ABSTRACT
The outstanding simplicity of this model is its constant hazard rate. We display some p.d.f.’s and survivor functions for three different values of λ in Figure 3.1. The relationship between the cumulative hazard and the survivor
function (1.6) is
log ( H(t)
) = log
(− log(S(t))) = log(λ) + log(t) or, equivalently expressed with log(t) on the vertical axis,
log(t) = − log(λ) + log (− log(S(t))). (3.1) Hence, the plot of log(t) versus log
(− log(S(t))) is a straight line with slope 1 and y-intercept − log(λ). At the end of this section we exploit this linear relationship to construct a Q-Q plot for a graphical check of the goodness of fit of the exponential model to the data. Since the hazard function, h(t) = λ, is constant, plots of both empirical hazards, h˜(ti) and ĥ(t) (page 32), against time provide a quick graphical check. For a good fit, the plot patterns should resemble horizontal lines. Otherwise, look for another survival model. The parametric approach to estimating quantities of interest is presented in Section 3.4. There we first illustrate this with an uncensored sample. Then the same approach is applied to a censored sample. The exponential is a special case of both the Weibull and gamma models, each with their shape parameter equal to 1.
