On a non-monotonic ageing class based on the failure rate average

On some properties of distributions possessing a bathtub-shaped failure rate average

“…Proof. Using [12,Theorem 3.2] we get finiteness of moments of F n . Now, following Theorem 3.1 we conclude that F is BFRA.…”

“…Now, following Theorem 3.1 we conclude that F is BFRA. If F is BFRA with a finite change point then again an application of [12,Theorem 3.2] yields finiteness of moments of F. Now we consider the case when F is BFRA with a change point at infinity, i.e. DFRA.…”

“…Attempts to model such scenarios have led to several non-parametric classes of distributions such as bathtub failure rate (BFR) [23], increasing initially then decreasing mean residual life (IDMRL) [24], new worse then better than used in expectation (NWBUE) [47], increasing initially then decreasing mean time to failure (IDMTTF) [27] etc. [46] observed a trend change in the failure rate average which results in the family of bathtub failure rate average (BFRA) distributions, subsequently developed and studied in detail in [12].…”

“…Tests of exponentiality against nonexponential IFRA distributions have been proposed by several authors [5,37,39,43,57,59]. [46] and [12] both showed that if a twice-differentiable life distribution function F is BFR(t 0 ), then F is BFRA(x 0 ) where x 0 ≥ t 0 . The relationship between the discrete versions of BFR and BFRA was also given in [45].…”

“…The relationship between the discrete versions of BFR and BFRA was also given in [45]. [12] proposed a non-parametric test of exponentiality against BFRA alternatives by assuming that the proportion of early failures is known, and investigated the closure of BFRA property under some reliability operations.…”