In this paper, Burr-type XII $$ar{X}$$ X ˉ synthetic schemes are proposed as an alternative to the classical $$ar{X}$$ X ˉ synthetic schemes when the assumption of normality fails to hold. First, the basic design of the Burr-type XII $$ar{X}$$ X ˉ synthetic scheme is developed and its performance investigated using exact formulae. Secondly, the non-side-sensitive and side-sensitive Burr-type XII $$ar{X}$$ X ˉ synthetic schemes are introduced and their zero-state and steady-state performances, in terms of the average run-length and expected extra quadratic loss values, are investigated using a Markov chain approach. Thirdly, the proposed schemes are compared to the existing classical runs-rules and synthetic $$ar{X}$$ X ˉ schemes. It is observed that the proposed schemes have very interesting properties and outperform the competing schemes in many cases under symmetric and skewed underlying process distributions. Finally, an illustrative real-life example is given to demonstrate the design and implementation of the proposed Burr-type XII $$ar{X}$$ X ˉ synthetic schemes.
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机译：在本文中，提出了伯尔型XII $$ bar {X} $$ Xˉ合成方案作为经典的$$ bar {X} $$ Xˉ合成方案的替代方案，当正态性假设不成立时。首先，开发了Burr型XII $$ bar {X} $$ Xˉ合成方案的基本设计，并使用精确公式对其性能进行了研究。其次，介绍了非侧面敏感和侧面敏感的Burr型XII $$ bar {X} $$ Xˉ合成方案，并根据平均游程长度给出了其零态和稳态性能。使用马尔可夫链方法研究预期的和额外的二次损失值。第三，将拟议的方案与现有的经典运行规则和综合的$$ bar {X} $$ Xˉ方案进行比较。可以看出，在对称和偏斜的基础过程分布下，所提出的方案具有非常有趣的性能，并且在许多情况下都优于竞争方案。最后，给出了一个真实的示例来说明所提出的Burr型XII $$ bar {X} $$ X X合成方案的设计和实现。
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