Method of Moments

While Maximum Likelihood Estimation (MLE) is the gold standard for efficiency, it can sometimes be computationally difficult or mathematically intractable. The Method of Moments (MoM) is an alternative technique that is often simpler to derive and compute.

The core idea of Method of Moments is to equate the sample moments (calculated from data) with the theoretical population moments (functions of parameters) and solve the resulting system of equations.

1. Defining Moments

Population Moments

The k-th population moment is the expected value of the k-th power of the random variable X. μk = E[Xk]

First Moment: μ1 = E[X] = μ (Mean)
Second Moment: μ2 = E[X2] = σ2 + μ2

Sample Moments

The k-th sample moment is the average of the k-th power of the observed data points. mk = (1/n) Σ xik

First Sample Moment: m1 = x̄ (Sample Mean)
Second Sample Moment: m2 = (1/n) Σ xi2

2. The Algorithm

To estimate k unknown parameters (θ1, θ2, … θk):

Calculate the first k sample moments from the data.
Write down the expressions for the first k population moments in terms of the parameters.
Set the sample moments equal to the population moments.
Solve the resulting system of equations for θ.

Example: Gamma Distribution

The Gamma distribution has two parameters: shape α and scale β. The theoretical moments are:

E[X] = αβ
Var(X) = αβ2 ⇒ E[X2] = Var(X) + (E[X])2 = αβ2 + α2β2

Step 1: Calculate sample mean x̄ and sample variance s2. Step 2: Equate moments:

x̄ = αβ
s2 = αβ2 (Using variance directly is equivalent and easier)

Step 3: Solve for α and β. From equation 2: β = s2 / (x̄ / β) … let’s simply divide equation 2 by equation 1: s2 / x̄ = (αβ2) / (αβ) = β

So, β̂ = s2 / x̄.

Substitute back into equation 1: α̂ = x̄ / β̂ = x̄2 / s2.

These are the Method of Moments estimators for the Gamma distribution. They are simple closed-form solutions!

3. System Design Perspective: MoM vs MLE

Why would a Systems Engineer care about MoM if MLE is “better”?

Feature	Method of Moments (MoM)	Maximum Likelihood (MLE)
Computational Cost	Low. Usually closed-form algebraic solutions. O(1) after computing stats.	High. Often requires iterative optimization (Gradient Descent, Newton-Raphson).
Data Requirements	Streaming Friendly. Can easily update running means/variances.	Batch Dependent. Often needs the full dataset or complex online approximations.
Precision	Lower. Higher variance (less efficient).	Highest. Asymptotically achieves the Cramer-Rao Lower Bound.
Initialization	Excellent. Used to initialize MLE algorithms.	Needs Init. Iterative solvers need a good starting point to avoid local optima.

[!TIP] Production Pattern: Use MoM to get a “quick and dirty” estimate in real-time (e.g., for monitoring dashboards), and use MLE for offline batch processing where precision matters more than speed. Also, use MoM estimates as the starting point for your MLE optimization loop.

4. Interactive: Moment Matcher

Adjust the parameters α and β of a Gamma distribution to match the observed sample mean and variance (shown as dashed lines).

Moment Matching (Gamma)

Target Mean: 5.0

Target Var: 2.0

α (Shape): 2.0

β (Scale): 1.0

Not Matched

5. Code Examples: Estimating Gamma Parameters

Here we compare how to estimate the parameters of a Gamma distribution using both Method of Moments (simple formulas) and MLE (iterative optimization).

Python

import numpy as np
from scipy.stats import gamma

# 1. Generate Data (True Alpha=5, Beta=2)
np.random.seed(42)
true_alpha, true_beta = 5.0, 2.0
data = np.random.gamma(true_alpha, scale=true_beta, size=1000)

# 2. Method of Moments
sample_mean = np.mean(data)
sample_var = np.var(data, ddof=1) # Unbiased variance

# Derived formulas: mean = a*b, var = a*b^2
# b = var / mean
mom_beta = sample_var / sample_mean
mom_alpha = sample_mean / mom_beta

print(f"True Params: Alpha={true_alpha}, Beta={true_beta}")
print(f"MoM Estimates: Alpha={mom_alpha:.4f}, Beta={mom_beta:.4f}")

# 3. MLE (using Scipy)
# Scipy fits (alpha, loc, scale). We fix loc=0.
mle_alpha, _, mle_beta = gamma.fit(data, floc=0)

print(f"MLE Estimates: Alpha={mle_alpha:.4f}, Beta={mle_beta:.4f}")

Java

public class MomGamma {
    public static void main(String[] args) {
        // Sample data (small sample for demo)
        double[] data = {9.5, 10.2, 8.8, 11.5, 9.9, 10.1};

        double sum = 0;
        for (double x : data) sum += x;
        double sampleMean = sum / data.length;

        double sumSqDiff = 0;
        for (double x : data) sumSqDiff += Math.pow(x - sampleMean, 2);
        double sampleVar = sumSqDiff / (data.length - 1); // Unbiased

        // MoM Formulas:
        // Mean = alpha * beta
        // Var = alpha * beta^2
        // beta = Var / Mean
        double momBeta = sampleVar / sampleMean;
        double momAlpha = sampleMean / momBeta;

        System.out.printf("Sample Mean: %.2f, Var: %.2f%n", sampleMean, sampleVar);
        System.out.printf("MoM Alpha: %.4f%n", momAlpha);
        System.out.printf("MoM Beta: %.4f%n", momBeta);
        System.out.println("Note: MLE would require an iterative solver (e.g. Apache Commons Math).");
    }
}

Go

package main

import (
    "fmt"
    "math"
)

func main() {
    // Sample data
    data := []float64{9.5, 10.2, 8.8, 11.5, 9.9, 10.1}

    sum := 0.0
    for _, x := range data {
        sum += x
    }
    sampleMean := sum / float64(len(data))

    sumSqDiff := 0.0
    for _, x := range data {
        sumSqDiff += math.Pow(x-sampleMean, 2)
    }
    sampleVar := sumSqDiff / float64(len(data)-1)

    // MoM Formulas
    momBeta := sampleVar / sampleMean
    momAlpha := sampleMean / momBeta

    fmt.Printf("Sample Mean: %.2f, Var: %.2f\n", sampleMean, sampleVar)
    fmt.Printf("MoM Alpha: %.4f\n", momAlpha)
    fmt.Printf("MoM Beta: %.4f\n", momBeta)
    fmt.Println("Note: MLE would require numerical optimization (e.g. Gonum).")
}

6. Summary

The Method of Moments is a powerful tool in your statistical toolkit. While not always the most efficient, its simplicity makes it an excellent starting point for parameter estimation and streaming applications.