Strict And Inclusive Loss Thresholds

Introduction

A threshold statement can look simple but hide an important distinction: does “loss of 500” belong in the count? For an observed loss value, the answer changes the empirical probability because one or more records may equal the threshold. The difference is especially visible when several losses are tied there.

This tutorial translates “less than,” “less than or equal to,” “greater than,” and “greater than or equal to” into counts from a finite set of exact loss records. The emphasis is on choosing the correct boundary, not on fitting a probability model. The examples use equally weighted observations; no interest rate, mortality assumption, or parametric distribution is required.

Empirical Counts at a Threshold

Suppose $X_1,\ldots,X_n$ are $n$ exact loss records. Each record receives weight $1/n$, so an empirical probability is the number of records meeting a condition divided by $n$. For a threshold $x$, the four relevant counts are the numbers of records strictly below $x$, at or below $x$, strictly above $x$, and at or above $x$.

Definition: For exact loss records $X_1,\ldots,X_n$, the empirical distribution function is $$ F_n(x)=\frac{1}{n}\sum_{i=1}^{n} I(X_i\le x), $$ where $I(\cdot)$ equals 1 when its condition is true and 0 otherwise. Thus $nF_n(x)$ counts records at or below $x$. The empirical probability of a condition is its count divided by $n$.

The strict version uses $X_i<x$, not $X_i\le x$. Write $F_n(x-)$ for the empirical proportion strictly below $x$, that is, the value immediately to the left of $x$ in the step-shaped empirical distribution. Then the four probabilities are:

\begin{aligned} P_n(X<x)&=F_n(x-), & P_n(X\le x)&=F_n(x),\\ P_n(X>x)&=1-F_n(x), & P_n(X\ge x)&=1-F_n(x-). \end{aligned}

Here $P_n$ denotes probability under the empirical distribution. These are proportions of the recorded losses, not claims about an unknown population unless further statistical assumptions are made. The complementary relationships work because the records not satisfying $X\le x$ are exactly those satisfying $X>x$; similarly, the records not satisfying $X<x$ are exactly those satisfying $X\ge x$.

Why Ties Make the Boundary Matter

If no record equals $x$, then “less than” and “less than or equal to” give the same count. The same is true of “greater than” and “greater than or equal to.” At an observed value, however, the records equal to $x$ separate the strict and inclusive counts.

Key result: If $m_x$ of the $n$ records equal $x$, then $$ F_n(x)-F_n(x-)=\frac{m_x}{n}. $$ The empirical probability at the exact threshold is the size of the jump in the empirical distribution. In count form, the inclusive count is the strict count plus $m_x$.

This result follows directly from the observations counted on each side of the boundary. Every record strictly below $x$ contributes to both counts. The count at or below $x$ includes those same records and also every record equal to $x$. No record strictly above $x$ contributes to either count. Therefore the difference between the two counts is exactly $m_x$; dividing by $n$ gives the formula.

Identify the condition.
Translate the words carefully: “below” means $X<x$, while “at most” means $X\le x$.

Count records at the boundary.
Check whether any losses equal $x$. If so, strict and inclusive counts differ by the number of ties.

Use a complement when convenient.
Count $X>x$ as $n-\#\{X_i\le x\}$, and count $X\ge x$ as $n-\#\{X_i<x\}$.

Divide by the full sample size.
For equally weighted exact records, divide the selected count by $n$, not by the number of distinct loss values.

Worked Threshold Calculations

Worked Example: A Threshold That Is an Observed Loss

A hypothetical insurer has eight exact claim amounts, in dollars: 12, 18, 18, 25, 31, 31, 31, and 44. Find the empirical probabilities of a claim strictly below 31, at most 31, strictly above 31, and at least 31. Also identify the probability mass at 31.

Method. Count the records on each side of the threshold, then divide by $n=8$. The threshold 31 is observed three times.

Strictly below 31. The qualifying amounts are 12, 18, 18, and 25, so the count is 4 and $P_n(X<31)=4/8=0.5000$.

At most 31. The three records equal to 31 are included, giving a count of $4+3=7$. Thus $P_n(X\le31)=7/8=0.8750$.

Strictly above 31. Only 44 qualifies, so $P_n(X>31)=1/8=0.1250$.

At least 31. The three records equal to 31 and the record 44 qualify, giving $P_n(X\ge31)=4/8=0.5000$.

The empirical probability mass at 31 is $3/8=0.3750$. As a check, the strict and inclusive probabilities differ by $0.8750-0.5000=0.3750$, and the two complementary pairs sum to 1. Probabilities are shown to four decimal places.

Common mistake avoided: Treating “at most 31” as if it meant “below 31” would give $4/8$, incorrectly excluding all three claims equal to 31.

Worked Example: “Exceeds” Versus “At Least”

A hypothetical portfolio contains ten exact losses, in thousands of dollars: 0, 0, 250, 250, 400, 600, 600, 900, 1,200, and 1,200. Calculate the proportions of losses that exceed 600 thousand dollars and are at least 600 thousand dollars. Then calculate the proportions below and at most 600 thousand dollars.

Method. Keep the threshold in the stated units and separate the two losses equal to 600 from the losses strictly above it. There are $n=10$ records.

The amounts strictly below 600 are 0, 0, 250, 250, and 400. Their count is 5, so the proportion below the threshold is $5/10=0.5000$.

Including the two records equal to 600 gives a count of $5+2=7$ at or below the threshold. The proportion is $7/10=0.7000$.

Strictly above 600 are 900, 1,200, and 1,200. Therefore, the proportion that exceeds 600 is $3/10=0.3000$.

At least 600 includes the two records equal to 600 as well as those three larger amounts. Its count is $2+3=5$, so the proportion is $5/10=0.5000$.

The results check: $0.5000+0.5000=1$ for below versus at least, and $0.7000+0.3000=1$ for at most versus above. The mass exactly at 600 is $2/10=0.2000$, matching $0.7000-0.5000$.

Common mistake avoided: “Exceeds 600” means strictly greater than 600; it does not include the two losses equal to 600. “At least 600” does include them.

Worked Example: Combining Threshold Counts to Count a Band

A hypothetical set of twelve exact repair losses, in dollars, is 0, 250, 500, 500, 750, 1,000, 1,000, 1,000, 1,250, 1,500, 1,500, and 2,000. Find the proportions below 1,000, at most 1,000, above 1,000, and at least 1,000. Then find the proportion in the band from 500 inclusive to 1,000 exclusive.

Method. First count each threshold condition among all $n=12$ records. For the band, subtract the count below 500 from the count below 1,000.

The records strictly below 1,000 are 0, 250, 500, 500, and 750: five records. Thus $P_n(X<1{,}000)=5/12$.

There are three records equal to 1,000, so the at-most count is $5+3=8$, giving $P_n(X\le1{,}000)=8/12$. The records strictly above 1,000 are 1,250, 1,500, 1,500, and 2,000, so $P_n(X>1{,}000)=4/12$. At least 1,000 includes those four records and the three ties, giving $P_n(X\ge1{,}000)=7/12$.

For $500\le X<1{,}000$, there are five records below 1,000 and two records below 500. Subtracting removes the losses that are too small: $$ \#\{500\le X_i<1{,}000\} =\#\{X_i<1{,}000\}-\#\{X_i<500\} =5-2=3. $$ The three qualifying losses are 500, 500, and 750. The band proportion is $3/12=0.2500$.

As checks, the at-most and above proportions sum to $8/12+4/12=1$; the below and at-least proportions sum to $5/12+7/12=1$. The three losses equal to 1,000 account for the difference $8/12-5/12=3/12$.

Common mistake avoided: For the band $500\le X<1{,}000$, subtracting the count at most 500 would incorrectly exclude the two losses equal to 500. The lower boundary is inclusive, so subtract only the count strictly below 500.

Common Boundary Errors

Threshold wording determines which records belong in the count. A reliable solution makes the inequality explicit before scanning the data. This is particularly important when a record equals the threshold, when several records are tied there, or when a band has one inclusive and one exclusive endpoint.

Confusing “exceeds” with “at least.” “Exceeds $x$” is $X>x$; “at least $x$” is $X\ge x$. The latter includes equality.
Using the wrong complement. The complement of $X\le x$ is $X>x$, not $X\ge x$. The complement of $X<x$ is $X\ge x$.
Ignoring ties. At an observed threshold, count the records equal to it explicitly. The difference between the inclusive and strict counts must equal the number of ties.
Dividing by the number of distinct values. The denominator is the number of loss records, including repeated values, because each record contributes its own empirical weight.
Applying exact-record counts to censored observations. These formulas require exact loss values for the records being counted. A record known only to exceed a reporting limit cannot be assigned to a precise side of every threshold; censored-data methods are needed for that setting.

Key takeaway: Translate the inequality first, count equality cases deliberately, and divide by the total number of exact records. At an observed value $x$, the gap between “at most $x$” and “below $x$” is exactly the proportion of records equal to $x$. This threshold discipline is aligned with the SOA Exam ASTAM (Advanced Short-Term Actuarial Mathematics) syllabus.

Check Your Understanding

For each problem, write the inequality indicated by the wording before counting. The records in every question are exact and equally weighted.

A hypothetical sample of six losses is 4, 7, 7, 10, 13, 13. Find the proportions below 7, at most 7, above 7, and at least 7. (Answer: $2/6,\ 3/6,\ 3/6,\ 4/6$, respectively.)
Eight exact losses are 0, 100, 100, 200, 350, 500, 500, 900. Find the proportions strictly above 500 and at least 500, and state the empirical mass at 500. (Answer: $1/8,\ 3/8,\ 2/8$, respectively.)
Seven exact losses are 5, 10, 10, 15, 20, 20, 25. At the threshold 12, find the proportions below, at most, above, and at least the threshold. (Answer: $3/7,\ 3/7,\ 4/7,\ 4/7$, respectively.)
Ten exact losses are 100, 200, 200, 400, 600, 600, 600, 800, 1,000, 1,200. Find the count and proportion satisfying $200\le X<600$, and compare the counts for $X\ge600$ and $X>600$. (Answer: the band count is 3 and its proportion is $3/10$; the two threshold counts are 6 and 3.)
Nine exact losses are 0, 0, 150, 300, 300, 450, 600, 600, 900. At 300, calculate $P_n(X<300)$, $P_n(X\le300)$, $P_n(X>300)$, and $P_n(X\ge300)$, and verify the jump. (Answer: $3/9,\ 5/9,\ 4/9,\ 6/9$; the jump is $2/9$, equal to the mass at 300.)

What You'll Learn