Back to notes
Define the Method of Moments in the context of estimator design.
Press to flip
The Method of Moments involves equating sample moments to theoretical moments from the distribution to solve for unknown parameters.
How is the first sample moment M₁ calculated for a given n IID samples?
M₁ = (1/n) ΣXᵢ, which is the sample mean.
What are the main steps in using the Method of Moments procedure?
1. Calculate sample moments from the data. 2. Equate these sample moments to the theoretical moments of the distribution. 3. Solve the resulting equations for the unknown parameters.
Why might you need multiple moments to estimate parameters using the Method of Moments?
If a single moment isn’t sufficiently informative about all parameters, additional moments can provide the necessary equations to solve for multiple unknowns.
For a Bernoulli-distributed random variable, what is the Method of Moments estimator of parameter p?
The estimator of p is the sample mean, i.e., p̂ = M₁.
What practical applications can you calculate using the Method of Moments estimators for Bernoulli, Poisson, and Normal distributions?
For Bernoulli: Count successes in samples. For Poisson: Count events over intervals. For Normal: Compute mean and variance from samples.
For an Exponential distribution, how do you estimate the parameter λ using the Method of Moments?
The estimator for λ is the reciprocal of the sample mean, i.e., λ̂ = 1/M₁.
What are the Method of Moments estimators for the parameters α and β of a Gamma distribution?
The estimators are α̂ = (M₁²)/(M₂ - M₁²) and β̂ = (M₁)/(M₂ - M₁²).
What system of equations do you solve to estimate μ and σ² for a Normal distribution using the Method of Moments?
μ = M₁ and σ² = M₂ - M₁².
What is the Method of Moments estimator for the parameter λ of a Poisson distribution?
The estimator for λ is the sample mean, i.e., λ̂ = M₁.
Explain why the Method of Moments is straightforward to use in parameter estimation.
The Method of Moments directly uses moment calculations from the sample data, allowing easy substitution into the theoretical moment equations.
Describe the Method of Moments estimator for parameters n and p of a Binomial distribution.
Use E[X] = np and E[X²] = n²p² + np(1 - p) solved with sample moments M₁ and M₂ to estimate n and p.
How do the Central Limit Theorem and Weak Law of Large Numbers support the use of sample moments?
These theorems ensure that as the sample size increases, the sample moments will converge to the true moments of the distribution.
What are the characteristics of estimators discussed in the previous lecture?
Bias, Variance, Squared error risk (mean squared error)
What is the second sample moment M₂ and how is it calculated?
M₂ is the sample second moment calculated as (1/n) ΣXᵢ².
Previous
Next