Constructing The Empirical Loss Distribution

Introduction

A set of exact loss records can be summarized by asking what fraction of the observed losses are no greater than a selected amount. Answering that question at one threshold is a count; answering it at every possible threshold defines the empirical cumulative distribution function. It provides a direct, data-based description of the observed loss distribution without requiring a fitted parametric model.

The construction is straightforward but sensitive to details. Records must be sorted or counted accurately, every observation must have the same weight, and a loss exactly equal to the threshold must be handled consistently with the definition. This tutorial builds the function from exact losses and evaluates it at specified thresholds.

Definition: Let $X_1,\ldots,X_n$ be $n$ exact loss observations, each recorded in the same units. The empirical cumulative distribution function is $$ \widehat F_n(x)=\frac{1}{n}\sum_{i=1}^{n} I(X_i\le x) =\frac{\#\{i:X_i\le x\}}{n}, $$ where $I(X_i\le x)$ equals 1 when $X_i\le x$ and 0 otherwise. Thus, $\widehat F_n(x)$ is the proportion of the observed losses that are less than or equal to $x$.

The symbol $F_X(x)$ commonly denotes the distribution function of a loss variable $X$, while $\widehat F_n(x)$ denotes the empirical version computed from the observed sample. The empirical function describes the sample exactly. It may also be used as an estimate of an underlying loss distribution when the observations are considered representative, but that interpretation depends on the data collection and sampling context.

From Sorted Records to Cumulative Proportions

Write the losses in increasing order as $X_{(1)}\le X_{(2)}\le\cdots\le X_{(n)}$. The parentheses indicate order, not a new observation. Once the observations are sorted, evaluating the empirical distribution at $x$ amounts to finding how many ordered values are at most $x$, then dividing that count by $n$.

If exactly $r$ of the sorted observations are no greater than $x$, then $\widehat F_n(x)=r/n$. If $x$ falls between two consecutive ordered observations, the count is the number of observations below that interval. If $x$ equals an observed loss, that loss is included because the definition uses $\le$.

The result is a proportion between 0 and 1. A useful basic check is that $\widehat F_n(x)=0$ for any $x$ below the smallest observation, and $\widehat F_n(x)=1$ for any $x$ at least as large as the greatest observation. These checks do not replace the count, but they can identify an arithmetic or sorting error.

Worked Example: Evaluating the Function at Two Thresholds

A hypothetical insurer has six exact reported losses, in dollars: $18{,}000,\ 4{,}000,\ 12{,}000,\ 7{,}000,\ 25{,}000,\ 9{,}000$. Construct the sorted observations and calculate $\widehat F_6(10{,}000)$ and $\widehat F_6(18{,}000)$.

Method. Sort the records, count those at or below each threshold, and divide by the six observations.

The sorted losses are $4{,}000,\ 7{,}000,\ 9{,}000,\ 12{,}000,\ 18{,}000,\ 25{,}000$. Therefore $n=6$.

At $x=10{,}000$, the losses no greater than the threshold are $4{,}000,\ 7{,}000,$ and $9{,}000$. The count is 3, so $$ \widehat F_6(10{,}000)=\frac{3}{6}=0.5000. $$

At $x=18{,}000$, the losses no greater than the threshold are $4{,}000,\ 7{,}000,\ 9{,}000,\ 12{,}000,$ and $18{,}000$. The count is 5, so $$ \widehat F_6(18{,}000)=\frac{5}{6}=0.8333, $$ rounded to four decimal places.

Check. The second threshold includes one more observed loss than the first, so its empirical proportion is larger. The common mistake avoided here is excluding the loss equal to $18{,}000$: it is included by the $\le$ in the definition.

Why the Count Formula Works

The empirical distribution treats the observed records as equally weighted. At a selected threshold, each record contributes either one unit to the count if it is at or below the threshold, or zero otherwise. Dividing the total contribution by $n$ converts the count into the fraction of records meeting the condition.

This also explains how sorting supports the calculation. If $r$ ordered records are no greater than $x$, then the indicators for the first $r$ records equal 1, and the remaining $n-r$ indicators equal 0. Their sum is therefore $r$, giving the following equivalent form:

\widehat F_n(x) =\frac{1}{n}\left( \underbrace{1+\cdots+1}_{r\text{ contributions}} +\underbrace{0+\cdots+0}_{n-r\text{ contributions}} \right) =\frac{r}{n}.

No assumption about a particular severity distribution is needed to construct this sample summary. The loss records must, however, be exact observations for this calculation: an incomplete record that only establishes a lower or upper bound does not provide an exact $X_i$ to compare with every threshold. Such incomplete-data methods require additional treatment.

Confirm the data and units.
Identify the exact loss records to be used and check that each loss and each requested threshold are expressed in the same units.

Count the records.
Set $n$ equal to the number of observations. Do not reduce the denominator because some records are below the threshold.

Sort the losses.
Arrange the observations in increasing order to make the threshold count visible and easier to verify.

Apply the threshold condition.
Count each observation satisfying $X_i\le x$, including an observation equal to $x$.

Divide and check.
Calculate the count divided by $n$, report the requested precision, and check that the answer is between 0 and 1.

Several Thresholds and a Cumulative Table

When several values are requested, it is usually more reliable to sort once and reuse the ordered list than to restart the calculation from the original data for every threshold. A cumulative table can record the ordered loss, its position, and the proportion accumulated through that position. For a threshold between listed losses, use the cumulative count reached before the next larger loss.

The table below is not a new model or a fitted distribution. It is a convenient accounting device for the observed records. The cumulative proportion at a row is the row's cumulative count divided by the total number of observations.

Worked Example: Building and Using a Cumulative Table

A hypothetical portfolio has eight exact losses recorded in thousands of dollars: $8.1,\ 3.1,\ 9.4,\ 1.5,\ 5.0,\ 2.7,\ 6.8,\ 4.2$. Construct the cumulative table and calculate $\widehat F_8(4.5)$ and $\widehat F_8(5.0)$. Give proportions to four decimal places.

Method. Sort the data, assign successive cumulative counts, and divide each count by $n=8$. All losses and thresholds in the calculation are in thousands of dollars.

The ordered losses are $1.5,\ 2.7,\ 3.1,\ 4.2,\ 5.0,\ 6.8,\ 8.1,\ 9.4$. The cumulative table is:

Ordered loss (thousands of dollars)	Cumulative count	Cumulative proportion
1.5	1	1/8 = 0.1250
2.7	2	2/8 = 0.2500
3.1	3	3/8 = 0.3750
4.2	4	4/8 = 0.5000
5.0	5	5/8 = 0.6250
6.8	6	6/8 = 0.7500
8.1	7	7/8 = 0.8750
9.4	8	8/8 = 1.0000

For $x=4.5$, the first four ordered losses are no greater than the threshold, while $5.0$ and all larger losses are above it. Thus $$ \widehat F_8(4.5)=\frac{4}{8}=0.5000. $$

For $x=5.0$, the first five losses are no greater than the threshold, including the loss equal to $5.0$. Thus $$ \widehat F_8(5.0)=\frac{5}{8}=0.6250. $$

Check. The threshold increases from 4.5 to 5.0 and passes one observed loss, so the cumulative proportion increases by $1/8=0.1250$. The common mistake avoided is comparing a threshold stated in dollars with losses recorded in thousands without first converting one of them.

Interpretation and Common Errors

An empirical value such as $\widehat F_n(x)=0.6250$ means that 62.50% of the observed loss records are no greater than $x$. It does not, by itself, establish that exactly 62.50% of losses in a broader population are no greater than $x$. That population interpretation treats the sample as informative about the population and should be made with appropriate care.

The construction is simple enough that most errors are procedural rather than conceptual. Use the complete observation count in the denominator, compare losses and thresholds in consistent units, and apply the stated inequality carefully. Sorting makes the calculation easier to inspect, but the defining formula still applies if one counts directly from an unsorted list.

Using the wrong denominator. The denominator is $n$, the total number of exact loss observations, not the number below the threshold.
Forgetting the boundary. The definition counts $X_i=x$ as satisfying $X_i\le x$.
Sorting inaccurately. A missed or misplaced observation can change the cumulative count. Check that the sorted list contains exactly the original records.
Mixing units. Convert the threshold or the losses before comparing them; a numerical comparison is meaningless if one value is in dollars and the other is in thousands of dollars.
Interpreting a sample proportion too broadly. The empirical function summarizes observed records; population conclusions require a justified connection between the sample and the population.

Worked Example: Checking a Distribution at Multiple Cutoffs

A hypothetical equipment insurer records nine exact losses, in thousands of dollars: $42,\ 12,\ 28,\ 7,\ 35,\ 19,\ 51,\ 23,\ 15$. Find the empirical distribution values at $x=20$, $x=40$, and $x=51$, and check the overall endpoints.

Method. Sort the nine losses and use the cumulative counts at the specified thresholds. The common mistake to avoid is using the sorted position as the proportion without dividing by the full sample size.

The sorted observations are $7,\ 12,\ 15,\ 19,\ 23,\ 28,\ 35,\ 42,\ 51$, so $n=9$.

At $x=20$, four losses ($7,\ 12,\ 15,\ 19$) are no greater than 20: $$ \widehat F_9(20)=\frac{4}{9}=0.4444. $$

At $x=40$, seven losses ($7,\ 12,\ 15,\ 19,\ 23,\ 28,\ 35$) are no greater than 40: $$ \widehat F_9(40)=\frac{7}{9}=0.7778. $$

At $x=51$, all nine observations are no greater than the threshold: $$ \widehat F_9(51)=\frac{9}{9}=1.0000. $$

For an endpoint check, any threshold below 7 has empirical value 0 because no observation is that small; at 51, the value is 1 because every observation is included. The values $4/9$, $7/9$, and $9/9$ are all between 0 and 1, and they increase as the thresholds increase.

Key takeaway: For exact losses, the empirical cumulative distribution at $x$ is the number of observed losses satisfying $X_i\le x$, divided by the total number $n$. Sorting the observations turns that definition into a transparent cumulative count.

Check Your Understanding

For each problem, sort the records where helpful, count losses at or below the stated threshold, and divide by the full number of observations.

A hypothetical claims file contains exact losses of $11,\ 4,\ 16,\ 7$ thousand dollars. Calculate $\widehat F_4(10)$. (Answer: $2/4=0.5000$)
Exact losses, in thousands of dollars, are $2.4,\ 6.0,\ 1.2,\ 4.8,\ 3.6,\ 9.0$. Calculate $\widehat F_6(4.8)$. (Answer: $4/6=0.6667$)
A small portfolio has exact losses $5,\ 8,\ 10,\ 14,\ 21,\ 27,\ 40$, in thousands of dollars. Calculate $\widehat F_7(14)$ and $\widehat F_7(20)$. (Answer: $4/7=0.5714$ at 14, and $4/7=0.5714$ at 20)
Exact losses are $32,\ 12,\ 24,\ 8,\ 40,\ 16,\ 28,\ 20$, in thousands of dollars. Calculate $\widehat F_8(15)$ and $\widehat F_8(24)$. (Answer: $2/8=0.2500$ at 15, and $5/8=0.6250$ at 24)
A hypothetical data set has nine exact losses $3,\ 6,\ 9,\ 12,\ 15,\ 18,\ 21,\ 24,\ 27$, in thousands of dollars. Calculate $\widehat F_9(17)$ and $\widehat F_9(24)$. Which value is greater? (Answer: $5/9=0.5556$ at 17; $8/9=0.8889$ at 24; the value at 24 is greater.)

What You'll Learn