Performance of Phase II Control Chart for Location When One Parameter is Estimated in Terms of Run Length Distribution and Percentiles
Keywords:
average run length (ARL), false alarm rate (FAR), median run length (MRL), parameter estimation, stochastic orderingAbstract
It is well known that the median run length measures chart performance better than the average run-length. Some authors have advocated more representative measures like the percentiles for assessing charts performance and called for an examination of the percentiles of the entire run-length distribution. An earlier work studied the percentiles of Shewhart chart when the process mean and variance are known (Case KK), later, another when the process mean and variance are unknown (Case UU). Here, we consider when only the process mean is unknown (Case UK) and when only the process variance is unknown (Case KU) by evaluating and plotting their exact run-length cumulative distribution functions for some reference samples (m) of size 5 at a given false alarm rate. We compare the results with those for Cases KK and UU. Unlike in Case UU, the cumulative distribution function curves for Case UK for small to moderate m are stochastically ordered relative to that of the geometric distribution and dominance is a function of δ, however, in line with Case UU, in Case KU, the curves cross that for the geometric distribution at some points and for at least m = 500 and
n = 5, the curves for Cases UK and KU converge with that for Cases KK and UU.
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