Concept Architecture
Concept
Theoretically, the Method of Moments (MoM) is a statistical parameter estimation technique that estimates unknown population parameters by equating sample moments to their corresponding theoretical population moments. The method is founded on the principle that, for sufficiently large samples, sample moments converge to population moments according to the Law of Large Numbers. It exists as a general estimation framework applicable to a wide range of probability distributions and statistical models, particularly when maximum likelihood estimation is computationally difficult or unavailable.
Mathematically, the Method of Moments estimates parameters by solving a system of equations obtained by matching the first k sample moments to the first k theoretical moments, where k equals the number of unknown parameters. Sample moments are calculated directly from the observed data, while theoretical moments are expressed as functions of the unknown parameters. Solving these equations yields the moment estimators. Although generally less statistically efficient than maximum likelihood estimators, Method of Moments estimators are often simple to derive and compute.
In practice, the Method of Moments is used to estimate parameters of probability distributions, epidemiological models and stochastic processes. In health economics, it is applied in econometric modelling, estimation of healthcare utilisation distributions, survival models, generalised method of moments estimation and calibration of probabilistic models. Moment estimators are frequently used as initial parameter values for iterative optimisation procedures such as Maximum Likelihood Estimation.
Purpose
Used to estimate unknown model parameters by matching sample moments to theoretical moments, providing computationally straightforward estimators for probability distributions and statistical models used in health economics.
Mathematical Formulae
Primary Formula
For the r-th moment:
(1/n) ? ????� x?? = E(X?)
where:
- n = sample size
- x? = observed value
- E(X?) = theoretical r-th population moment
For k unknown parameters:
m? = ??(?)
m? = ??(?)
?
m? = ??(?)
where:
- m? = sample moment
- ??(?) = theoretical moment
- ? = vector of unknown parameters
Supporting Formulae
First sample moment:
m? = (1/n) ? ?x?
Second sample moment:
m? = (1/n) ? ?x?�
Example (Normal distribution):
?? = x?
�?� = (1/n) ? ?(x? ? x?)�
Related Mathematical Methods
- Generalised Method of Moments
- Maximum Likelihood Estimation
- Bayesian Estimation
- Estimating Equations
- Least Squares Estimation
- Generalised Linear Models
- Method of Simulated Moments
Example
A health economist models annual healthcare expenditure assuming a Gamma distribution with unknown shape (�) and scale (?) parameters.
From a sample of 800 patients:
- Sample mean = �4,500
- Sample variance = �9,000,000
For the Gamma distribution:
Mean = �?
Variance = �?�
Equating sample and theoretical moments:
�? = Mean� / Variance
�? = 4,500� / 9,000,000
�? = 2.25
?? = Variance / Mean
?? = 9,000,000 / 4,500
?? = 2,000
These parameter estimates provide initial values for subsequent maximum likelihood estimation of the healthcare cost model.
Excel Implementation
| Function | Example Formula | Health Economics Application |
|---|---|---|
| AVERAGE | =AVERAGE(B2:B801) | Calculate the first sample moment (sample mean). |
| VAR.P | =VAR.P(B2:B801) | Calculate the second central moment (population variance estimate). |
| POWER | =POWER(B2,2) | Calculate higher-order sample moments. |
| SUMPRODUCT | =SUMPRODUCT(B2:B801^2)/COUNT(B2:B801) | Calculate raw sample moments. |
| Solver | Solve moment equations by varying unknown parameters. | Estimate distribution parameters using the Method of Moments. |
VBA (Optional)
A VBA routine can automatically calculate sample moments, estimate distribution parameters using the Method of Moments and export parameter estimates for subsequent modelling.
Sources
- Casella G, Berger RL. Statistical Inference. Cengage Learning.
- Kendall MG, Stuart A. The Advanced Theory of Statistics. Charles Griffin.
- Greene WH. Econometric Analysis. Pearson.
- Hall AR. Generalized Method of Moments. Oxford University Press.
- Hansen LP. Large Sample Properties of Generalized Method of Moments Estimators. Econometrica. 1982;50(4):1029?1054.
- Briggs A, Claxton K, Sculpher M. Decision Modelling for Health Economic Evaluation. Oxford University Press.
Related Concepts (2)
Library
Publications
1
Statistical Analysis of Cost-Effectiveness Data — Willan & Briggs, 1st Edition ed., 2006 (John Wiley & Sons)
A synthesis of statistical methods for analysing cost-effectiveness data, including net-benefit regression, confidence intervals for the ICER, cost-effectiveness acceptability curves, and covariate adjustment. Part of the Wiley Statistics in Practice series.
BookView source →
Frequently Asked Questions (6)
What is the method of moments?
A technique estimating a distribution's parameters by equating sample moments, such as mean and variance, to their theoretical population equivalents.
Source: Casella G, Berger RL. Statistical Inference. 2nd ed. Duxbury; 2002.
How does the method of moments estimate parameters?
The method of moments estimates a distribution's parameters by matching quantities calculated from the sample, such as its mean and variance, to the theoretical expressions for those quantities in terms of the unknown parameters, then solving for the parameters. This makes it simple and quick, requiring no search for a maximum, which is why it is often used to provide starting values or when likelihood methods are awkward. It is generally less efficient than maximum likelihood, using the data less fully. Equating sample and theoretical moments is its method. Kirkwood and Sterne (2003) describe this technique.
Source: Kirkwood & Sterne 2003
How does the method of moments work?
The method of moments works by writing the population moments, such as the mean and variance, as functions of the unknown parameters, computing the corresponding sample moments from the data, setting the sample moments equal to the population expressions, and solving the resulting equations for the parameters. So the method of moments works by matching a sufficient number of sample and theoretical moments and solving for the parameters, which yields the estimates directly, requiring as many moment equations as there are parameters, and this simple matching makes it easy to apply, though the resulting estimators may not be as efficient as those from maximum likelihood.
Source: Casella & Berger 2002
What are the advantages of the method of moments?
The advantages of the method of moments include its simplicity and ease of computation, since it involves only matching moments and solving equations, often without iterative optimisation; and its generality, applying whenever the moments can be expressed in terms of the parameters. So the method of moments is advantageous for its straightforwardness and low computational demand, which makes it useful for quick estimates and as starting values for more complex methods such as maximum likelihood, and it can be applied without fully specifying the distribution, though these advantages come at the cost of efficiency, since it does not use all the information in the data as maximum likelihood does.
Source: Casella & Berger 2002
What are the limitations of the method of moments?
The limitations of the method of moments include that its estimators are generally less efficient than maximum likelihood estimators, having larger variance, since they use only certain moments rather than the full likelihood; that they can sometimes give estimates outside the valid parameter range; and that they may perform poorly for some distributions. So the method of moments is used with awareness that it trades efficiency for simplicity, which is why it is often superseded by maximum likelihood when efficiency matters, though it remains useful for its ease and as a source of initial estimates, with its limitations meaning it is not preferred when the most precise estimation is required.
Source: Casella & Berger 2002
How does the method of moments compare with maximum likelihood?
The method of moments estimates parameters by matching sample and population moments, while maximum likelihood chooses parameters to maximise the probability of the observed data. The method of moments is simpler and computationally easier but generally less efficient, whereas maximum likelihood uses all the information in the data and has better large-sample properties. So the two differ in approach and performance, with the method of moments offering simplicity and maximum likelihood greater efficiency, which is why maximum likelihood is usually preferred when precise, efficient estimates are wanted, and the method of moments is used for its ease or to provide starting values for the more demanding maximum likelihood computation.
Source: Fisher 1922
Trust Record
Verified by Dr Darrin Baines
British health economist
Professional identity: darrinbaines.org
Verification date: 18 Dec 2025
Content version: 1.0.0
Canonical Identity
- Persistent URI
- https://healtheconomics.wiki/concept/method-of-moments
- Term code
- HE-ES-SA-117
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