EWMA charts: ARL considerations in case of changes in location and scale
Widely spread tools within the area of Statistical Process Control are control charts of various designs. Control chart applications are used to keep process parameters (e.g., mean <InlineEquation ID="IEq1"> <EquationSource Format="TEX">$$\mu $$</EquationSource> <EquationSource Format="MATHML"> <math xmlns:xlink="http://www.w3.org/1999/xlink"> <mi mathvariant="italic">μ</mi> </math> </EquationSource> </InlineEquation>, standard deviation <InlineEquation ID="IEq2"> <EquationSource Format="TEX">$$\sigma $$</EquationSource> <EquationSource Format="MATHML"> <math xmlns:xlink="http://www.w3.org/1999/xlink"> <mi mathvariant="italic">σ</mi> </math> </EquationSource> </InlineEquation> or percent defective <InlineEquation ID="IEq3"> <EquationSource Format="TEX">$$p$$</EquationSource> <EquationSource Format="MATHML"> <math xmlns:xlink="http://www.w3.org/1999/xlink"> <mi>p</mi> </math> </EquationSource> </InlineEquation>) under surveillance so that a certain level of process quality can be assured. Well-established schemes such as exponentially weighted moving average charts (EWMA), cumulative sum charts or the classical Shewhart charts are frequently treated in theory and practice. Since Shewhart introduced a <InlineEquation ID="IEq4"> <EquationSource Format="TEX">$$p$$</EquationSource> <EquationSource Format="MATHML"> <math xmlns:xlink="http://www.w3.org/1999/xlink"> <mi>p</mi> </math> </EquationSource> </InlineEquation> chart (for attribute data), the question of controlling the percent defective was rarely a subject of an analysis, while several extensions were made using more advanced schemes (e.g., EWMA) to monitor effects on parameter deteriorations. Here, performance comparisons between a newly designed EWMA <InlineEquation ID="IEq5"> <EquationSource Format="TEX">$$p$$</EquationSource> <EquationSource Format="MATHML"> <math xmlns:xlink="http://www.w3.org/1999/xlink"> <mi>p</mi> </math> </EquationSource> </InlineEquation> control chart for application to continuous types of data, <InlineEquation ID="IEq6"> <EquationSource Format="TEX">$$p=f(\mu ,\sigma )$$</EquationSource> <EquationSource Format="MATHML"> <math xmlns:xlink="http://www.w3.org/1999/xlink"> <mrow> <mi>p</mi> <mo>=</mo> <mi>f</mi> <mo stretchy="false">(</mo> <mi mathvariant="italic">μ</mi> <mo>,</mo> <mi mathvariant="italic">σ</mi> <mo stretchy="false">)</mo> </mrow> </math> </EquationSource> </InlineEquation>, and popular EWMA designs (<InlineEquation ID="IEq7"> <EquationSource Format="TEX">$$\bar{X}$$</EquationSource> <EquationSource Format="MATHML"> <math xmlns:xlink="http://www.w3.org/1999/xlink"> <mover accent="true"> <mrow> <mi>X</mi> </mrow> <mrow> <mo stretchy="false">¯</mo> </mrow> </mover> </math> </EquationSource> </InlineEquation>, <InlineEquation ID="IEq8"> <EquationSource Format="TEX">$$\bar{X}$$</EquationSource> <EquationSource Format="MATHML"> <math xmlns:xlink="http://www.w3.org/1999/xlink"> <mover accent="true"> <mrow> <mi>X</mi> </mrow> <mrow> <mo stretchy="false">¯</mo> </mrow> </mover> </math> </EquationSource> </InlineEquation>-<InlineEquation ID="IEq9"> <EquationSource Format="TEX">$$S^2$$</EquationSource> <EquationSource Format="MATHML"> <math xmlns:xlink="http://www.w3.org/1999/xlink"> <msup> <mi>S</mi> <mn>2</mn> </msup> </math> </EquationSource> </InlineEquation>) are presented. Thus, isolines of the average run length are introduced for each scheme taking both changes in mean and standard deviation into account. Adequate extensions of the classical EWMA designs are used to make these specific comparisons feasible. The results presented are computed by using numerical methods. Copyright Springer-Verlag Berlin Heidelberg 2014
