Hazard-Rate Comparisons for Exponential and Weibull Laws

Why Compare Hazards?

For a lifetime \(T\), the hazard at time \(t\) describes the instantaneous event rate among individuals who have survived to \(t\). Comparing hazards can therefore reveal how the relative event rates evolve over time. Exponential and Weibull laws make this comparison especially transparent: their hazards are constants or simple power functions.

The direction of the comparison matters. If \(X\) has a higher hazard than \(Y\) at every positive time, then \(X\) is smaller in hazard-rate order: its survival is relatively less favorable as time passes. If the hazards cross, neither lifetime has a hazard advantage that holds at all times. The calculations below make these claims precise for the two distribution families.

Hazards and Survival Ratios

Write \(S_T(t)=P(T>t)\) for the survival function of a nonnegative lifetime \(T\). For the smooth survival functions used here, define its hazard by the negative derivative of the log survival function. This definition specifies a pointwise hazard without ambiguity about which version of a density is chosen.

For two positive, differentiable survival functions, the survival ratio \(S_Y(t)/S_X(t)\) connects directly to the hazards. This is the pointwise criterion used throughout the tutorial.

To prove the criterion, differentiate the ratio using the product and quotient rules:

The ratio is positive, so its derivative is nonnegative exactly when \(h_X-h_Y\ge0\). If \(h_X\ge h_Y\) everywhere, the derivative is nonnegative everywhere, and the mean value theorem implies that the ratio is nondecreasing. Conversely, a differentiable nondecreasing function has a nonnegative derivative wherever its derivative exists; applying this fact to \(S_Y/S_X\) gives \(h_X\ge h_Y\). \(\square\)

Exponential and Weibull Hazard Functions

Use the rate parameterization \(T\sim\operatorname{Exp}(\lambda)\), with \(\lambda>0\), for the exponential law. Its survival function is \(S_T(t)=e^{-\lambda t}\) for \(t\ge0\). A Weibull law will be written with shape \(k>0\) and scale \(\eta>0\), so its survival function is \(S_T(t)=\exp[-(t/\eta)^k]\) for \(t\ge0\).

For the exponential case, differentiating \(\log S_T(t)=-\lambda t\) gives the constant hazard \(\lambda\). For the Weibull case, \(\log S_T(t)=-t^k/\eta^k\), so differentiating and changing sign gives \(k t^{k-1}/\eta^k\). Equivalently, differentiating the Weibull distribution function gives \[ f_T(t)=\frac{k}{\eta^k}t^{k-1}\exp[-(t/\eta)^k], \] and division by its positive survival function gives the same hazard. These expressions are pointwise valid for every \(t>0\), including when the hazard tends to zero or infinity as \(t\) approaches zero.

Parameter Cases for Hazard-Rate Order

For two exponential lifetimes, the hazard comparison is determined entirely by their rates. For two Weibull lifetimes with the same shape, it is determined by the coefficients multiplying the common power of time. If the shapes differ, the powers themselves determine the behavior near zero and at large times.

For exponentials, the hazards are the constants \(\lambda_X\) and \(\lambda_Y\); the hazard-rate criterion therefore gives the first equivalence. The survival ratio makes the same comparison explicit:

This ratio is nondecreasing exactly when \(\lambda_X\ge\lambda_Y\). For equal-shape Weibull laws, the hazard difference is \[ h_X(t)-h_Y(t)=k t^{k-1}\left(\frac{1}{\eta_X^k}-\frac{1}{\eta_Y^k}\right). \] Since \(k t^{k-1}>0\) for \(t>0\), it is nonnegative everywhere exactly when \(\eta_X\le\eta_Y\). This proves the second equivalence.

Different Weibull Shapes: Hazards Must Cross

Write the Weibull hazard as \(h(t)=A t^{k-1}\), where \(A=k/\eta^k>0\). For two different shapes, the ratio of the hazards is a nonconstant power of \(t\). That ratio tends either to zero or to infinity near zero and has the opposite limiting behavior as \(t\) grows without bound. Consequently, the hazard ordering reverses exactly once.

To derive the crossing time, set the hazards equal and define \(A_X=k_X/\eta_X^{k_X}\), \(A_Y=k_Y/\eta_Y^{k_Y}\): \[ A_X t^{k_X-1}=A_Y t^{k_Y-1} \quad\Longleftrightarrow\quad t^{k_X-k_Y}=\frac{A_Y}{A_X}. \] The right side is positive, and the nonzero power has exactly one positive solution, \(t_*\) as displayed. Moreover, \[ \frac{h_X(t)}{h_Y(t)} =\frac{A_X}{A_Y}t^{k_X-k_Y}. \] If \(k_X>k_Y\), this ratio increases strictly from zero to infinity, so it equals one once and gives the stated direction on either side. If \(k_X<k_Y\), it decreases strictly from infinity to zero, giving the reversed direction. Since each hazard is higher on one side of the crossing and lower on the other, neither can dominate for every \(t>0\). \(\square\)

Common Mistake and Interpretation

Common mistake: “Both hazards are positive, so one lifetime must be smaller in hazard-rate order.” Positivity only says that each hazard is nonnegative; it does not compare the hazards. In the last example, both hazards are positive for every \(t>0\), yet their difference changes sign at \(1/2\). The correction is to compare \(h_X(t)-h_Y(t)\) over the full time range, or equivalently check whether \(S_Y(t)/S_X(t)\) is nondecreasing over that range.

A second point of care concerns pointwise claims. An arbitrary density can be altered at an individual point without changing its distribution, so an equality between \(f/S\) and a derivative-based hazard need not hold pointwise for every density version. Here the exponential and Weibull survival functions are differentiable on \((0,\infty)\), and their displayed hazards are defined as \(-S'/S\); their smooth density versions agree with these hazards there. This regularity is why the pointwise comparisons and crossing statements are justified.

The parameter patterns are useful when selecting a comparison: exponential rates decide the constant-hazard case, a common Weibull shape reduces the comparison to scales, and different Weibull shapes force a crossing. The hazard-rate order records an all-times comparison; a crossing means that this particular order does not hold in either direction.