Method-of-Moments Estimation From Population Equations

Introduction

A probability model describes a population through quantities such as its mean and variance. In practice, those quantities may depend on unknown parameters. The method of moments estimates the parameters by matching moments calculated from observed data to the corresponding moments implied by the model.

The method is direct: write the model's moment equations, calculate the required sample moments, and solve for the unknown parameters. Its simplicity does not remove the need to specify the sampling model correctly. In particular, the equations below use complete observations from the distribution being modeled, not values altered by truncation, censoring, or a reporting rule.

Definition: Let $X_1,\ldots,X_n$ be an independent and identically distributed sample from a parametric model with parameter $\theta$. The $r$th sample raw moment is $$ m'_r=\frac{1}{n}\sum_{i=1}^n X_i^r. $$ The corresponding population raw moment is $$ \mu'_r(\theta)=E_\theta[X^r], $$ provided this expectation exists. A method-of-moments estimate equates selected sample raw moments to their population counterparts and solves the resulting equations for the unknown parameter or parameters.

From a Model to Estimating Equations

A raw moment is taken about zero. For example, the first raw moment is $E[X]$, and the second is $E[X^2]$. The second raw moment is not the variance: when the mean exists, $$ \operatorname{Var}(X)=E[X^2]-\{E[X]\}^2. $$ That distinction matters when constructing equations for a model with both an unknown mean and an unknown variance.

For a model with $k$ unknown parameters, a common setup uses the first $k$ raw moments. The equations are formed by setting each selected $m'_r$ equal to $\mu'_r(\theta)$. This is a general recipe, not a guarantee that the equations have one useful solution. Here the emphasis is on setting up and applying the equations; the choice of moments and more involved questions about solutions require separate care.

State the sampling model.
Specify the distribution, its unknown parameter or parameters, and the assumption that the observations are independent and identically distributed complete values.

Write the population moment equations.
For each selected order $r$, calculate $E_\theta[X^r]$ from the stated model before substituting any data.

Calculate sample raw moments.
Use the observed values and the denominator $n$ in $m'_r$.

Equate and solve.
Set the sample moments equal to their corresponding population moments, solve for the parameter estimates, and substitute them back to check the equations.

Why the Matching Principle Is Reasonable

The method uses a population quantity that the model says should describe the observations, then substitutes the observed sample analogue for that unknown quantity. This matching is justified as a natural estimating rule; it does not say that a parameter estimate must itself be unbiased.

For a fixed order $r$, suppose the sample is independent and identically distributed and $E[X^r]$ exists. Linearity of expectation gives

E[m'_r] =E\left[\frac{1}{n}\sum_{i=1}^n X_i^r\right] =\frac{1}{n}\sum_{i=1}^n E[X_i^r] =\frac{1}{n}\,n\mu'_r(\theta) =\mu'_r(\theta).

Thus, the sample raw moment has the population raw moment as its expectation under these assumptions. The method-of-moments step sets the observed value of that sample moment equal to the model expression. That reasoning supports the construction of the equations, but it does not establish that every resulting parameter estimate is unbiased or that it is the best estimator for a particular purpose.

Worked Examples

Worked Example: Estimating a Claim Probability

A hypothetical portfolio has eight independent policies, each with the same probability $p$ of producing at least one claim during the year. Model each policy's claim indicator as a Bernoulli random variable: $X_i=1$ if the policy produces a claim and $X_i=0$ otherwise. The observed indicators are $1,0,1,0,0,1,0,0$. Estimate $p$ by the method of moments.

Model and assumptions. The $X_i$ are independent and identically distributed Bernoulli random variables with $P(X_i=1)=p$. For a Bernoulli variable, $X_i^2=X_i$, and the first population raw moment is

\mu'_1(p)=E[X_i]=1\cdot p+0\cdot(1-p)=p.

Sample moment. There are three ones among eight observations, so

m'_1=\frac{1+0+1+0+0+1+0+0}{8} =\frac{3}{8} =0.375.

Match and solve. Equating the first raw moments gives $m'_1=\mu'_1(p)$, hence

0.375=p \quad\Longrightarrow\quad \widehat p_{\mathrm{MOM}}=0.375.

The estimate is a probability between zero and one, and substituting it into the population first moment gives $E[X_i]=0.375=m'_1$, as required. Method: match the sample proportion of claim indicators to the model's expected indicator. Common mistake avoided: the number of claims, three, is not the estimated probability; it must be divided by the eight policies.

Worked Example: Estimating a Uniform Severity Endpoint

Suppose five complete hypothetical loss severities, recorded in thousands of dollars, are $1.2, 2.0, 2.4, 3.1,$ and $4.3$. Assume they are independent observations from a uniform distribution on $[0,b]$, where $b$ is an unknown endpoint measured in thousands of dollars. Estimate $b$ by the method of moments.

Model and population moment. Under the stated model, the density is $1/b$ on $0\leq x\leq b$, with $b>0$. Therefore,

\mu'_1(b)=E[X] =\int_0^b x\frac{1}{b}\,dx =\frac{1}{b}\left[\frac{x^2}{2}\right]_0^b =\frac{b}{2}.

Calculate the sample moment. The sum of the five values is $1.2+2.0+2.4+3.1+4.3=13.0$ thousand dollars. Thus,

m'_1=\frac{13.0}{5}=2.6 \quad\text{thousand dollars}.

Match and solve. Set the sample mean equal to the model mean:

2.6=\frac{b}{2} \quad\Longrightarrow\quad \widehat b_{\mathrm{MOM}}=5.2 \quad\text{thousand dollars}.

The fitted model has mean $5.2/2=2.6$ thousand dollars, matching the calculated sample mean. The largest observation, $4.3$, is below the estimated endpoint $5.2$, so it is also consistent with the fitted support. Method: use the first raw moment of the uniform model. Common mistake avoided: using the sample maximum $4.3$ as the method-of-moments estimate would use a different statistic; the moment equation here gives $5.2$, not $4.3$.

Worked Example: Estimating Both Parameters of a Normal Model

A hypothetical set of four independent, complete measurements of a signed portfolio adjustment, in thousands of dollars, is $4,6,7,$ and $9$. Assume a normal model $X_i\sim N(\mu,\sigma^2)$, where both $\mu$ and $\sigma^2$ are unknown. Estimate both parameters by matching the first two raw moments.

Model and population moments. For this normal model,

\mu'_1=E[X]=\mu, \qquad \mu'_2=E[X^2]=\operatorname{Var}(X)+\{E[X]\}^2 =\sigma^2+\mu^2.

Calculate the first sample raw moment.

m'_1=\frac{4+6+7+9}{4} =\frac{26}{4} =6.5 \quad\text{thousand dollars}.

Calculate the second sample raw moment. First square each observation: $4^2=16$, $6^2=36$, $7^2=49$, and $9^2=81$. Their sum is $16+36+49+81=182$. Therefore,

m'_2=\frac{182}{4}=45.5 \quad\text{(thousand dollars)}^2.

Match and solve. The first equation is $6.5=\mu$, so $\widehat\mu_{\mathrm{MOM}}=6.5$. The second is $45.5=\sigma^2+\mu^2$. Substituting the first estimate gives

45.5=\widehat{\sigma^2}_{\mathrm{MOM}}+(6.5)^2, \qquad (6.5)^2=42.25, \qquad \widehat{\sigma^2}_{\mathrm{MOM}}=45.5-42.25=3.25.

The estimates are $\widehat\mu_{\mathrm{MOM}}=6.5$ thousand dollars and $\widehat{\sigma^2}_{\mathrm{MOM}}=3.25$ squared thousand dollars. As a check, the fitted second raw moment is $3.25+(6.5)^2=45.5$, exactly the calculated $m'_2$. The corresponding estimated standard deviation is $\sqrt{3.25}=1.8028$ thousand dollars, rounded to four decimal places. Method: match the first and second raw moments, then recover the variance from the second equation. Common mistake avoided: substituting the sample variance with denominator $n-1$ is not the prescribed raw-moment matching calculation.

Common Mistakes and Practical Limits

The mechanics are short, but errors often enter before the equations are solved. Check the model statement and the definition of each moment before doing arithmetic.

Using the wrong denominator. The sample raw moment is $n^{-1}\sum X_i^r$. For the second raw moment, do not divide the sum of squares by $n-1$.
Confusing raw and central moments. $m'_2$ averages squared observations about zero. It is not the average squared deviation from the sample mean.
Matching unlike quantities. Match order $r$ to order $r$: $m'_2$ corresponds to $E[X^2]$, not directly to $\operatorname{Var}(X)$.
Ignoring units. If observations are recorded in thousands of dollars, a first moment is in thousands of dollars and a second moment is in squared thousands of dollars.
Treating altered observations as complete losses. If a value is censored at a limit, omitted below a reporting threshold, or otherwise modified by the observation process, the raw average of the recorded values need not estimate the raw moment of the underlying loss distribution. The observation mechanism must be accounted for rather than silently treating those records as complete $X_i$ values.

Method-of-moments equations are useful because they translate model properties into calculations that can often be checked directly. Their interpretation still depends on the model and data assumptions: an equation based on the wrong distribution, units, or observation process does not become valid merely because its algebra is correct. This foundational procedure is aligned with the SOA Exam ASTAM (Advanced Short-Term Actuarial Mathematics) syllabus.

Key takeaway: State the model, calculate the sample raw moments with denominator $n$, equate them to the corresponding population raw moments, solve, and substitute the estimates back into the equations. The resulting estimates are only meaningful under the sampling and observation assumptions used to form those equations.

Check Your Understanding

For each problem, write the population moment equation before substituting the sample values.

A hypothetical group of 20 independent policies has seven claim indicators equal to one and the rest equal to zero. Under a common Bernoulli claim probability $p$, find the first-moment method-of-moments estimate of $p$. (Answer: $\widehat p_{\mathrm{MOM}}=0.35$)
Four complete hypothetical severities, in thousands of dollars, are $2,3,3,$ and $4$. Assume a uniform model on $[0,b]$. Estimate $b$ by the first raw moment. (Answer: $\widehat b_{\mathrm{MOM}}=6$ thousand dollars)
Five independent observations from a normal model are summarized by $\sum X_i=25$ and $\sum X_i^2=145$. Use the first two raw moments to estimate $\mu$ and $\sigma^2$. (Answer: $\widehat\mu_{\mathrm{MOM}}=5,\ \widehat{\sigma^2}_{\mathrm{MOM}}=4$)
A uniform model has support $[a,a+6]$, with known width 6 and unknown lower endpoint $a$. Four complete observations have sample mean $11.4$. Use the first raw moment to estimate $a$. (Answer: $\widehat a_{\mathrm{MOM}}=8.4$)

What You'll Learn