Roller-Coaster Failure Rates and Mean Residual Life Functions

The investigation in this paper was motivated by an extended generalized inverse Gaussian (EGIG) distribution which has more than one turning point of the failure rate for certain values of the parameters. We present some general results for studying the reationship between the change points of Glaser's eta function, the failure rate and the MRLF. Also, we establish an ordering between the number of change points of Glaser's eta function, the hazard rate and the MRLF. These results are used to investigate, in detail, the monotonicity of the three functions in the case of the EGIG. The EGIG model has one additional parameter, S, than the generalized inverse Gaussian (GIG) model's three parameters; see (Jorgensen, 1982). For the EGIG model, the maximum likelihood estimation of the four parameters is discussed and a score test is developed for testing the importance of the additional parameter, S. An example is provided to illustrate that the EGIG model fits the data better than the GIG of (Jorgensen, 1982).