Estimating turning points of the failure rate of the extended Weibull distribution

Marshall and Olkin (1997) proposed a way of introducing a parameter, called the tilt parameter, to expand a family of distributions. In this paper we compare the extended distribution and the original distribution with respect to some stochastic orderings. Also we investigate thoroughly the monotonicity of the failure rate of the resulting distribution when the baseline distribution is taken as Weibull. It turns out that the failure rate is increasing, decreasing, or non-monotonic with one or two turning points depending on the parameters. For non-monotonic types, the turning points of the failure rate are estimated and their confidence intervals are provided. Simulation studies are carried out to examine the performance of these intervals. An example is provided to illustrate the procedure.