Data-Quality Diagnostics for Empirical Estimates

Why Data Quality Comes Before Estimation

An empirical estimate can be calculated exactly and still be misleading if its records do not represent the intended experience. A duplicated claim may inflate a count; a missing zero-claim policy-year may shrink the denominator; a payment date used in place of an incurred date may move losses between years. These errors affect estimates in different ways, so the first task is to define the quantity being estimated and check whether the data support it.

The checks in this tutorial concern the integrity and alignment of the records used for an estimate. They do not establish that the observed experience is representative of a different population or coverage basis. A clean file is necessary for a sound empirical estimate, but cleanliness alone does not prove that the estimate is appropriate for every intended use.

Definition: A claim record is a row describing a claim or claim transaction; it need not represent a unique claim unless the data design says it does. A zero-claim period is an observed policy or exposure period with no claims. Exposure is the amount of time or insured risk included in the frequency denominator, measured in a defined unit such as policy-years. A data-quality diagnostic checks whether these records, periods, amounts, dates, and exposure units are internally consistent with the estimate being calculated.

For $n$ observed periods with claim counts $N_j$ and exposures $e_j$, a basic exposure-based empirical frequency is

\widehat{\lambda}=\frac{\sum_{j=1}^{n}N_j}{\sum_{j=1}^{n}e_j}.

Here $e_j$ must use the same exposure unit for every period, and the periods included in the numerator and denominator must refer to the same experience base. The estimate is not the average count among only the periods that happened to have claims. For positive claim amounts $X_1,\ldots,X_m$, the empirical mean severity is $\bar{X}=\sum_{k=1}^{m}X_k/m$. If the same claim records and exposure basis are used, the empirical aggregate loss per exposure unit is total claim amount divided by total exposure.

What Each Diagnostic Protects

Diagnostic	What to check	Possible effect if unresolved
Duplicate observations	Repeated claim identifiers, matching amounts and dates, and transaction-versus-claim definitions	May inflate claim counts and total losses; severity effects depend on which records are duplicated
Missing zero-claim periods	Whether the period file includes all eligible periods, including those with no claims	Can understate total exposure and overstate estimated frequency or loss per exposure
Dates	Valid calendar dates, consistent date meaning, and period assignment	Can move counts or amounts between periods and distort period-level comparisons
Amounts	Missing, negative, implausible, or inconsistent amounts and units	Can corrupt severity, total loss, and aggregate summaries
Exposure	Policy-period completeness, units, overlaps, gaps, and agreement with the claim population	Can distort rates even when the claim numerator is correct

A flag is a reason to investigate, not automatic proof that a record is wrong. Two identical amounts on the same date might be separate valid claims. A negative transaction might be a legitimate recovery or adjustment rather than an impossible claim amount. Record definitions and source documentation determine how the observation should be treated.

Worked Examples: Diagnosing and Quantifying Defects

Worked Example: A Duplicated Claim Record

A hypothetical portfolio has four policy-years of exposure and three unique claims with amounts $600, $900, and $1,500. A data extract contains the $900 claim twice. Calculate the correct and uncorrected empirical frequency, mean severity, and aggregate loss per exposure unit. Assume each listed unique claim is a single claim and the exposure total is four policy-years.

Method. First identify the duplicate using the claim identifier and source record. Then calculate each summary using the unique claims, and compare it with the result from treating every row as a separate claim.

Correct frequency. There are $3$ unique claims and $4$ policy-years, so

\widehat{\lambda}_{\mathrm{correct}}=\frac{3}{4}=0.75 \quad\text{claims per policy-year}.

Correct mean severity. The unique-claim total is $600+900+1{,}500=3{,}000$, and there are $3$ claims:

\bar{X}_{\mathrm{correct}}=\frac{3{,}000}{3}=\$1{,}000.

Correct aggregate loss per exposure. Divide the total unique-claim loss by the four policy-years:

\frac{3{,}000}{4}=\$750\text{ per policy-year}.

If the duplicate is left in the file, the recorded count is $4$, and the recorded total is $3{,}000+900=3{,}900$. Thus the uncorrected frequency is $4/4=1.00$ claim per policy-year, the uncorrected mean severity is $3{,}900/4=\$975$, and the uncorrected aggregate loss per policy-year is $3{,}900/4=\$975$.

Check. The uncorrected row-level totals are consistent: $1.00\times\$975=\$975$ per policy-year. The duplicate raises frequency and aggregate loss per exposure, while it lowers mean severity in this particular example because the repeated amount is below the correct mean. Common mistake avoided: deleting a repeated-looking row without confirming whether rows represent claims or transactions. Confirm the data definition before deduplicating.

Worked Example: Restoring Zero-Claim Periods

A hypothetical book has five complete one-policy-year periods with claim counts $0,1,0,2,0$. The three claims have amounts $400, $800, and $600. A report contains only the two periods with at least one claim. Calculate the frequency and aggregate loss per policy-year using the full data and the incomplete report.

Method. Use all observed exposure periods in the denominator, including periods with zero claims. The full count is $0+1+0+2+0=3$, and total exposure is $5$ policy-years:

\widehat{\lambda}_{\mathrm{full}}=\frac{3}{5}=0.60 \quad\text{claims per policy-year}.

The claim total is $400+800+600=\$1{,}800$, so the aggregate loss per policy-year is $1{,}800/5=\$360$.

The incomplete report has two periods and the same three claims. Its apparent frequency is $3/2=1.50$ claims per represented policy-year, and its apparent aggregate loss is $1{,}800/2=\$900$ per represented policy-year. The reported rates are too high for the full five-year experience because three zero-claim policy-years were omitted.

Check. The full-period mean claim count is $3/5$, not the mean of the two positive counts, $(1+2)/2$. Both calculations are arithmetically correct, but they answer different questions. Common mistake avoided: interpreting a file of claim-bearing periods as though it were a complete exposure-period file.

Worked Example: Dates and Period Assignment

A hypothetical book has one policy-year of exposure in 2024 and one in 2025. Three claims have these incurred dates and amounts: March 12, 2024, $700; December 28, 2024, $500; and January 15, 2025, $900. The $500 claim was paid on February 3, 2025. Calculate annual aggregate loss per exposure using incurred year, then show what happens if the $500 claim is assigned by payment year instead.

Method. The requested basis is incurred year, so assign each claim using its incurred date, not the date money was paid. On that basis, 2024 losses total $700+500=\$1{,}200$, and 2025 losses total $\$900$. With one policy-year of exposure in each year, the annual losses per exposure are

\frac{\$1{,}200}{1}=\$1{,}200\text{ in 2024}, \qquad \frac{\$900}{1}=\$900\text{ in 2025}.

If the $500 claim is instead assigned to its payment year, 2024 contains only the $700 claim and 2025 contains $500+900=\$1{,}400$. The corresponding amounts are $700 and $1,400 per policy-year. Across both years the total remains $2,100, but the year-by-year pattern changes.

Check. Incurred-year totals sum to $1{,}200+900=\$2{,}100$; payment-year totals sum to $700+1{,}400=\$2{,}100$. The defect is period assignment, not the two-year total. An impossible date, such as February 30, should be flagged and checked against the source; it should not be silently assigned to a convenient year. Common mistake avoided: mixing incurred-year exposure with payment-year claims and interpreting the resulting annual comparisons as consistent.

Worked Example: An Exposure-File Mismatch

A hypothetical portfolio has four policy periods with claim counts $2,1,0,3$ and exposures $1.0,0.5,1.5,1.0$ policy-years, respectively. An exposure extract mistakenly records the last period as $0.5$ policy-year. Find the correct and mismatched frequency estimates.

Method. Sum the claims and exposures separately, checking that every period in the claim-count file has its matching exposure. The total count is $2+1+0+3=6$, and the correct exposure is $1.0+0.5+1.5+1.0=4.0$ policy-years. Therefore,

\widehat{\lambda}_{\mathrm{correct}}=\frac{6}{4.0}=1.50 \quad\text{claims per policy-year}.

With the erroneous exposure, the denominator is $1.0+0.5+1.5+0.5=3.5$ policy-years. The resulting estimate is $6/3.5=12/7\approx1.7143$ claims per policy-year. Relative to the correct estimate, this is an overstatement of $(1.7143/1.50)-1\approx14.29\%$.

Check. The claim numerator is unchanged; the entire difference comes from reducing exposure by $0.5$ from $4.0$ to $3.5$. Common mistake avoided: checking claim records while assuming the denominator file is automatically complete and aligned.

A Short Justification: Why the Data Must Align

Suppose the claim records in a defined set of periods have amounts $X_1,\ldots,X_m$, and those same periods have total exposure $E=\sum_j e_j$. The total observed loss is $L=\sum_{k=1}^{m}X_k$. For $m>0$, the empirical frequency is $m/E$ and the empirical mean severity is $L/m$. Multiplying gives

\left(\frac{m}{E}\right)\left(\frac{L}{m}\right)=\frac{L}{E}.

Thus, frequency times mean severity equals observed aggregate loss per exposure when both summaries use the same unique claims and the same exposure base. This is an arithmetic identity, not a claim that a theoretical frequency and severity model must apply. A duplicate changes $m$ and $L$; omitted periods change $E$; a date error can change which periods contribute to $L$ or $E$. If the inputs refer to different records or period definitions, the identity no longer describes the intended experience.

A Practical Diagnostic Sequence

A defensible workflow makes each check traceable and avoids turning suspicion into an undocumented correction.

Define the estimate.
State the target measure, claim definition, amount basis, period basis, and exposure unit. For example, distinguish a claim from a payment transaction before counting rows.

Reconcile the period frame.
List expected policy periods and exposures, including periods with no claims. Check for missing periods, duplicate periods, overlaps, gaps, and inconsistent units.

Review claim identity and dates.
Look for repeated identifiers and suspiciously similar rows. Validate date formats and calendar dates, and use one documented date basis for period assignment.

Validate amounts and reconcile totals.
Check for missing values, unexpected signs, unit errors, and totals that do not agree with source records. Investigate unusual values rather than deleting them solely because they are large.

Quantify and document the impact.
Recalculate the estimate before and after each confirmed correction. Record what was changed, why it was changed, and which unresolved records were excluded or retained.

This sequence matters because the same defect can affect several summaries differently. A duplicated claim can raise frequency and total loss while raising or lowering mean severity, depending on the duplicate amount. Missing zero-claim periods usually leave claim totals unchanged but reduce exposure. A date error can preserve an overall total while changing annual estimates. Reporting only one summary can conceal these distinctions.

Common Diagnostic Traps

Equating a repeated row with a duplicate claim. A claim can generate multiple valid transactions. Use identifiers and the record definition to determine whether the repeated row is an error.
Dropping periods with no claims. Zero is an observed count, not missing data. Keep the corresponding exposure when the period was in scope.
Treating all dates as interchangeable. Incurred, reported, and paid dates answer different timing questions. State which date controls the analysis.
Silently replacing an implausible amount. A negative amount might be a recovery or adjustment. Confirm its meaning and the intended treatment before including or excluding it.
Assuming exposure is just a row count. A partial policy-year is not necessarily one full policy-year. Sum the specified exposure measure and reconcile it to the population represented by the claims.
Reporting a corrected value without an audit trail. Keep the original record, correction rationale, and sensitivity of the estimate visible to a reviewer.

Key takeaway: An empirical estimate is only as interpretable as the records and denominator behind it. Check claim identity, zero-claim periods, dates, amounts, and exposure alignment; then show how confirmed defects change the estimate. This diagnostic discipline is aligned with the SOA Exam FAM (Fundamentals of Actuarial Mathematics) syllabus.

Check Your Understanding

For each problem, identify the relevant data-quality issue and calculate the requested empirical quantity.

A two-policy-year hypothetical experience has three unique claims of $300, $500, and $700. The $500 claim is duplicated in the extract. Find the correct frequency, mean severity, and aggregate loss per policy-year, then find the aggregate loss per policy-year if the duplicate is retained. (Answer: Correct frequency $1.5$, mean severity $\$500$, and aggregate loss $\$750$ per policy-year; with the duplicate retained, aggregate loss is $\$1{,}000$ per policy-year.)
Five complete one-year periods have counts $0,2,0,1,0$. Find the exposure-based frequency and the rate obtained if only claim-bearing periods are retained. (Answer: $0.6$ claims per policy-year using all periods; $1.5$ using only the two claim-bearing periods.)
A book has one policy-year in each of 2023 and 2024. Claims are incurred in 2023 for $250 and $450, with the $450 claim paid in 2024; a $350 claim is incurred and paid in 2024. Find annual aggregate loss per exposure using incurred year and using payment year. (Answer: Incurred-year amounts are $\$700$ in 2023 and $\$350$ in 2024; payment-year amounts are $\$250$ in 2023 and $\$800$ in 2024, each per policy-year.)
Two periods have counts $1$ and $3$, and exposures $0.5$ and $1.5$ policy-years. The second exposure is mistakenly recorded as $1.0$. Find both frequency estimates and the percentage overstatement from the mismatch. (Answer: Correct estimate $2.0$ claims per policy-year; mismatched estimate $8/3\approx2.6667$; overstatement $33.33\%$.)
Three claim records show amounts $200, $400, and $-$100. If investigation confirms that the negative entry is an erroneous record rather than a valid recovery, calculate the mean severity using the two valid claims and compare it with the raw three-row average. (Answer: Valid mean severity \(\$300$; raw average $\$166.67$.)

What You'll Learn