Preservation of Life Distribution Classes Under Reliability Operations

Let G be a continuous life distribution with support [0, c] and consider the following classes of life distributions: 𝒷 = {F: F(0) = 0 and G⁻¹F is convex on (0, F⁻¹(1))}, 𝒱 = {F: G⁻¹F is concave on (0, ∞)}, 𝒻 = {F: G⁻¹F(θx) ≤ θG⁻¹F(x) for 0 < θ < 1, 0 < x < F⁻¹(l)}, and 𝒜 = {F: G⁻¹F(θx) ≥ G⁻¹F(x) for 0 < θ < 1, 0 < x < ∞}. In the case where G is exponential, 𝒷, 𝒱, 𝒻, and 𝒜 are, respectively, the classes of life distributions with increasing failure rate, decreasing failure rate, increasing failure rate average, and decreasing failure rate average. In this paper we study the closure properties of the classes 𝒷, 𝒱, 𝒻, and 𝒜 under the formation of coherent systems, convolutions, and mixtures. Our results can be used to obtain information about the reliability of systems of components on the basis of component data.