Gamma Severity With Known Shape

Why the Known-Shape Gamma Model Is Useful

A gamma distribution can represent positive, right-skewed loss amounts. Its shape parameter controls the form of the distribution, while its rate parameter controls the scale of the amounts. In some estimation problems, the shape is specified and the available complete loss observations are used to estimate only the rate. This makes the likelihood maximization particularly direct: the estimate is determined by the sample mean.

The distinction between a rate and a scale is essential. A larger rate produces smaller expected losses; it is not interchangeable with a scale parameter. The model and parameter convention must therefore be stated before writing a likelihood.

Definition: Let $X_1,\ldots,X_n$ be independent and identically distributed complete observations from a gamma severity distribution with known shape $\alpha>0$ and unknown rate $\beta>0$. The density is $$ f_X(x;\alpha,\beta)=\frac{\beta^\alpha}{\Gamma(\alpha)}x^{\alpha-1}e^{-\beta x},\qquad x>0, $$ where $\Gamma(\alpha)=\int_0^\infty t^{\alpha-1}e^{-t}\,dt$ is the gamma function. Under this rate parameterization, $E[X]=\alpha/\beta$ and $\operatorname{Var}(X)=\alpha/\beta^2$.

The observations must be complete loss amounts, each recorded on the same scale, and the model assumes independent observations from the stated gamma distribution. In particular, the likelihood below is not the likelihood for records subject to truncation or censoring: incomplete observations require accounting for how they were observed. Here the shape $\alpha$ is treated as fixed, not estimated.

Likelihood and Rate Estimate

For observed values $x_1,\ldots,x_n>0$, independence means that the joint density is the product of the individual densities. As a function of the unknown rate, the likelihood is

L(\beta)=\prod_{i=1}^n\frac{\beta^\alpha}{\Gamma(\alpha)}x_i^{\alpha-1}e^{-\beta x_i} =\frac{\beta^{n\alpha}}{\Gamma(\alpha)^n} \left(\prod_{i=1}^n x_i^{\alpha-1}\right)e^{-\beta\sum_{i=1}^n x_i},\qquad \beta>0.

Taking logarithms simplifies the product and leaves an expression that is easier to differentiate. Terms involving only the observations and the known shape are constant with respect to $\beta$, but they remain part of the log-likelihood.

\ell(\beta)=n\alpha\ln\beta-n\ln\Gamma(\alpha) +(\alpha-1)\sum_{i=1}^n\ln x_i-\beta\sum_{i=1}^n x_i.

The maximum likelihood estimate is found by differentiating with respect to $\beta$, setting the score equal to zero, and checking that the resulting value is a maximum.

Differentiate.
The score is $d\ell/d\beta=n\alpha/\beta-\sum_{i=1}^n x_i$.

Solve the score equation.
Setting the score to zero gives $n\alpha/\beta=\sum_{i=1}^n x_i$, so $\widehat{\beta}=n\alpha/\sum_{i=1}^n x_i$.

Check the maximum.
The second derivative is $-n\alpha/\beta^2<0$ for every $\beta>0$, so the log-likelihood is strictly concave and the score root is its unique maximum.

Theorem: For independent, complete gamma observations with known shape $\alpha>0$ and unknown rate $\beta>0$, the maximum likelihood estimate is $$ \widehat{\beta}=\frac{n\alpha}{\sum_{i=1}^n x_i} =\frac{\alpha}{\overline{x}}, $$ where $\overline{x}=\sum_{i=1}^n x_i/n$ is the sample mean. The estimate has units inverse to the units used for the losses.

Why the Sample-Mean Formula Appears

The likelihood calculation explains the result, and the model mean gives a useful interpretation. Under the stated rate parameterization, the expected severity is $\alpha/\beta$. The estimate sets this model mean equal to the observed sample mean:

\frac{\alpha}{\widehat{\beta}}=\overline{x} \quad\Longleftrightarrow\quad \widehat{\beta}=\frac{\alpha}{\overline{x}}.

This equality is not an assumption that every sample mean exactly equals the true population mean. It is a feature of this particular maximum likelihood estimate: maximizing the likelihood produces the rate whose model mean matches the observed mean. Because the shape is fixed, the sample mean alone determines the rate estimate. A different parameterization or a different set of observation conditions can lead to a different likelihood expression.

Worked Examples

Worked Example: Maximizing a Likelihood for Three Losses

A hypothetical portfolio has three complete claim severities, measured in thousands of dollars: $1$, $2$, and $3$. Assume independent gamma observations with known shape $\alpha=2$ and unknown rate $\beta$, measured per thousand dollars. Find the maximum likelihood estimate and verify it directly from the likelihood.

Method and common mistake avoided. Use the complete-data gamma likelihood with the shape held at 2. Do not estimate the shape from these observations or confuse the rate with its reciprocal.

The sample size is $n=3$, and the total observed severity is $1+2+3=6$ thousand dollars. For shape 2, $\Gamma(2)=1$, so the likelihood is

L(\beta)=\prod_{i=1}^3\beta^2x_i e^{-\beta x_i} =\beta^6(1)(2)(3)e^{-\beta(1+2+3)} =6\beta^6e^{-6\beta}.

The log-likelihood is $\ell(\beta)=\ln 6+6\ln\beta-6\beta$. Its derivative is $6/\beta-6$. Setting this to zero gives $6/\beta=6$, hence $\widehat{\beta}=1$ per thousand dollars. The second derivative is $-6/\beta^2$, which is negative for every positive $\beta$, confirming a maximum.

The sample mean is $6/3=2$ thousand dollars, and the formula gives $\widehat{\beta}=2/2=1$ per thousand dollars. Substitution into the likelihood gives $L(1)=6e^{-6}$. The direct likelihood and sample-mean calculations agree.

Worked Example: Keeping the Rate Units Consistent

A hypothetical insurer records three complete claim amounts of $\$4{,}000$, $\$5{,}000$, and $\$9{,}000$. Assume a gamma model with known shape $\alpha=3$. Find the rate estimate per dollar and express the equivalent estimate per thousand dollars.

Method and common mistake avoided. Apply the estimate using one consistent unit at a time. A rate calculated from dollar-valued observations is per dollar, not per thousand dollars.

There are $n=3$ claims, and their sum in dollars is $\$4{,}000+\$5{,}000+\$9{,}000=\$18{,}000$. Thus

\widehat{\beta} =\frac{n\alpha}{\sum x_i} =\frac{3(3)}{18{,}000} =\frac{9}{18{,}000} =0.0005\text{ per dollar}.

The sample mean is $\$18{,}000/3=\$6{,}000$, and the alternative form gives $3/6{,}000=0.0005$ per dollar. To express the same model using losses in thousands of dollars, the observations are $4$, $5$, and $9$, with sum $18$. The estimate on that scale is $9/18=0.5$ per thousand dollars.

These rates describe the same distribution because $(0.0005)(1{,}000)=0.5$. Reporting $0.0005$ per thousand dollars would be a unit error: the numerical rate must change when the loss unit changes.

Worked Example: Checking the Fitted Mean and Variance

A hypothetical risk pool has four complete severities, in thousands of dollars: $8$, $10$, $11$, and $11$. Treat the severities as independent gamma observations with known shape $\alpha=4$. Estimate the rate and calculate the model mean and variance using that estimate.

Method and common mistake avoided. First estimate the rate from the total or sample mean, then substitute it into the rate-parameterized gamma moments. Do not use the rate as though it were a scale.

The total is $8+10+11+11=40$ thousand dollars, so $n=4$ and $n\alpha=4(4)=16$. Therefore $\widehat{\beta}=16/40=0.4$ per thousand dollars. The sample mean is $40/4=10$ thousand dollars, and $\alpha/\widehat{\beta}=4/0.4=10$ thousand dollars, which checks the fitted mean.

Using $\operatorname{Var}(X)=\alpha/\beta^2$, the fitted variance is $4/(0.4)^2=4/0.16=25$ square-thousands of dollars. The corresponding standard deviation is $\sqrt{25}=5$ thousand dollars. The mean and variance are expressed on different powers of the loss unit, as required.

Common Exam Traps and Scope of the Result

The rate estimate is simple, but it depends on reading the model and data conditions correctly. Before calculating, check that the shape is given as known, the observations are complete, and every amount uses the same unit.

Using the reciprocal formula. For the rate parameterization used here, $\widehat{\beta}=\alpha/\overline{x}$. The expression $\overline{x}/\alpha$ is the corresponding estimate of a scale parameter, not the rate.
Dropping the known shape from the score. The shape is not estimated, but it still appears in the likelihood as the exponent $n\alpha$ and in the numerator of the estimate.
Mixing loss units. If severities are converted from dollars to thousands of dollars, the rate must be expressed per thousand dollars. The likelihood and estimate must use a consistent measurement scale.
Using the complete-data likelihood for incomplete records. The product density above applies when each loss amount is observed in full. Truncation and censoring affect the likelihood and cannot be ignored.

The concavity check is also a useful safeguard. The score equation identifies a candidate, while the negative second derivative establishes that the candidate is the unique maximum on $\beta>0$. A positive rate estimate alone does not establish that it maximizes the likelihood.

Key takeaway: With a known gamma shape and independent complete observations, maximize the likelihood to obtain $\widehat{\beta}=n\alpha/\sum x_i=\alpha/\overline{x}$. This rate is positive, is the unique likelihood maximum, and has units inverse to the loss measurement unit. This treatment is aligned with the SOA Exam ASTAM (Advanced Short-Term Actuarial Mathematics) syllabus.

Check Your Understanding

For each problem, assume independent, complete observations from the stated gamma model and use the rate parameterization defined above.

A hypothetical sample has severities $2$, $3$, and $5$, in thousands of dollars, with known shape $\alpha=2.5$. Find the rate estimate per thousand dollars. (Answer: $0.75$ per thousand dollars)
Three complete hypothetical claim amounts are $\$800$, $\$1{,}200$, and $\$2{,}000$, and the known shape is $\alpha=1$. Find the rate estimate per dollar. (Answer: $0.00075$ per dollar)
A hypothetical data set contains severities $2$, $4$, and $6$, in thousands of dollars, and the known shape is $\alpha=3.5$. Find the rate estimate and the fitted model mean, both on the thousand-dollar scale. (Answer: rate $0.875$ per thousand dollars; fitted mean $4$ thousand dollars)
Five complete hypothetical severities have a total of $\$125{,}000$. The known shape is $\alpha=4$. Find the rate estimate per thousand dollars. (Answer: $0.16$ per thousand dollars)
A hypothetical sample contains $\$3{,}000$, $\$5{,}000$, and $\$7{,}000$ severities, with known shape $\alpha=2.5$. A candidate reports a rate of $0.5$ per dollar. Determine the correct rate per dollar and the equivalent rate per thousand dollars. (Answer: $0.0005$ per dollar, equivalent to $0.5$ per thousand dollars)

What You'll Learn