Why Plot a Fitted Severity Model?
A fitted severity distribution can summarize the center of a loss sample reasonably well while missing its spread or upper tail. A plot compares the ordered observed losses with what the fitted model predicts at corresponding probability levels. The comparison is visual rather than a formal acceptance test: it can flag a pattern worth investigating, but it does not establish that a model is correct or incorrect.
Probability plots are especially useful because the direction and shape of departures are interpretable. A nearly straight pattern supports the model’s distributional shape over the range shown. A consistent shift can suggest a location mismatch; a change in slope can suggest a scale mismatch; and systematic curvature or a tail departure can suggest a shape mismatch. These are diagnostic clues, not unique explanations.
This tutorial’s plotting methods are aligned with the SOA Exam FAM (Fundamentals of Actuarial Mathematics) syllabus. Assume throughout that losses are positive and observed as a complete sample, unless a problem states otherwise. Censoring, truncation, rounding, and grouping can alter what information the observations provide; do not treat them as ordinary complete observations without accounting for that sampling structure.
Constructing the Two Plot Types
Let \(x_1,\ldots,x_n\) be a complete sample of losses, and write the ordered values as \(x_{(1)}\leq\cdots\leq x_{(n)}\). A plotting position \(p_i\) assigns an approximate cumulative probability to \(x_{(i)}\). One common choice is the Weibull plotting position \(p_i=i/(n+1)\), for \(i=1,\ldots,n\). It keeps all positions strictly between zero and one, which is useful when calculating quantiles of distributions whose endpoint quantiles are infinite.
Plotting-position conventions are not unique. Another convention can give slightly different positions, particularly in a small sample. Use the convention specified in a problem; if none is specified, state the one used and apply it consistently.
| Plot | Horizontal coordinate | Vertical coordinate | Reference |
|---|---|---|---|
| Quantile plot | Fitted quantile \(F^{-1}(p_i)\) | Observed loss \(x_{(i)}\) | Diagonal \(y=x\) |
| Probability plot | Plotting position \(p_i\) | Fitted cumulative probability \(F(x_{(i)})\) | Diagonal \(y=x\) |
Here \(F\) denotes the fitted cumulative distribution function and \(F^{-1}(p)\) its \(p\)-quantile. Because a fitted CDF is increasing, these two plots encode the same comparison in different coordinates. In the quantile plot, a point above the diagonal means \(x_{(i)}>F^{-1}(p_i)\). Equivalently, in the probability plot it means \(F(x_{(i)})>p_i\): the observed loss is above the model quantile corresponding to its plotting position. This direction is easy to reverse when switching between plot types.
Why the Probability-Plot Comparison Works
The relationship follows directly from the definition of a quantile and the monotonicity of a continuous CDF. For a continuous, strictly increasing fitted CDF \(F\), its quantile \(F^{-1}(p)\) satisfies \(F(F^{-1}(p))=p\). Therefore:
Thus the observed loss lying above its fitted quantile corresponds to a probability-plot point above the diagonal. If the fitted distribution has a discontinuity or a non-unique inverse, quantile definitions require additional care; the equivalence in this simple form is stated for continuous, strictly increasing \(F\).
Worked Example: A Shifted Exponential Fit
Worked Example: Reading Both Plots for Exponential Severity
A hypothetical portfolio produces five complete losses, in thousands of dollars: \(12.0,\ 13.3,\ 15.0,\ 17.8,\ 24.0\). Suppose a fitted shifted exponential model has location \(\theta=10\) and scale \(\beta=5\), also in thousands of dollars. Its CDF is \(F(x)=1-\exp(-(x-\theta)/\beta)\) for \(x\geq\theta\), and its quantile is \(F^{-1}(p)=\theta-\beta\ln(1-p)\). Use Weibull positions. Construct both plot coordinates and comment on the pattern.
Method. Order the losses (they are already ordered), assign positions \(i/(n+1)=i/6\), calculate the fitted quantiles, and compare observations with those quantiles. This separates a point’s direction from the size of its departure.
Plotting positions. For \(n=5\), the positions are \(1/6,\ 2/6,\ 3/6,\ 4/6,\ 5/6\), or \(0.1667,\ 0.3333,\ 0.5000,\ 0.6667,\ 0.8333\), to four decimal places.
Fitted quantiles. Using the exact positions before rounding, \(F^{-1}(p)=10-5\ln(1-p)\). The five quantiles, in thousands, are \(10-5\ln(5/6)=10.9116\), \(10-5\ln(4/6)=12.0273\), \(10-5\ln(3/6)=13.4657\), \(10-5\ln(2/6)=15.4931\), and \(10-5\ln(1/6)=18.9588\), each rounded to four decimals.
Quantile-plot coordinates and residuals. The horizontal coordinates are those fitted quantiles, and the vertical coordinates are the observed losses. Observed minus fitted quantile residuals are \(12.0-10.9116=1.0884\), \(13.3-12.0273=1.2727\), \(15.0-13.4657=1.5343\), \(17.8-15.4931=2.3069\), and \(24.0-18.9588=5.0412\), in thousands of dollars. All points are above the diagonal, and the departures grow in the upper part of the sample.
Probability-plot coordinates. Evaluate the fitted CDF at each observed loss. The resulting probabilities are \(F(12)=1-e^{-0.4}=0.3297\), \(F(13.3)=1-e^{-0.66}=0.4831\), \(F(15)=1-e^{-1}=0.6321\), \(F(17.8)=1-e^{-1.56}=0.7899\), and \(F(24)=1-e^{-2.8}=0.9392\), to four decimal places. Pair these vertical coordinates with the plotting positions above. Each fitted probability exceeds its plotting position, consistent with every observed loss exceeding its corresponding fitted quantile.
Conclusion and check. The increasingly large upper-end residuals suggest that this fitted model may understate the upper losses in this small sample. They do not, by themselves, prove a particular alternative model is appropriate. As a direction check, the largest loss is \(24.0\), greater than its fitted \(0.8333\)-quantile \(18.9588\), and \(F(24)=0.9392>0.8333\). Both plot conventions therefore place the point above the diagonal.
Location, Scale, and Shape Patterns
Interpret a plot by looking for a pattern across several ordered observations rather than reacting to one point. In a quantile plot, the vertical difference \(x_{(i)}-F^{-1}(p_i)\) is a convenient diagnostic residual. Its pattern can suggest what feature of a fitted model deserves review.
- Location mismatch: residuals that are roughly constant across the plot suggest that the observations are shifted relative to the fitted quantiles. For a location-scale family, a location parameter shifts all quantiles by the same additive amount.
- Scale mismatch: residuals that tend to be negative at lower quantiles and positive at upper quantiles suggest the observed sample is more spread out than the fitted quantiles. The reverse pattern suggests less spread. For a location-scale family, increasing scale moves lower quantiles down and upper quantiles up relative to the center.
- Shape mismatch: systematic curvature, changing residual patterns, or a pronounced tail departure can indicate that the distributional shape is not capturing the sample. The precise interpretation depends on the model and the plot’s axes; curvature is not a unique diagnosis.
For some distributions, a transformation gives a familiar straight-line diagnostic. If \(X\) has a lognormal distribution with \(\ln X\sim N(\mu,\sigma^2)\), then the theoretical log quantile is \(\ln F^{-1}(p)=\mu+\sigma\Phi^{-1}(p)\), where \(\Phi\) is the standard normal CDF. Plotting \(\ln x_{(i)}\) against \(\Phi^{-1}(p_i)\) should therefore produce an approximately straight line if the lognormal model is suitable. Its intercept corresponds to \(\mu\), and its slope to \(\sigma\). This linearity follows from the lognormal model definition; it does not make the plot a formal test.
Worked Example: A Lognormal Plot with an Upper-Tail Departure
Consider hypothetical losses in dollars \(301.0,\ 606.5,\ 1000.0,\ 1648.7,\ 6049.6\). A fitted lognormal model has \(\ln(X/1000)\sim N(0,1)\). Use Weibull positions and examine the normal-score plot, with horizontal coordinate \(z_i=\Phi^{-1}(p_i)\) and vertical coordinate \(y_i=\ln(x_{(i)}/1000)\). Values are rounded for display; the calculations use the stated unrounded model and data-generating expressions \(1000e^{-1.2},1000e^{-0.5},1000,1000e^{0.5},1000e^{1.8}\).
Positions and model scores. The positions are \(0.1667,\ 0.3333,\ 0.5000,\ 0.6667,\ 0.8333\). Their standard normal quantiles are approximately \(-0.9674,\ -0.4307,\ 0,\ 0.4307,\ 0.9674\), to four decimal places.
Observed log values and departures. By construction, the observed log values \(y_i\) are \(-1.2,\ -0.5,\ 0,\ 0.5,\ 1.8\). Since the fitted model has \(\mu=0\) and \(\sigma=1\), its fitted log quantile at each position is \(z_i\). Thus \(y_i-z_i\) is approximately \(-0.2326,\ -0.0693,\ 0,\ 0.0693,\ 0.8326\). The first four points lie fairly close to the reference line, while the largest observation has a much larger positive departure.
Conclusion and check. The plot suggests an upper-tail departure from this fitted lognormal model; it does not establish why that point is large or whether another model will fit better. The last observed log value is \(1.8\), compared with a fitted log quantile of \(0.9674\), so it is above its fitted quantile, as its positive residual indicates. In dollars, the observed value is about \(6049.6\), while the fitted quantile is \(1000e^{0.9674}\), approximately \(2631.2\).
Worked Example: Recognizing a Location Shift
A hypothetical sample has four ordered losses, in dollars: \(109.4629,\ 115.2165,\ 123.3258,\ 137.1888\). A fitted shifted exponential model uses \(\theta=100\) and \(\beta=20\). Its quantile is \(100-20\ln(1-p)\). Use the Weibull positions for \(n=4\), and decide whether the departures look more like a location or a scale mismatch.
Positions and quantiles. The positions are \(0.2,\ 0.4,\ 0.6,\ 0.8\). The fitted quantiles are \(100-20\ln(0.8)=104.4629\), \(100-20\ln(0.6)=110.2165\), \(100-20\ln(0.4)=118.3258\), and \(100-20\ln(0.2)=132.1888\), to four decimals.
Residuals. Subtract each fitted quantile from the corresponding observation: \(109.4629-104.4629=5.0000\), \(115.2165-110.2165=5.0000\), \(123.3258-118.3258=5.0000\), and \(137.1888-132.1888=5.0000\), to the displayed precision. The nearly constant positive residuals are consistent with an additive location shift, rather than a spread mismatch that would typically produce residuals changing systematically across quantiles.
Probability-plot check. For this model, \(F(x)=1-\exp(-(x-100)/20)\). Since each observation is approximately its fitted quantile plus \(5\), its fitted probability is \(1-(1-p)e^{-5/20}\). At positions \(0.2,\ 0.4,\ 0.6,\ 0.8\), these probabilities are approximately \(0.3770,\ 0.5327,\ 0.6885,\ 0.8442\). Each exceeds its plotting position, matching the positive quantile residuals. The sample is deliberately small, so the pattern is illustrative rather than conclusive.
Common Traps and Practical Limits
- Reversing the direction. If \(F(x_{(i)})>p_i\), then \(x_{(i)}>F^{-1}(p_i)\) for a continuous, strictly increasing fitted CDF. The observed loss is above, not below, its corresponding fitted quantile.
- Mixing plot axes. State the axes before interpreting a point. The quantile plot uses fitted quantiles and observed losses; the probability plot uses plotting positions and fitted CDF values.
- Using endpoints. Positions of zero or one can require infinite quantiles for common severity distributions. Positions such as \(i/(n+1)\) avoid those endpoints.
- Overinterpreting one extreme point. A single large loss may reflect sampling variability or data issues as well as a tail mismatch. Check data quality and the broader pattern before changing the model.
- Ignoring the fitting process. When parameters were estimated from the same observations being plotted, the plot reflects both sampling variation and parameter estimation. Visual closeness is not a calibrated test of fit.
Probability plots matter because they make model discrepancies visible on the scale of losses or probabilities, including discrepancies that a center-focused summary can conceal. They are most informative when the sample is correctly ordered, the plotting-position convention is explicit, the fitted model’s support is respected, and the observed pattern is described before proposing an explanation.
Check Your Understanding
Use the stated plotting-position convention and distinguish a visual clue from a formal conclusion.
- A sample has three ordered losses and uses Weibull positions. For a fitted exponential model with location \(0\) and scale \(4\), calculate the fitted quantiles at the three positions to four decimals. (Answer: Positions \(0.25,\ 0.50,\ 0.75\); fitted quantiles \(1.1507,\ 2.7726,\ 5.5452\).)
- A fitted lognormal model has \(\ln(X/100)\sim N(0,0.04)\). Find its \(0.90\)-quantile, using \(\Phi^{-1}(0.90)=1.2816\), to the nearest cent. (Answer: \(100e^{0.2(1.2816)}\approx \$129.22\).)
- On a probability plot with horizontal coordinate \(p_i=0.60\), an observation has fitted probability \(F(x_{(i)})=0.75\). State which side of the diagonal the point is on and compare the observation with its fitted quantile. (Answer: Above the diagonal; \(x_{(i)}>F^{-1}(0.60)\).)
- In a quantile plot, residuals are approximately \(-4,-2,0,2,4\) from the lowest to highest plotted quantile. What broad feature does this pattern suggest? (Answer: The observed sample is more spread out than the fitted quantiles, suggesting a possible scale mismatch.)
- A shifted exponential quantile plot shows residuals close to \(+3\) dollars at every plotted position. What mismatch is suggested, and what limitation should be stated? (Answer: An additive location mismatch is suggested; the pattern is only a visual clue and does not establish that the model is wrong.)