The Design of Generalized Likelihood Ratio Control Chart for Monitoring the von Mises Distributed Data

Document Type : Research Paper

Authors

1 Department of Statistics, Faculty of Mathematical Sciences, Ferdowsi University of Mashhad, Mashhad, Iran

2 Khorasan Razavi Agricultural and Natural research and Education center, Mashhad, Iran

Abstract

‎In this paper‎, ‎a new generalized likelihood ratio (GLR) control chart based on sequentially probability ratio test (SPRT) is introduced to monitor the directional mean of von Mises distribution‎. ‎Different window size of past samples are utilized to construct the GLR chart statistic‎, ‎and the performance of this chart in detecting a wide range of parameter shift is evaluated‎. ‎A simulation study is carried out to investigate the performance of the proposed control chart in comparison with cumulative sum (CUSUM) control chart‎. ‎To guide practitioners‎, ‎a real example is provided‎.

Keywords


[1] Abbasi Ganji Z, Sadeghpour Gildeh B (2022) A likelihood control chart for monitoring bivariate lifetime processes. Journal of Mahani Mathematical Research center, 11 (2):97-118.
[2] Apley DW, & Shi J (1999) The GLRT for statistical process control of auto correlated processes. IIE Transactions, 31 (12): 1123-1134.
[3] Best DJ, and Fisher NI (1981) The BIAS of the maximum likelihood estimators of the von Mises-Fisher concentration parameters: the BIAS of the maximum likelihood estimators. Communications in Statistics-Simulation and Computation, 10 (5): 493-502.
[4] Capizzi G (2001) Design of change detection algorithms based on the generalized likelihood ratio test. Environ metrics: The ocial journal of the International Environ metrics Society, 12 (8): 749-756.
[5] Fisher NI (1995) Statistical analysis of circular data. Cambridge University Press.
6] Gadsden RJ, and Kanji GK (1981) Sequential analysis for angular data. The Statistician,
[7] Gombay E (2000) Sequential change-point detection with likelihood ratios. Statistics & probability letters, 49 (2): 195-204.
[8] Han D, Tsung F, Hu X, and Wang, K (2007) CUSUM and EWMA multi-charts for detecting a range of mean shifts. Statistica Sinica, 17 (3): 1139-1164.
[9] Hawkins DM, and Lombard F (2017) Cusum control for data following the von Mises distribution.Journal of Applied Statistics, 44 (8): 1319-1332.
[10] Hawkins DM, and Zamba KD (2005) Statistical process control for shifts in mean or variance using a change point formulation. Technimetrics, 47 (2): 164-173.
[11] KazemiNia A, Sadeghpour Gildeh B, and Abbasi Ganji Z (2018) The design of geometric generalized likelihood ratio control chart. Quality and Reliability Engineering International, 34 (5): 953-965.
[12] Laha AK, Gupta A (2011) Statistical Quality Control of Directional Data 2nd IIMA International Conference on Advanced Data Analysis, Business Analytics and Intelligence,Ahmedabad.
[13] Lai TL (1995) Sequential change point detection in quality control and dynamical systems. Journal of the Royal Statistical Society Series B (Methodological), 57 (4): 613-658.
[14] Lee J, Peng Y, Wang N, and Reynolds MR (2017) A GLR control chart for monitoring a multinomial process. Quality and Reliability Engineering, 33 (8): 1773-1782.
[15] Lombard F (1986) The change{point problem for angular data: a nonparametric approach. Technometrics, 28 (4): 391-397.
[16] Mardia KV (1975) Statistics of directional data. Journal of the Royal Statistical Society. Series B (Methodological), 37 (3): 349-393.
[17] Mardia KV, Jupp PE (2009) Directional statistics. John Wiley & Sons, vol. 494.
[18] Reynolds Jr. MR, and Lou J (2010) An evaluation of a GLR control chart for monitoring the process mean. Journal of quality technology, 42 (3): 287-310.
[19] Peker K O, Bacanli S (2007) a sequential test for the mean direction applied to circular data and an application. Eskisehir Osmangazi Universitesi Muh.Mim.Fak.Dergisi, C.XX,S.2.
[20] Sengupta A, and Laha AK (2008) A Bayesian analysis of the change-point problem for directional data. Journal of Applied Statistics, 35 (6): 693-700.
[21] Siegmund D, and Venkatraman ES (1995) Using the generalized likelihood ratio statistic for sequential detection of a change-point. The Annals of Statistics, 23 (1): 255-271.
[22] Willsky A, Jones H (1976) A generalized likelihood ratio approach to the detection and estimation of jumps in linear systems. IEEE Transactions on Automatic control, 21 (1):108-112.