Statistics - Method of Moments Estimators

Question

A random sample of n observations, 𝑌_1,𝑌_2,…,𝑌_𝑛 is selected from a population in which 𝑌_𝑖, 𝑖=1,2,…𝑛, possesses a gamma probability density function with parameters 𝛼 and 𝛽.

How many parameters ? What are they?

How many equations you have to build up according to the population and sample moments?

Find method of moments estimators for the unknown parameters 𝛼 and 𝛽.

Answer 1

Penn State offers as very thorough description of method of moments estimation on its website for its undergraduate math stats class,

https://online.stat.psu.edu/stat415/lesson/1/1.4.

The resources is free to use for all and to disseminate. It gives several worked examples including solving for the alpha and beta parameters of the gamma distribution. alpha is the shape parameter and beta is the scale parameter

In running method of moments estimation, you simply substitute your sample moments for the corresponding population moments. For instance, in the gamma distribution, the theoretical mean is alpha*Beta. You may use bar-X as your estimate of the mean. The second moment is your variance, but you will need to divide by n instead of n - 1. The theoretical variance of the gamma distribution = alpha*beta^2. You may substitute the sum(x - bar-x)^2 /N for it.

If you need to estimate parameters that are not one of the moments, as is the case with the scale and shape parameters of the gamma distribution, you use as many moments as you have parameters to solve for. So you will need the first two moments of the gamma function in order to solve for alpha and beta.

Solve for alpha in terms of beta

1. The theoretical mean = alpha*beta = bar-X, hence alpha = bar-X/beta

Solve for beta

alpha*beta^2 = (bar-X/beta)*beta^2 = sum(x - bar-x)^2 /N

etc.

Substitute value for beta into estimate of alpha = bar-X/beta