Exact Loss Records And Empirical Data Notation

Why the Record Type Matters

A loss-data table may contain dollar amounts that look alike but do not convey the same information. An exact observation says what the loss was. A censored record says only that the loss was above, below, or within some range. Treating a bound as if it were the exact loss can change which observations belong in an exact-loss sample and can lead to incorrect counts.

This tutorial establishes a consistent way to describe those records. It focuses on identifying the information each record contains, stating the sample size, and counting the distinct values that are actually observed exactly. It does not construct an empirical distribution or calculate summaries from one.

Definition: An exact loss observation is a record that gives the realized loss amount $x_i$ without ambiguity. A censored record gives incomplete information about the loss, such as a lower bound, an upper bound, or an interval containing it. A bound or interval is not itself an exact loss amount.

For example, a record of “$720” states an exact loss of $720. A record of “at least $720” does not: the loss might be $720, $900, or another amount. The two records may both contain the number 720, but only the first gives an exact loss.

For a data set of $n$ exact loss observations, write the observations as $x_1,x_2,\ldots,x_n$, where $x_i$ is the $i$th recorded exact loss. The sample size $n$ is the number of exact loss observations in that sample. If the data set contains repeated amounts, let $z_1,\ldots,z_k$ denote the distinct exact values, and let $r_j$ be the number of times $z_j$ occurs. The number of distinct exact values is $k$, which need not equal $n$.

\text{Number of exact observations}=n,\qquad \text{number of distinct exact values}=k,\qquad \sum_{j=1}^{k} r_j=n.

The labels $z_j$ need not be arranged in increasing order unless the problem asks for that. They name the different values; $r_j$ counts how often each appears. Keep those roles separate from the original observation labels $x_i$, which identify individual records.

Counting Exact Observations and Distinct Values

A data file can contain several kinds of records. To avoid ambiguity, state the total number of records and, when the file includes censored records, the number of exact observations separately. Use $n_E$ for the number of exact loss observations and $n_C$ for the number of censored records in the file. These counts describe the records as provided; they do not turn censored losses into exact ones.

n_{\text{records}}=n_E+n_C.

When writing $x_1,\ldots,x_{n_E}$ for a data file containing censored records, make clear that these indexed values refer only to the exact losses. The censored records remain part of the file, but they are not members of that exact-loss sample. The number $k$ of distinct exact values is counted among the $n_E$ exact observations, not among every number printed in the file.

Key distinction: Sample size counts observations under the stated sampling rule. Distinct-value count counts different exact amounts within that sample. Repeated exact amounts count repeatedly toward the sample size but only once toward the distinct-value count.

Worked Example: Counting Repeated Exact Losses

A hypothetical administrator records eight fully settled losses, in dollars: 420, 760, 760, 1,150, 1,150, 2,040, 2,040, and 2,040. Treat every listed amount as exact. State the sample size, the distinct exact values, and their multiplicities.

Method. First count the records, then group equal exact amounts. No interest or mortality assumption is relevant; the data are observed loss amounts, and no parametric distribution is assumed.

There are eight listed exact amounts, so $n=8$. The different amounts are $420$, $760$, $1{,}150$, and $2{,}040$, so $k=4$. Their multiplicities are $1,2,2,$ and $3$, respectively:

(z_1,z_2,z_3,z_4)=(420,760,1150,2040),\qquad (r_1,r_2,r_3,r_4)=(1,2,2,3).

The count checks: $1+2+2+3=8=n$. Thus the sample contains eight exact observations but only four distinct exact values. The grouping is a way to describe the records, not a change in the number of observations.

Common mistake avoided. Reporting $k=8$ because there are eight records confuses the number of observations with the number of different amounts. The correct distinct-value count is four.

What Censoring Does to the Record

A record is censored when the available information does not specify a unique loss amount. For a right-censored record at $c$, the recorded information is that the loss is at least $c$; for a left-censored record, the loss is known to be no greater than a stated threshold. An interval-censored record places the loss within a stated interval. In each case the loss itself is not recorded as an exact amount.

The precise boundary convention should follow the data description. For example, if a problem says “exceeds $900,” the information is $X>900$; if it says “at least $900,” it is $X\geq 900$. Neither statement gives an exact value. Do not silently replace one inequality with the other.

Worked Example: Separating Exact and Censored Records

A hypothetical file contains five claim records: $310 exact, $640 exact, at least $900, $1,120 exact, and $1,450 exact. Count all records, exact observations, censored records, and distinct exact values.

Method. Classify each record by the information it provides. No interest or mortality assumption is involved, and no probability distribution or parameters are specified or needed.

The file has five records in total. Four give exact amounts: $310, $640, $1,120, and $1,450. One record says only that the loss is at least $900, so it is censored. Therefore $n_{\text{records}}=5$, $n_E=4$, and $n_C=1$.

(x_1,x_2,x_3,x_4)=(310,640,1120,1450),\qquad n_E=4,\qquad k=4.

All four exact amounts differ, so the number of distinct exact values is $k=4$. The number 900 does not enter that list: it is the bound in a censored record, not a recorded exact loss. The count also checks because $n_E+n_C=4+1=5$.

Common mistake avoided. Counting “at least $900” as an exact $900 loss would put a value into the exact sample that the file does not establish. The correct exact sample has four observations, not five.

Payments, Limits, and the Underlying Loss

Actuarial data may report a payment rather than the underlying loss. These are different quantities. If a hypothetical policy pays the lesser of the loss and a limit, a payment equal to the limit does not reveal whether the underlying loss equaled the limit or exceeded it. The payment may be exact as a payment amount even though the loss is censored at the limit.

This distinction matters whenever the question asks for an exact sample of losses. First identify the target variable: underlying loss $X$, payment per loss $Y^L$, or payment per payment $Y^P$, as applicable. A reported amount is exact only for the quantity it actually measures. A payment record should not be relabelled as an exact underlying loss merely because it is a precise dollar amount.

Worked Example: A Capped Payment Is Not an Exact Loss

A hypothetical policy pays $Y=\min(X,1000)$, where $X$ is the underlying loss and $1,000 is the payment limit. Five observed payments are $450, $1,000, $1,000, $780, and $450. Assuming no deductible, identify the exact underlying losses, the censored underlying losses, and the distinct exact underlying values.

Method. Compare each payment with the limit. A payment below $1,000 identifies the underlying loss exactly because the payment then equals $X$. A payment of $1,000 only establishes $X\geq 1000$. No interest or mortality assumption applies; no distribution or parameters are assumed.

The first payment is $450, below the limit, so its underlying loss is exactly $450. The second payment is $1,000, so its underlying loss is at least $1,000 and is censored. The third payment is also $1,000, giving a second censored underlying loss. The fourth payment is $780, below the limit, so its underlying loss is exactly $780. The fifth payment is $450, below the limit, so its underlying loss is exactly $450.

There are five underlying-loss records: three exact and two censored. The exact-loss sample is $(450,780,450)$, so $n_E=3$. It has two distinct exact values, $450 and $780; their multiplicities are 2 and 1. The five payments themselves have three distinct payment amounts: $450, $780, and $1,000. That payment-value count is not the distinct-value count for exact underlying losses.

Check. The exact and censored counts add to $3+2=5$, the total number of claims. The two occurrences of $450 contribute two exact observations but one distinct exact value.

Common mistake avoided. Treating both $1,000 payments as exact $1,000 losses incorrectly includes the policy limit as an observed underlying loss. The data establish only that each of those losses reached or exceeded the limit.

A Reliable Notation Workflow

A short record-classification process prevents most counting errors. In particular, keep the record count, exact-sample count, and distinct-value count in separate statements.

Name the quantity.
Decide whether the question concerns underlying losses, payments, or another recorded amount. Do not substitute one for another.

Classify each record.
Mark it exact only if the stated information identifies one value for the quantity of interest. Otherwise retain its bound or interval as a censored record.

Count the records.
State the total file count and, if records are mixed, report $n_E$ and $n_C$ separately.

Write and group exact values.
Use $x_1,\ldots,x_{n_E}$ for the exact sample. List its distinct values $z_1,\ldots,z_k$ and, when useful, give their multiplicities $r_1,\ldots,r_k$.

Check the counts.
Verify $n_{\text{records}}=n_E+n_C$ for a mixed file and $\sum_{j=1}^{k}r_j=n_E$ for the exact observations.

The multiplicity identity follows directly from how the groups are formed. Every exact observation has one value, so it belongs to exactly one group of equal values. The groups are disjoint, and together they contain every exact observation. Counting all observations by those groups therefore gives $r_1+\cdots+r_k=n_E$. If every record is exact, $n_E=n$, recovering $\sum r_j=n$.

This is a bookkeeping identity, but it is a useful diagnostic. If the multiplicities sum to more than the exact-sample size, a record has been counted twice. If they sum to less, at least one exact observation has been omitted or misclassified.

Common Exam Traps

Counting a limit amount as the loss. A capped payment may be exact as a payment but only a lower bound on the underlying loss.
Including censoring thresholds among distinct exact values. A threshold describes what is known about a loss; it is not necessarily an observed loss amount.
Using the file count as the exact-sample size. If some records are censored, report the total number of records separately from $n_E$.
Confusing repeats with distinct values. Repeated exact amounts each count toward $n_E$, but each amount appears only once in the set of $z_j$ values.
Ignoring the inequality in a bound. “Greater than” and “at least” are not identical statements. Preserve the wording and its boundary condition.

Careful notation is the foundation for later empirical work: it makes clear which values are truly observed and which records contain only partial information. The key habit is to define the target quantity, classify each record before counting, and check that the exact-value groups account for all and only the exact observations. This treatment is aligned with the SOA Exam ASTAM (Advanced Short-Term Actuarial Mathematics) syllabus.

Check Your Understanding

For each problem, identify the requested counts and preserve the distinction between exact values and bounds. Assume all records not explicitly described as censored are exact.

A hypothetical data set lists losses of $125, $300, $125, $500, and $300. State the exact sample size, the number of distinct exact values, and each distinct value’s multiplicity. (Answer: $n=5$, $k=3$; $125 occurs 2 times, $300 occurs 2 times, and $500 occurs 1 time.)
A file contains six records: $210 exact, $400 exact, at least $600, $750 exact, at least $900, and $210 exact. Give the total record count, $n_E$, $n_C$, and the distinct exact values. (Answer: 6 records, $n_E=4$, $n_C=2$, and 3 distinct exact values: $210, $400, and $750.)
A hypothetical policy pays $Y=\min(X,800)$. Five payments are $250, $800, $630, $800, and $250, with no deductible. For underlying loss $X$, state the exact sample size, censored count, and number of distinct exact losses. (Answer: 3 exact losses, 2 censored losses, and 2 distinct exact losses: $250 and $630.)
A file contains exact losses $90, $140, $140, $260, and $310, plus one record stating “at least $200.” How many total records are there, how many exact observations, and how many distinct exact values? (Answer: 6 records, $n_E=5$, $n_C=1$, and $k=4$; the distinct exact values are $90, $140, $260, and $310.)
A five-record file contains exact losses of $500, $500, and $700; one loss known only to lie between $500 and $700; and one loss known only to be at least $900. State the record count, exact-sample size, censored count, and distinct exact-value count. (Answer: 5 records, $n_E=3$, $n_C=2$, and $k=2$, with distinct exact values $500 and $700.)

What You'll Learn