Materials for “ Economic - statistical Design of Synthetic Max Chart

The standard max chart gives an out-of-control signal when the chart's statistic M exceeds the upper control limit UCL. The probability of M falling below the UCL is given as [3] 2 2 2) UCL (2 2 2) UCL (2 1 2 1 UCL UCL δ χ δ χ δ δ δ δ H H n n , where UCL > 0,) (⋅ Φ is the cumulative distribution function of a standard normal random variable,) (⋅ H is the cumulative distribution function of a chi-square random variable with (n – 1) degrees of freedom and 2 β χ is the β quantile of a chi-square random variable with (n – 1) degrees of freedom. When the process is in control, 0 1 = δ and 1 2 = δ. Thus, () 2) UCL (Φ χ H =) UCL (Φ and () 2) UCL (− Φ χ H =) UCL (− Φ. Consequently, F(UCL;n,0,1) = {Φ(UCL) − Φ(UCL)} 2. If F(UCL;n,0,1) = 1 − α, where α is the type I error probability of the standard max chart, solving this equation will yield UCL = 2 1 , 1 α χ − ,where 2 1 , r χ is the r quantile of a chi-square random variable with degree of freedom of 1. Then, the in-control average run length (i.e. the expected number of samples required to give an out-of-control signal when 0 1 = δ and 1 2 = δ) of the standard max chart can be calculated as [3]) 1 , 0 , ; UCL (1 1 ARL 0 n F − = { } 2) UCL () UCL (1 1 − Φ − Φ − = .