Mixtures of distributions with increasing linear failure rates

Abstract

Populations of specific components are often heterogeneous and consist of a small number of different sub-populations. For example there are often two groups: defective components with shorter lifetimes and standard components with longer lifetimes. Another heterogeneous population results when components produced by two different manufacturing lines are combined. In either case a mixture results. The resulting population can be described using the statistical concept of a mixture. It is a well-known result that a mixture of distributions with decreasing failure rates has a decreasing failure rate. However, little is known about the monotonicity of a mixture when the various subpopulations have failure rates which are not necessarily decreasing. In this paper we study and attempt to determine the shape as well as the overall behavior of the failure rate of a mixture from two subpopulations each of which has increasing linear failure rate.

On the behavior and shape of mixture failure rates from a family of IFR Weibull distributions

“…However, changing the value of p in both examples might result to different shape of r m (t). Recently, Block, Savits, and Wondmagegnehu [7] studied the behavior of mixture of distributions with increasing linear failure rates. According to their findings, the shape of mixture failure rate varies from IFR to a shape with four changes of monotonicity.…”

“…However, the shape of the mixture in the intermediate intervals for heterogeneous populations is not well understood. As an example of this fact, see Block, Li, and Savits [3] and Block et al [7]. In Block et al [2,5], the asymptotic behavior of a mixture failure rate was discussed in detail.…”

A Note on the Efficiency Based Reliability Measures for Heterogeneous Populations

“…Navarro and Hernandez (2004) state that the mixture failure rate of two truncated normal distributions, depending on parameters involved, can also be increasing, BT(bathtub)-shaped or MBT-shaped. Block et al (2003) give explicit conditions which describe the possible shapes of the mixture failure rate for two increasing linear failure rates.…”

On asymptotic failure rates in bivariate frailty competing risks models

“…As monotonicity properties of the mixture failure rate can differ dramatically from those of the baseline failure rate, this topic was thoroughly investigated in the literature (see Badia et al (2001), Block et al (1993), Finkelstein and Esaulova (2001) and Lynch (1999), to name a few). Considerable attention was also paid to the asymptotic behavior of mixture failure rates (Block et al, 2003;Finkelstein and Esaulova, 2006;Shaked and Spizzichino, 2001).…”