Some tests for mean residual life criteria based on the total time on test transform

Let F be a life distribution with survival function $\text{F}\text{= 1 −}\text{F}$ and finite mean $\text{μ = ∫}{}_0{}^{\mathrm{\infty }}\text{F}\text{(}\text{x}\text{)d}\text{x}$. The scaled total time on test transform is defined by . In this paper, the properties of $\text{φ}{}_{\mathrm{F}}\text{(t)}$ for some criteria of the mean residual life are investigated. These criteria include decreasing mean residual life average, decreasing harmonic mean residual life average, new better than used harmonic mean residual and new better than used mean residual life. Some test statistics for testing exponentially against all these mean residual life criteria are proposed. The distributions of the test statistics are investigated for small statistics samples. Powers of some of these tests are estimated by simulation.