Testing whether survival function is harmonic new better than used in expectation (Q1085938)

The concept of HNBUE (harmonic new better than used in expectation) is defined as follows: \(\int^{+\infty}_{t}\bar F(x)dx<\mu \exp \{-t/\mu \}\) for every \(t>0\), where \(\bar F=1-F\), \(\mu =\int^{+\infty}_{0}\bar F(x)dx<\infty\). This concept extends the concept of NBUE distributions. The authors obtain two test statistics \(A_ n\) and \(B_ n\) (n being the sample size) for testing \(H_ 0: F(x)\) exponential versus \(H_ 1: F(x)\) HNBUE (and not exponential). These statistics are based on the empirical scaled total time on test (TTT). Under the null hypothesis the exact distributions of \(A_ n\) and \(B_ n\) \((n=2,3,...)\) are obtained. Furthermore, the asymptotic distributions for \(A_ n\) and \(B_ n\) are obtained and some statistical properties of \(B_ n\) are shown.