find an estimator for theta from gamma distribution using method od moment
To find an estimator for the parameter theta (τ) in a gamma distribution using the method of moments, follow these steps:
Step 1: Define the moments
- For a gamma distribution, the kth moment is given by m_k = E(X^k), where X represents the random variable following the gamma distribution and k is a positive integer.
Step 2: Calculate the sample moments
- Calculate the sample moments from the given data set. Suppose you have n observations: x_1, x_2, ..., x_n.
- To estimate theta (τ), you need to calculate the first moment (m_1) and second moment (m_2) using the sample data.
Step 3: Set up equations using moments
- Equate the sample moments (m_1 and m_2) to their corresponding population moments (E(X) and E(X^2)).
Step 4: Solve the equations
- Solve the equations obtained in Step 3 for the unknown parameter theta (τ).
Step 5: Compute the estimator
- Once you have solved the equations, substitute the values of m_1 and m_2 into the estimator equation to get the estimator for theta.
The estimator for theta (τ) obtained using the method of moments is the solution to the equations derived in Step 4.
Note: The gamma distribution has two parameters: shape parameter (k) and scale parameter (θ). The method of moments can be used to estimate either of these parameters, depending on which parameter you are interested in estimating.