Method of moments

The Method of Moments (MoM) is a statistical technique used for estimating the parameters of a probability distribution. It is one of the oldest and simplest methods of parameter estimation. Developed by Karl Pearson in the early 20th century, this method involves equating the theoretical moments of a distribution (i.e., its expected values of powers of the random variable) to the sample moments (i.e., the corresponding averages computed from the sample data).

Overview

The Method of Moments is based on the principle that the population moments (like mean, variance) should be equal to the corresponding sample moments. For a given random variable X with a probability distribution depending on parameters θ₁, θ₂, ..., θ_k, the method equates the first k theoretical moments of X to the first k sample moments.

Mathematical Formulation

Given a random variable X with unknown parameters θ₁, θ₂, ..., θ_k, the j-th moment of X about the origin is given by:

E(X^j) = μ_j(θ)

where μ_j is a function of the parameters θ.

The sample moments are calculated from a sample x₁, x₂, ..., x_n of X as:

m_j = (1/n) ∑_i=1ⁿ x_i^j

The method of moments estimates θ by solving the equations:

μ_j(θ) = m_j for j = 1, 2, ..., k

Applications

The Method of Moments is used in various fields including Economics, Statistics, and Engineering. It is particularly useful when the moment equations are simple enough to solve analytically. It has been applied in contexts such as estimating the parameters of the Normal distribution, Exponential distribution, and Gamma distribution.

Advantages and Disadvantages

Advantages

• Simplicity: The method is straightforward and easy to implement.
• Analytical solutions: In some cases, the method allows for analytical solutions, avoiding the need for numerical optimization.

Disadvantages

• Inefficiency: MoM estimates are not always efficient, especially if the moments do not contain enough information about the parameters.
• Bias: The method can produce biased estimates, particularly in small samples.
• Solvability: The moment equations may not always have a unique or real solution.

Comparison with Other Methods

The Method of Moments is often compared to the Maximum Likelihood Estimation (MLE). While MLE is generally more efficient in terms of the variance of the estimates, MoM is simpler and does not require assumptions about the distribution of data, such as the existence of a likelihood function.

See Also

• Estimator
• Sample mean
• Sample variance
• Karl Pearson

References

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