10.14489/hb.2026.01.pp.063-088

2026, 01 январь (January)

DOI: 10.14489/hb.2026.01.pp.063-088

Борбаць Н. М., Школина Т. В.
ТАБЛИЦЫ СТАНДАРТИЗИРОВАННЫХ КОНТРОЛЬНЫХ ГРАНИЦ КАРТЫ СРЕДНИХ ДЛЯ РАСПРЕДЕЛЕНИЯ ВЕЙБУЛЛА
(с. 63-88)

Аннотация. Важной практической задачей контроля качества является построение и ведение контрольных карт для мониторинга среднего значения контролируемой характеристики. В контрольных картах Шухарта, используемых в подобных ситуациях, границы определяются в предположении нормального распределения характеристики, однако на практике может быть известно, что контролируемая характеристика описывается распределением Вейбулла. В этом случае необходимо иметь справочные таблицы, позволяющие определить контрольные границы карты для наиболее часто используемых значений вероятности ложной тревоги. В данной работе приводятся подобные таблицы, полученные на основе разложения Корниша–Фишера и статистических испытаний методом Монте-Карло. Приведенные таблицы могут использоваться инженерами-практиками, занимающимися вопросами статистического управления процессами, для нахождения границ контрольных карт в ситуациях, когда исследуемая характеристика качества описывается распределением Вейбулла.

Ключевые слова: распределение Вейбулла; контрольная карта; метод Монте-Карло.

Borbats N. M., Shkolina T. V.
TABLES OF STANDARDIZED CONTROL LIMITS OF CHARTS OF AVERAGES FOR WEIBULL DISTRIBUTION
(pp. 63-88)

Abstract. An important practical task of quality control is the construction and maintenance of control charts for monitoring the average value of a controlled characteristic. Schewhart control charts used in such situations, determine the limits under the assumption of a normal distribution of the characteristic, but in practice it may be known that the controlled characteristic is described by Weibull distribution. In this case, it is necessary to have reference tables that allow you to determine the control limits of the chart for the most commonly used false alarm probability values. This article presents similar tables obtained on the basis of Cornish–Fisher expansion and statistical tests by the Monte Carlo method.

Keywords: Weibull distribution; Control chart; Monte Carlo method.

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Н. М. Борбаць, Т. В. Школина (Брянский государственный технический университет, Брянск, Россия) E-mail: Данный адрес e-mail защищен от спам-ботов, Вам необходимо включить Javascript для его просмотра.

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N. M. Borbats, T. V. Shkolina (Bryansk State Technical University, Bryansk, Russia) E-mail: Данный адрес e-mail защищен от спам-ботов, Вам необходимо включить Javascript для его просмотра.

+ - Библиографический список (References) Click to collapse

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1. Barlow R.E., Proschan F. Mathematical theory of reliability. Philadelphia: Society for Industrial and Applied Mathematic, 1987. 277 p.
2. Birolini A. Reliability Engineering: Theory and Practice. 8th ed, Berlin: Springer-Verlag, 2017. 651 p.
3. Vonta I., Ram M. Reliability Engineering: Theory and Applications. CRC Press: New York, 2019. 241 p.
4. Ali S., Lee S.-M., Jang C.-M. Statistical analysis of wind characteristics using Weibull and Rayleigh distributions in Deokjeok-do Island – Incheon, South Korea // Renewable Energy. 2018. Vol. 123, pp. 652 – 663, https://doi.org/ 10.1016/j.renene.2018.02.087.
5. Aslam M. Testing average wind speed using sampling plan for Weibull distribution under indeterminacy // Scientific Reports. 2021, No. 11, URL: https://www.nature.com/articles/s41598-021-87136-8 (дата обращения: 08.10.2025).
6. Bayazit M., Önöz B. Sampling variances of regional flood quantiles affected by intersite correlation // Journal of Hydrology. 2004. Vol. 291, No. 1 – 2, pp. 42 – 51, https://doi.org/10.1016/j.jhydrol.2003.12.009.
7. Koehler K.J., McGovern P.G. An application of the LFP survival model to smoking cessation data // Statistics in Medicine. 1990. Vol. 9, pp. 409 – 421, https://doi.org/10.1002/sim.4780090412.
8. Johnson N.L., Kotz S., Balakrishnan N. Continuous Univariate Distributions. 2th ed, New York: Wiley, 1995. 769 p.
9. Nelson P.R. Control charts for Weibull processes with standards given // IEEE Transactions on Reliability. 1979. Vol. R-28, No. 4, pp. 283 – 288, https://doi.org/10.1109/TR.1979.5220605.
10. Ramalhoto M.F., Morais M. Shewhart control charts for the scale parameter of a Weibull control variable with fixed and variable sampling intervals // Journal of Applied Statistics. 1999. Vol. 26, No. 1, pp. 129 – 160, https://doi.org/10.1080/02664769922700.
11. Pascual F. EWMA charts for the Weibull shape parameter // Journal of Quality Technology. 2010 Vol. 42, No. 4, pp. 400 – 416, https://doi.org/10.1080/ 00224065.2010.11917836.
12. Pascual F., Zhang H. Monitoring the Weibull shape parameter by control charts for the sample range // Quality and Reliability Engineering International. 2011. Vol. 27, No. 1, pp. 15 – 25, https://doi.org/10.1002/qre.1099.
13. Faraz A., Saniga E.M., Heuchenne C. Shewhart control charts for monitoring reliability with Weibull lifetimes // Quality and Reliability Engineering International. 2015. Vol. 31, No. 8, pp. 1565 – 1574, https://doi.org/10.1002/qre.1692.
14. Fernandes F.H., Lee Ho L., Bourguignon M. About Shewhart control charts to monitor the Weibull mean // Quality and Reliability Engineering. 2019. Vol. 35, No. 7, pp. 2343 – 2357, https://doi.org/10.1002/qre.2515.
15. Lee Ho L., Fernandes F.H., Quinino R.C. and Bourguignon M. Improving Shewhart control chart performance for monitoring the Weibull mean // Quality and Reliability Engineering. 2020. Vol. 37, No. 3, pp. 984 – 996, https://doi.org/10.1002/ qre.2777.
16. Montgomery D.C. Introduction to Statistical Quality Control. 7th ed, New York: John Wiley & Sons, 2013. 774 p.
17. Wheeler D.J., Chambers D.S. Understanding Statistical Process Control. Knoxville: SPC Press, 1992
18. Mittag H.-J. and Rinne H. () Statistical Methods of Quality Assurance, London: Chapman & Hall, 1993. 339 p.
19. Harter H.L. and Dubey S.D. Theory and tables for tests of hypotheses concerning the mean and the variance of a Weibull population // Aerospace Research Laboratories ARL 67-0059, Wright-Patterson Air Force Base, OH, 1967. URL: https://apps.dtic.mil/sti/citations/tr/AD0653593 (дата обращения: 08.10.2025).

Eng

1. Barlow, R. E., & Proschan, F. (1987). Mathematical theory of reliability. Society for Industrial and Applied Mathematics.
2. Birolini, A. (2017). Reliability engineering: Theory and practice (8th ed.). Springer-Verlag.
3. Vonta, I., & Ram, M. (2019). Reliability engineering: Theory and applications. CRC Press.
4. Ali, S., Lee, S.-M., & Jang, C.-M. (2018). Statistical analysis of wind characteristics using Weibull and Rayleigh distributions in Deokjeok-do Island – Incheon, South Korea. Renewable Energy, 123, 652–663. https://doi.org/10.1016/j.renene.2018.02.087
5. Aslam, M. (2021). Testing average wind speed using sampling plan for Weibull distribution under indeterminacy. Scientific Reports, 11, 7533. Retrieved October 8, 2025, from https://www.nature.com/articles/s41598-021-87136-8
6. Bayazit, M., & Önöz, B. (2004). Sampling variances of regional flood quantiles affected by intersite correlation. Journal of Hydrology, 291(1–2), 42–51. https://doi.org/10.1016/j.jhydrol.2003.12.009
7. Koehler, K. J., & McGovern, P. G. (1990). An application of the LFP survival model to smoking cessation data. Statistics in Medicine, 9(4), 409–421. https://doi.org/10.1002/sim.4780090412
8. Johnson, N. L., Kotz, S., & Balakrishnan, N. (1995). Continuous univariate distributions (2nd ed., Vol. 2). Wiley.
9. Nelson, P. R. (1979). Control charts for Weibull processes with standards given. IEEE Transactions on Reliability, R-28(4), 283–288. https://doi.org/10.1109/TR.1979.5220605
10. Ramalhoto, M. F., & Morais, M. (1999). Shewhart control charts for the scale parameter of a Weibull control variable with fixed and variable sampling intervals. Journal of Applied Statistics, 26(1), 129–160. https://doi.org/10.1080/02664769922700
11. Pascual, F. (2010). EWMA charts for the Weibull shape parameter. Journal of Quality Technology, 42(4), 400–416. https://doi.org/10.1080/00224065.2010.11917836
12. Pascual, F., & Zhang, H. (2011). Monitoring the Weibull shape parameter by control charts for the sample range. Quality and Reliability Engineering International, 27(1), 15–25. https://doi.org/10.1002/qre.1099
13. Faraz, A., Saniga, E. M., & Heuchenne, C. (2015). Shewhart control charts for monitoring reliability with Weibull lifetimes. Quality and Reliability Engineering International, 31(8), 1565–1574. https://doi.org/10.1002/qre.1692
14. Fernandes, F. H., Lee Ho, L., & Bourguignon, M. (2019). About Shewhart control charts to monitor the Weibull mean. Quality and Reliability Engineering International, 35(7), 2343–2357. https://doi.org/10.1002/qre.2515
15. Lee Ho, L., Fernandes, F. H., Quinino, R. C., & Bourguignon, M. (2020). Improving Shewhart control chart performance for monitoring the Weibull mean. Quality and Reliability Engineering International, 37(3), 984–996. https://doi.org/10.1002/qre.2777
16. Montgomery, D. C. (2013). Introduction to statistical quality control (7th ed.). John Wiley & Sons.
17. Wheeler, D. J., & Chambers, D. S. (1992). Understanding statistical process control (2nd ed.). SPC Press.
18. Mittag, H.-J., & Rinne, H. (1993). Statistical methods of quality assurance. Chapman & Hall.
19. Harter, H. L., & Dubey, S. D. (1967). Theory and tables for tests of hypotheses concerning the mean and the variance of a Weibull population (Report No. ARL 67-0059). Aerospace Research Laboratories, Wright-Patterson Air Force Base. Retrieved October 8, 2025, from https://apps.dtic.mil/sti/citations/tr/AD0653593

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