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Over the past few decades, the idea of the measurement of success or failure concerning reliability has developed, leading scientists to use this advantage to establish new branches of reliability such as classes of life distributions. In this book we shall construct a new class of life distributions and study the containment relation between our class and some other classes. Based on U-statistics three tests are proposed for testing exponentiality versus this class in case of non-censored and censored data. Our test statistics is applied to real data in different fields such as industry and medicine field.