkm در حضور داده‏هاي خودهمبسته‌ي AR(1) برتري دارد. البته، در شرايط عدم اطلاع از سطح خودهمبستگي، همواره روش بوت‌استرپ بلوكي حلقوي برتري دارد.

Process capability indices (PCIs) as quantitative measures of process performance in satisfying the customersʹ expectations have been widely used in industry. as comparative ratios between voice of the customer and voice of the process. The independency of observations over time is one of the assumptions of the most PCIs developed. However, the development of sampling technology has led to increasing the number of sampling as well as decreasing the time interval between sampling. This leads to occurrence of autocorrelation between successive observations and violation of the independency between observations. There are different point estimators and confidence intervals for the process capability indices of autocorrelated data in the literature. In this paper for the first time, confidence intervals of Cpm and Cpmk are estimated using Circular Block Bootstrap resampling technique when the data are autocorrelated and modeled by an AR(1) process. Estimating the PCIs sampling distribution through simulation studies showed that increasing autocorrelation coefficients leads to decreasing in sampling standard deviation and bias in the PCIs estimators. However, larger sample sizes have resulted in more accurate estimations and lower bias, skewness and kurtosis of estimators. In addition, performance of the proposed interval estimators are compared through numerical examples. Simulations results of two different methods of confidence interval indicate that 95% standard bootstrap (SB) method often outperforms the Biased-Corrected and accelerated (BCa) Percentile Bootstrap method without considering the magnitude of autocorrelation coefficient. Moreover, the effects of various autocorrelation coefficients, sample sizes, mean values and standard deviations on the proposed confidence intervals have been appraised. Finally, by comparing the results of the proposed method and existing methods in the literature, some advantages of the bootstrap have been mentioned. Generally, considerable volume of available data is brought about average coverage percentage (ACP) close to a nominal confidence level and lower average interval length (AIL) with more accuracy. In addition, the proposed methods have often lower AILs, and give acceptable ACPs particularly in weak autocorrelation coefficients.