### Journals Information

**
Mathematics and Statistics Vol. 10(2), pp. 397 - 409 DOI: 10.13189/ms.2022.100215 Reprint (PDF) (417Kb) **

## On The Unconditional Run Length Distribution and Percentiles for The -chart When The In-control Process Parameter Is Estimated

**Ugwu Samson. O ^{1}^{,*}, Uchenna Nduka .C ^{1}, Ezra Precious .N ^{1}, Ugwu Gibson .C ^{2}, Odoh Paschal .N ^{3}, Nwafor Cynthia. N ^{1}**

^{1}Department of Statistics, University of Nigeria Nsukka, Nigeria

^{2}Department of Statistics, Enugu-State, Institute of Management and Technology, Nigeria

^{3}Department of Industrial Mathematics and Statistics, Enugu-State of Science and Technology, Enugu-State, Nigeria

**ABSTRACT**

It is well known that the median is a better measure of location in skewed distributions. Run-length (RL) distribution is a skewed distribution, hence, median run-length measures chart performance better than the average run length. Some authors have advocated examination of the entire percentiles of the RL distribution in assessing chart performance. Such works already exist for Shewhart −chart, CUSUM chart, CUSUM and EWMA charts, Hotelling's chi-square, and the two simple Shewhart multivariate non-parametric charts. Similar work on -chart for one- and two-sided lacks in the literature. This work stands in the gap. Therefore, a detailed and comparative study of the one-sided upper and the two-sided -control charts for some m reference samples at fixed sample size and false alarm rate will be considered here using the information from the unconditional RL cdf curve and its percentiles (mainly median). The order of the RL cdf curves of the one-sided upper -chart is independent of the state of the process unlike in the two-sided one. The one-sided upper chart outperformed the two-sided one both in the in-control and in detecting positive shifts. The two-sided -chart is more sensitive in detecting incremental shifts than to decremental shifts.

**KEYWORDS**

Average Run Length (ARL), False Alarm Rate (FAR), Median Run Length (MRL), Stochastic Ordering, Pooled Variance

**Cite This Paper in IEEE or APA Citation Styles**

(a). IEEE Format:

[1] Ugwu Samson. O , Uchenna Nduka .C , Ezra Precious .N , Ugwu Gibson .C , Odoh Paschal .N , Nwafor Cynthia. N , "On The Unconditional Run Length Distribution and Percentiles for The -chart When The In-control Process Parameter Is Estimated," Mathematics and Statistics, Vol. 10, No. 2, pp. 397 - 409, 2022. DOI: 10.13189/ms.2022.100215.

(b). APA Format:

Ugwu Samson. O , Uchenna Nduka .C , Ezra Precious .N , Ugwu Gibson .C , Odoh Paschal .N , Nwafor Cynthia. N (2022). On The Unconditional Run Length Distribution and Percentiles for The -chart When The In-control Process Parameter Is Estimated. Mathematics and Statistics, 10(2), 397 - 409. DOI: 10.13189/ms.2022.100215.