A general formula for the failure-rate function when distribution information is partially specified

This paper presents a new formula for the failure-rate function (FRF), derived from a recently introduced 4-parameter family of distributions. The new formula can be expressed in terms of its Cdf, is characterized by algebraic simplicity, and can replace more complex hazard functions by using routine distribution fitting. When the actual Cdf is unknown and partial distribution-information is available (or can be extracted from sample data), new fitting procedures that use only first-degree or first- and second-degree moments are used to approximate the unknown FRF. This new approach is demonstrated for some commonly used Cdfs and shown to yield highly accurate values for the FRF. Relative to current practice, the new FRF has four major advantages: it does not require specification of an exact distribution thus avoiding errors incurred by the use of a wrong model; since estimates of only low-degree (at most first- or second-degree) moments are required to determine the parameters of the FRF, the associated mean-square-deviations are relatively small; the new FRF can be easily adapted for use with censored data; and simple maximum likelihood estimates can be developed.