Why Mixture Hazards Need Updated Weights
A severity mixture represents a population of losses drawn from different component distributions. The original mixture proportions describe the population before a loss amount is observed. But a loss known to exceed a large threshold is not a random member of that original population: components with greater survival at the threshold are more likely to be represented. Therefore, the hazard at a loss size depends both on each component’s hazard and on the component proportions among losses that have survived to that size.
This distinction matters in actuarial work because the mixture hazard describes the instantaneous rate of loss ending near a particular amount, conditional on the loss having exceeded that amount. Using the original component proportions in place of the survivor proportions generally gives an incorrect hazard.
The original component weights are \(p_j\). At loss amount \(x\), the component weights among losses exceeding \(x\) are instead \(P(j\mid X>x)=p_jS_j(x)/S_X(x)\). These updated weights depend on \(x\), unless the components have identical survival behavior. They sum to one because the component survival contributions \(p_jS_j(x)\) sum to \(S_X(x)\).
All formulas in this tutorial concern continuous severity distributions with densities. They apply at thresholds where the relevant survival probabilities are positive. If a severity model has point masses, a density-based hazard does not describe the mass at an atom; the mass and its conditional probability must be handled separately.
Deriving the Mixture Hazard
The derivation follows directly from the definition of hazard. At a given loss amount, each component contributes to the mixture density in proportion to its original weight and its density there. The total mixture density is divided by the total mixture survival. Rewriting each component density as its hazard times its survival reveals the updated survivor weights.
By definition, \(h_X(x)=\left(\sum_j p_j f_j(x)\right)/\left(\sum_j p_j S_j(x)\right)\), provided \(S_X(x)>0\).
For a component with \(S_j(x)>0\), its density satisfies \(f_j(x)=S_j(x)h_j(x)\).
Substitute the density identity and divide each term by the mixture survival. The resulting coefficient of \(h_j(x)\) is \(p_jS_j(x)/S_X(x)\).
Thus the mixture hazard is a weighted average of the component hazards, but the weights are the proportions in the surviving group at \(x\), not generally the original proportions. The result is an average with nonnegative weights summing to one. Consequently, when all component hazards are defined, \(h_X(x)\) lies between the smallest and largest component hazards at that same loss amount.
Worked Calculations
Worked Example: Combining a Weibull and an Exponential Component
A hypothetical severity mixture has two components. With probability \(0.40\), a loss follows a Weibull distribution with shape \(2\) and scale \(2\); with probability \(0.60\), it follows an exponential distribution with rate \(0.25\) per dollar. Find the mixture hazard at \(x=2\) dollars and the component proportions among losses exceeding \(2\) dollars. Give the hazard to four decimal places.
Method. Calculate each component’s survival and hazard at the specified threshold, use them to obtain the survivor proportions, and then take the survivor-weighted average of the component hazards.
For the Weibull component, \(S_1(x)=\exp[-(x/2)^2]\). Its hazard is \(h_1(x)=2x/2^2=x/2\). At \(x=2\), \(S_1(2)=e^{-1}=0.3678794\) and \(h_1(2)=1\) per dollar. For the exponential component, \(S_2(x)=e^{-0.25x}\) and \(h_2(x)=0.25\) per dollar, so \(S_2(2)=e^{-0.5}=0.6065307\).
The two weighted survival contributions are \(0.40(0.3678794)=0.1471518\) and \(0.60(0.6065307)=0.3639184\). Their sum is \(S_X(2)=0.5110702\). Therefore, the survivor proportions are \(0.1471518/0.5110702=0.2879\) for the Weibull component and \(0.3639184/0.5110702=0.7121\) for the exponential component.
The mixture hazard is \(0.2879(1)+0.7121(0.25)\), with the unrounded weights used in the calculation. More explicitly, the weighted density contributions are \(0.1471518(1)=0.1471518\) and \(0.3639184(0.25)=0.0909796\). Their sum, \(0.2381314\), divided by \(0.5110702\), gives \(h_X(2)=0.4659\) per dollar to four decimal places.
Check. The result lies between the component hazards \(0.25\) and \(1\), as it must. The common mistake avoided here is using the original \(0.40\) and \(0.60\) weights: those describe all losses before conditioning on exceeding \(2\), not the losses that remain above that threshold.
Worked Example: Tracking Survivor Weights at Two Loss Amounts
A hypothetical portfolio has equal prior proportions of two severity components. Component A is Weibull with shape \(2\) and scale \(1\), so \(S_A(x)=e^{-x^2}\) and \(h_A(x)=2x\). Component B is exponential with rate \(0.5\) per dollar, so \(S_B(x)=e^{-0.5x}\) and \(h_B(x)=0.5\). Calculate the survivor proportions and mixture hazard at \(x=1\) and \(x=2\).
At \(x=1\), the weighted survival contributions are \(0.5e^{-1}=0.1839397\) for A and \(0.5e^{-0.5}=0.3032653\) for B. Total survival is \(0.4872050\). Dividing each contribution by the total gives survivor proportions \(0.3775\) for A and \(0.6225\) for B (more precisely, approximately \(0.377541\) and \(0.622459\)). The component hazards are \(h_A(1)=2\) and \(h_B(1)=0.5\), so \(h_X(1)=0.377541(2)+0.622459(0.5)=1.0663\) per dollar.
At \(x=2\), the weighted survival contributions are \(0.5e^{-4}=0.0091578\) for A and \(0.5e^{-1}=0.1839397\) for B. Total survival is \(0.1930975\). Thus, the survivor proportions are \(0.0091578/0.1930975=0.0474\) for A and \(0.1839397/0.1930975=0.9526\) for B (more precisely, approximately \(0.047426\) and \(0.952574\)). The hazards are now \(h_A(2)=4\) and \(h_B(2)=0.5\). Therefore, \(h_X(2)=0.047426(4)+0.952574(0.5)=0.6660\) per dollar.
The original proportions remain \(0.50\) and \(0.50\), but the proportions among survivors change substantially as the threshold rises. For example, using the original proportions at \(x=2\) would give \(0.50(4)+0.50(0.5)=2.25\), not the correct \(0.6660\). The correct calculation conditions the component identity on survival past \(x\). This example also shows that the mixture hazard can change as the component composition changes; it does not establish a general rule that every mixture hazard must decrease.
Worked Example: A Three-Component Mixture at a Fixed Threshold
A hypothetical loss mixture has three exponential components with rates \(0.25\), \(0.50\), and \(1.00\) per dollar and prior proportions \(0.20\), \(0.30\), and \(0.50\), respectively. Find the proportions among losses exceeding \(1\) dollar and the mixture hazard there. Give the hazard to four decimal places.
For an exponential component with rate \(\lambda\), \(S(x)=e^{-\lambda x}\) and \(h(x)=\lambda\). At \(x=1\), the three component survival probabilities are \(e^{-0.25}=0.7788008\), \(e^{-0.50}=0.6065307\), and \(e^{-1}=0.3678794\). After multiplying by the prior proportions, the weighted survival contributions are \(0.1557602\), \(0.1819592\), and \(0.1839397\). Their sum is \(S_X(1)=0.5216591\).
Divide each contribution by \(0.5216591\). The survivor proportions are approximately \(0.298586\), \(0.348808\), and \(0.352606\), which sum to \(1.000000\) at the shown precision. The mixture hazard is therefore \(0.298586(0.25)+0.348808(0.50)+0.352606(1.00)\). The three hazard contributions are approximately \(0.074647\), \(0.174404\), and \(0.352606\); their sum is \(0.601657\), or \(0.6017\) per dollar to four decimal places.
As a direct check, the weighted density contributions are \(0.1557602(0.25)=0.0389400\), \(0.1819592(0.50)=0.0909796\), and \(0.1839397(1)=0.1839397\). Their sum is \(0.3138594\), and \(0.3138594/0.5216591=0.6017\) to four decimal places. The two methods agree.
Interpreting Changing Component Weights
The formula \(P(j\mid X>x)=p_jS_j(x)/S_X(x)\) gives a precise meaning to changing component weights. A component’s survivor proportion rises relative to another component when its survival probability declines less rapidly over the relevant range, relative to the other component’s survival probability. The threshold-specific weights, rather than the prior proportions alone, determine which component hazards have the greatest influence on \(h_X(x)\).
There is no general guarantee that a mixture hazard is increasing or decreasing. The component hazards themselves may change with \(x\), and the survivor weights also change with \(x\). Both effects enter the mixture hazard. Even if the component hazards are constant, changes in the component proportions among survivors can change the mixture hazard. Conversely, the formula does not imply that the mixture hazard must move in one direction for every choice of components and weights.
| Quantity | Meaning | Use in the mixture hazard |
|---|---|---|
| \(p_j\) | Original probability of component \(j\) | Weights the component survival and density when forming the mixture. |
| \(p_jS_j(x)\) | Probability contribution from component \(j\) to \(X>x\) | Numerator used to update the component proportion among survivors. |
| \(P(j\mid X>x)\) | Proportion from component \(j\) among losses exceeding \(x\) | Weight multiplying \(h_j(x)\) in \(h_X(x)\). |
| \(h_j(x)\) | Component \(j\)’s hazard at \(x\) | Contributes according to the component’s current survivor proportion. |
Common Mistakes and a Practical Check
The most common error is to write \(h_X(x)=\sum_j p_jh_j(x)\) as though the original weights remain appropriate at every loss amount. That expression is valid at a particular \(x\) only if the weights among survivors equal the original weights there; this occurs, for example, when all component survival probabilities at \(x\) are equal. In general, calculate \(p_jS_j(x)\), add those contributions to obtain \(S_X(x)\), and divide each contribution by the total before averaging the component hazards.
A second useful check is to calculate the hazard both ways: first as the survivor-weighted average of component hazards, and then as the mixture density divided by the mixture survival. Agreement checks both the updated weights and the component hazard calculations. Also verify that the updated weights are nonnegative and sum to one, and that the final hazard lies between the component hazards at the threshold.
Check Your Understanding
For each problem, calculate component proportions among survivors before averaging the component hazards.
- A hypothetical mixture has prior proportions \(0.40\) and \(0.60\). At \(x=1\), its two components have survival probabilities \(e^{-0.2}\) and \(e^{-0.25}\), and hazards \(0.2\) and \(0.5\) per dollar, respectively. Find the mixture hazard at \(x=1\). (Answer: \(0.3764\) per dollar.)
- A hypothetical mixture has equal prior proportions of a Weibull component with shape \(2\), scale \(1\), and an exponential component with rate \(0.5\) per dollar. Find the survivor proportions and mixture hazard at \(x=2\). (Answer: Proportions approximately \(0.0474\) and \(0.9526\); hazard \(0.6660\) per dollar.)
- A three-component exponential mixture has rates \(0.25\), \(0.50\), and \(1.00\) per dollar and prior proportions \(0.20\), \(0.30\), and \(0.50\). Find the mixture hazard at \(x=1\). (Answer: \(0.6017\) per dollar.)
- A hypothetical mixture has two exponential components with rates \(\ln 2\) and \(\ln 4\) per dollar and equal prior proportions. At \(x=1\), find the survivor proportions and mixture hazard. (Answer: Proportions \(2/3\) and \(1/3\); hazard \((4/3)\ln 2\approx0.9242\) per dollar.)
- At a specified threshold, two components have prior proportions \(0.70\) and \(0.30\), survival probabilities \(0.80\) and \(0.20\), and hazards \(0.40\) and \(1.50\) per dollar. Find the proportions among survivors and the mixture hazard. (Answer: Proportions approximately \(0.9032\) and \(0.0968\); hazard \(0.5065\) per dollar.)