The Cubic Hazard Rate Distribution: Properties, Inference, and Regression Models

Failure risks may vary significantly over time, or exhibit sudden changes, in lifetime data. The constant, linear and quadratic hazard functions may thus be too restrictive. This paper presents the four-parameter cubic hazard rate distribution (CHRD) that contains the exponential, Rayleigh, linear failure rate and quadratic hazard rate distributions as special cases. The coefficients are successive changes in the hazard and are used to decide if the cubic term provides useful flexibility. We define the admissible parameter space and derive the main distributional, reliability and information-theoretic properties. These include the density, distribution, survival and hazard functions, quantiles, moments, order statistics and entropy measures. Constrained maximum likelihood is used to estimate parameters. The observed Fisher information is used to approximate the confidence intervals for the parameters and selected reliability measures by the delta method. Several parameter settings and sample sizes are studied in the Monte Carlo study. The errors in estimation and the length of the intervals tend to be smaller as the sample size increases, and the empirical coverage is close to the specified confidence level. The lower-order coefficients are more precisely estimated than the quadratic and cubic coefficients. Four lifetime datasets are used to compare the CHRD with alternative distributions, and information criteria, goodness-of-fit statistics, fitted-distribution plots and pairwise likelihood surfaces are used. For Dataset III, both information criteria and discrepancy statistics favor the CHRD; in Datasets I, II, and IV, the best-fitting and complexity-adjusted choices differ. A proportional-hazards extension is also developed and fitted to right-censored mortality data. The cubic baseline is the one with the smallest AIC, and the degree-one baseline is the one with the smallest BIC. The difference illustrates that the preferred polynomial degree depends on the balance between improved fit and model complexity, rather than establishing that the cubic term is necessary under every criterion.