Why Category Boundaries Matter
A chi-square goodness-of-fit test compares observed counts with the counts expected under a candidate model. Before making that comparison, the loss observations must be placed into categories. The boundaries determine which observations are counted together, and therefore which departures from the model the test can detect.
There is no universally best partition. Candidate-model quantiles give a systematic way to create categories with equal expected probabilities. Actuarial thresholds, such as a deductible or a coverage limit, can make the categories easier to interpret and can focus attention on a financially important part of the loss distribution. The choice should be made with both the model probabilities and the purpose of the analysis in mind.
For nonnegative losses, the first category will often be \([0,b_1]\) rather than \((-\infty,b_1]\). What matters is that the categories cover the full support of the observations, do not overlap, and use a consistent rule for assigning observations exactly on a boundary. Under the candidate model, their probabilities must sum to one.
Using Candidate-Model Quantiles
If the candidate model is continuous and its quantiles are available, equal-probability categories are obtained by setting the internal boundaries at the corresponding quantiles. For \(k\) categories, choose \(b_j\) so that \(F_X(b_j)=j/k\), for \(j=1,\ldots,k-1\). This gives each category probability \(1/k\) and expected count \(n/k\).
Equal expected counts can make the comparison straightforward: every category has the same benchmark count, so unusually large or small observed counts are easy to spot. The boundaries also adapt to the candidate distribution. For a highly skewed severity model, for example, equal-probability intervals may be narrow among small losses and wide in the tail.
Worked Example: Four Equal-Probability Categories
A hypothetical portfolio has 100 independent positive losses. Under the null model, each loss follows an exponential distribution with mean 100 dollars, so \(F_X(x)=1-e^{-x/100}\) for \(x\geq 0\). Construct four equal-probability categories and calculate the Pearson statistic if the observed counts are 18, 27, 31, and 24.
Method. Use the model quartiles as the three internal boundaries. Solving \(F_X(b_j)=j/4\) gives \(b_j=-100\ln(1-j/4)\).
For the first boundary, \(b_1=-100\ln(0.75)=28.768\) dollars. For the second, \(b_2=-100\ln(0.50)=69.315\) dollars. For the third, \(b_3=-100\ln(0.25)=138.629\) dollars. These produce the categories \([0,28.768]\), \((28.768,69.315]\), \((69.315,138.629]\), and \((138.629,\infty)\), with boundary values rounded for display.
Each category has probability \(0.25\), so its expected count is \(100(0.25)=25\). The Pearson chi-square statistic is \(\sum_j (O_j-E_j)^2/E_j\), where \(O_j\) is the observed count. The category contributions are \( (18-25)^2/25=49/25=1.96\), \( (27-25)^2/25=4/25=0.16\), \( (31-25)^2/25=36/25=1.44\), and \( (24-25)^2/25=1/25=0.04\). Therefore, the statistic is \(1.96+0.16+1.44+0.04=3.60\).
Check. The expected counts sum to 100, as do the observed counts. The statistic is 3.600. The common mistake this method avoids is using equally spaced dollar boundaries and assuming they have equal probabilities under a skewed model.
Using Actuarial Thresholds
Quantiles are not the only defensible boundaries. A deductible, attachment point, limit, or other contract threshold may be more useful when the analysis is intended to assess a specific part of the loss distribution. For example, categories below and above a deductible can show whether the candidate model describes losses near that threshold adequately.
With actuarial thresholds, calculate each category probability from the candidate model rather than assuming the categories have equal probabilities. For thresholds \(c_1<c_2\), the probabilities are \(F_X(c_1)\), \(F_X(c_2)-F_X(c_1)\), and \(1-F_X(c_2)\). The corresponding expected counts are these probabilities multiplied by \(n\). The threshold changes the partition, not the candidate model.
Worked Example: Categories at Contract Thresholds
A hypothetical insurer reviews 200 independent positive claim severities. The candidate model is exponential with mean 100 dollars. For interpretation, an analyst uses thresholds of 50, 100, and 200 dollars. The observed counts in the four resulting categories are 85, 40, 45, and 30. Calculate the model probabilities, expected counts, and Pearson statistic.
Method. Use \([0,50]\), \((50,100]\), \((100,200]\), and \((200,\infty)\). Under this model, \(F_X(x)=1-e^{-x/100}\), so the category probabilities are \(1-e^{-0.5}\), \(e^{-0.5}-e^{-1}\), \(e^{-1}-e^{-2}\), and \(e^{-2}\).
Numerically, these probabilities are 0.393469, 0.238651, 0.232544, and 0.135335. They sum to 1.000000. Multiplying by 200 gives expected counts of 78.694, 47.730, 46.509, and 27.067. These expected counts also sum to 200, apart from displayed rounding.
The contributions are approximately \( (85-78.694)^2/78.694=0.505\), \( (40-47.730)^2/47.730=1.252\), \( (45-46.509)^2/46.509=0.049\), and \( (30-27.067)^2/27.067=0.318\). Their sum is \(2.124\), to three decimal places.
Check. The largest contribution is from the second category, where the observed count is below expectation. The common mistake this method avoids is assigning 50 observations to every category merely because there are four categories; the chosen thresholds do not have equal model probabilities.
How Boundaries Affect Sensitivity
A category-based test detects differences in category counts, not the exact locations of individual losses within a category. Two distinct loss patterns can therefore produce identical counts for one partition. Changing the boundaries can uncover a departure hidden by the first grouping, or make a departure less visible by combining losses with different behavior.
The following hypothetical comparison uses the same set of 100 losses and the same exponential candidate model with mean 100 dollars. It shows how a partition aligned with the model quantiles and one aligned with round-dollar thresholds can summarize the data differently.
Worked Example: Comparing Quantile and Threshold Partitions
Suppose the 100 losses consist of 30 values of 20 dollars, 10 of 40 dollars, 20 of 60 dollars, 10 of 80 dollars, 15 of 120 dollars, 10 of 150 dollars, and 5 of 250 dollars. Under the candidate model, the quartile boundaries are 28.768, 69.315, and 138.629 dollars. Compare the Pearson statistics for the quartile categories and for categories split at 50, 100, and 200 dollars.
Quantile partition. The observed counts are 30, 30, 25, and 15. For example, the second category contains the losses of 40 and 60 dollars, giving \(10+20=30\). The expected counts are 25 in each category. The contributions are \( (30-25)^2/25=1\), \( (30-25)^2/25=1\), \( (25-25)^2/25=0\), and \( (15-25)^2/25=4\). Thus the statistic is \(1+1+0+4=6.00\).
Threshold partition. At 50, 100, and 200 dollars, the observed counts are 40, 30, 25, and 5. The exponential model gives probabilities 0.393469, 0.238651, 0.232544, and 0.135335, so for 100 losses the expected counts are 39.347, 23.865, 23.254, and 13.534.
The corresponding contributions are approximately \( (40-39.347)^2/39.347=0.011\), \( (30-23.865)^2/23.865=1.577\), \( (25-23.254)^2/23.254=0.131\), and \( (5-13.534)^2/13.534=5.381\). The statistic is approximately \(7.100\), to three decimal places.
Check. Both partitions use four categories and account for all 100 observations, but the threshold partition gives greater weight to the shortfall above 200 dollars. That is why its statistic is larger for these data; it does not establish that threshold partitions are always more sensitive. The common mistake is to treat a larger statistic from one chosen grouping as proof that the grouping is universally superior.
Choosing and Reporting a Partition
A defensible choice begins with the question the test is intended to answer. Quantile boundaries are useful when a balanced model-based partition is desired. Contract thresholds are useful when the interpretation depends on losses relative to particular attachment points or limits. Either choice requires probabilities calculated from the same candidate model being tested.
Whenever possible, set boundaries before examining the observed category counts. If boundaries are repeatedly altered after seeing the results, the reported statistic reflects both the random data and the search for a partition that looks unusual. Its reference distribution may no longer be the one assumed for a prespecified test. If a partition is informed by the data, that choice needs to be addressed in the testing procedure rather than treated as though the boundaries were fixed in advance.
The categories must also be exhaustive and mutually exclusive, and the expected counts should be examined before relying on the chi-square approximation. How to handle small expected counts is a separate issue; choosing boundaries does not by itself guarantee that an approximation is appropriate. If parameters were estimated from the same data, the test’s degrees of freedom and calibration must also reflect that estimation, as covered in the earlier material on chi-square degrees of freedom after fitting.
Check Your Understanding
For each question, state the relevant probabilities or counts before giving your conclusion.
- A hypothetical severity model is exponential with mean 80 dollars, and the sample size is 120. Give the three boundaries for four equal-probability categories and the expected count in each. (Answer: Boundaries are approximately 23.014, 55.452, and 110.904 dollars; each expected count is 30.)
- A hypothetical exponential model has mean 50 dollars. For 100 losses, categories are \([0,50]\), \((50,100]\), and \((100,\infty)\). Give the expected counts, to two decimal places. (Answer: 63.21, 23.25, and 13.53.)
- For a candidate continuous model, the CDF values at three chosen boundaries are 0.2, 0.6, and 0.9. If \(n=240\), what are the four category probabilities and expected counts? (Answer: Probabilities 0.2, 0.4, 0.3, and 0.1; expected counts 48, 96, 72, and 24.)
- A four-category partition has expected count 20 in every category and observed counts 14, 18, 26, and 22. Calculate the Pearson statistic. (Answer: \(4.000\).)
- A hypothetical exponential model has mean 100 dollars and sample size 100. At boundaries 100 and 200 dollars, observed counts are 65, 25, and 10. Calculate the Pearson statistic to three decimals. (Answer: Approximately \(1.104\).)