Weibull Lifetimes and Hazard Shape

Why Weibull Hazard Shape Matters

A lifetime model describes not only when failures tend to occur, but also how the risk of failure changes as an item ages. A constant failure rate is a useful benchmark, but it cannot describe every setting: some components wear out, while others have a high early failure rate that declines among survivors. The Weibull distribution provides a simple model for these different patterns through one shape parameter.

We will use \(T\) for a nonnegative lifetime, \(\eta>0\) for the scale parameter, and \(k>0\) for the shape parameter. The scale \(\eta\) has the same units as time; the shape \(k\) is dimensionless. The exponential lifetime is the special case \(k=1\).

Weibull Survival, Density, and Hazard

The survival function gives the probability that a lifetime exceeds a specified time. Since \(S(0)=1\) and \(S(t)\) decreases to zero as \(t\) tends to infinity, the model places all its probability on nonnegative lifetimes. Differentiating \(F(t)\) for \(t>0\) gives the density.

To obtain the density, differentiate \(F(t)=1-\exp[-(t/\eta)^k]\). The chain rule gives $$ f(t)=\exp\left[-\left(\frac{t}{\eta}\right)^k\right]\frac{d}{dt}\left(\frac{t}{\eta}\right)^k =\frac{k}{\eta}\left(\frac{t}{\eta}\right)^{k-1}\exp\left[-\left(\frac{t}{\eta}\right)^k\right]. $$ The hazard is the instantaneous failure rate among items that have survived to time \(t\). More precisely, when the density and survival function are continuous and \(S(t)>0\), $$ h(t)=\lim_{\Delta\downarrow0} \frac{P(t\leq T<t+\Delta\mid T\geq t)}{\Delta} =\frac{f(t)}{S(t)}. $$ For the Weibull model, \(S(t)>0\) at every finite \(t\), so cancellation of the exponential terms yields the displayed hazard formula. Its units are inverse time: \(k\) and \(t/\eta\) are dimensionless, while \(1/\eta\) has units of inverse time.

Why These Formulas Define a Probability Distribution

The survival formula is decreasing and has the correct endpoint limits, but it is useful to check directly that the density is valid. For \(t>0\), \(f(t)\geq0\). For any \(b>0\), the fundamental theorem of calculus gives $$ \int_0^b f(t)\,dt=F(b)-F(0)=1-\exp\left[-\left(\frac{b}{\eta}\right)^k\right]. $$ As \(b\) tends to infinity, \((b/\eta)^k\) tends to infinity because \(k>0\); hence \(F(b)\) tends to one. Taking the improper-integral limit proves that the density integrates to one. There is no probability atom at zero, since \(F(0)=0\) and \(F\) is continuous there.

The density’s behavior at the endpoint depends on \(k\): as \(t\downarrow0\), \(f(t)\) tends to zero if \(k>1\), equals \(1/\eta\) if \(k=1\), and tends to infinity if \(0<k<1\). In the last case this is an integrable density singularity, not a point mass at zero.

Shape Parameter and Hazard Direction

For \(t>0\), differentiating the hazard gives $$ h'(t)=\frac{k(k-1)}{\eta^2}\left(\frac{t}{\eta}\right)^{k-2}. $$ All factors other than \(k-1\) are positive. Therefore the sign of the derivative depends only on whether \(k\) is below, equal to, or above one.

This classification concerns the hazard among survivors, not the density. For \(k>1\), the hazard begins at zero and increases for positive time. For \(k=1\), it is constant. For \(0<k<1\), it decreases from an unbounded limiting value as \(t\downarrow0\). These are statements about the hazard’s limit or behavior for positive \(t\); the hazard formula need not have a finite value at the endpoint.

Interpreting the Scale and Avoiding Common Errors

The scale parameter has a particularly direct survival interpretation: $$ S(\eta)=\exp(-1)\approx0.3679. $$ Thus \(\eta\) is the time by which about \(1-e^{-1}\approx0.6321\), or 63.21%, of the population has failed. It is not generally the mean or median lifetime. For example, the median depends on both parameters and is \(\eta(\log 2)^{1/k}\), as derived in the first worked example.

• Confusing density and hazard. The density describes probability per unit time near a point in the whole population; the hazard describes instantaneous failure rate conditional on survival to that time. The relation is \(h(t)=f(t)/S(t)\), not \(h(t)=f(t)\).
• Reversing the shape interpretation. A value \(k>1\) means increasing hazard, while \(0<k<1\) means decreasing hazard. The scale changes the time scale and hazard magnitude, but not the direction of the hazard trend.
• Calling the hazard a failure probability. A hazard has units of inverse time and may exceed one numerically. A probability over an interval is dimensionless; it must be calculated from \(F\) or from a survival difference.
• Ignoring endpoint behavior. When \(k<1\), the hazard diverges as \(t\downarrow0\), but the Weibull distribution still has no atom at zero. A divergent density or hazard at a single endpoint does not itself imply positive probability at that point.

The Weibull family is useful when a single smooth hazard trend is a reasonable approximation: increasing hazard can represent wear-out, decreasing hazard can represent early failures followed by a more robust surviving population, and constant hazard recovers the exponential model. The classification is a model property, not evidence by itself that a real system follows that model. Reliability conclusions should therefore be paired with an assessment of whether the Weibull assumption is appropriate for the system and time range.