Question 1179729: Let X1,...,Xn be a random sample from the beta distribution with parameters α and β. Find the method of moments estimator for α and β.
Thank you...
Answer by CPhill(1987)   (Show Source):
You can put this solution on YOUR website! Here's how to find the method of moments estimators for α and β in a beta distribution:
**1. Recall the Moments of the Beta Distribution:**
For a beta distribution with parameters α and β, the first two moments are:
* E[X] = α / (α + β)
* E[X²] = (α(α + 1)) / ((α + β)(α + β + 1))
**2. Calculate the Sample Moments:**
We need to calculate the first two sample moments:
* First sample moment (X̄): X̄ = (1/n) * Σ[i=1 to n] Xi
* Second sample moment (M2): M2 = (1/n) * Σ[i=1 to n] Xi²
**3. Set Up the System of Equations:**
Equate the population moments to the sample moments:
* α / (α + β) = X̄
* (α(α + 1)) / ((α + β)(α + β + 1)) = M2
**4. Solve for α and β:**
Let's solve this system of equations:
* From the first equation, we have: α = X̄(α + β) => α(1 - X̄) = X̄β => β = α(1 - X̄) / X̄.
* Substitute this expression for β into the second equation:
(α(α + 1)) / ((α + α(1 - X̄) / X̄)(α + α(1 - X̄) / X̄ + 1)) = M2
Simplify:
(α(α + 1)) / ((α/X̄)(α/X̄ + 1)) = M2
(α(α + 1)X̄²) / (α(α + X̄)) = M2
(α + 1)X̄² / (α + X̄) = M2
(α + 1)X̄² = M2(α + X̄)
αX̄² + X̄² = αM2 + M2X̄
α(X̄² - M2) = M2X̄ - X̄²
α = (M2X̄ - X̄²) / (X̄² - M2)
α = X̄²(M2/X̄ - 1) / X̄²(1 - M2/X̄²)
α = (M2/X̄ - 1) / (1 - M2/X̄²)
α = (X̄M2 - X̄²) / (X̄² - M2)
α = X̄(M2 - X̄) / (X̄² - M2)
* Now, substitute the expression for α back into the equation for β:
β = α(1 - X̄) / X̄
β = [(M2X̄ - X̄²) / (X̄² - M2)] * (1 - X̄) / X̄
β = (M2 - X̄)(1 - X̄) / (X̄² - M2)
**5. Express in Terms of X̄ and S² (Sample Variance):**
We can express M2 in terms of the sample mean (X̄) and sample variance (S²):
* S² = M2 - X̄²
* M2 = S² + X̄²
Substitute this into the expressions for α and β:
* α = X̄(S² + X̄² - X̄) / (X̄² - (S² + X̄²)) = X̄(S² + X̄(X̄-1)) / (-S²)
α = X̄(1-X̄)/S² * X̄
* β = (S² + X̄² - X̄)(1 - X̄) / (-S²)
β = (1-X̄)(1-(X̄(1-X̄)/S²))
* α = X̄(1 - X̄) / S² - X̄
* β = (1-X̄)(1- (X̄(1-X̄)/S²))
* α = X̄(1 - X̄) / S²
* β = (1 - X̄)(1 - (X̄(1 - X̄) / S²))
* β = (1-X̄)α/X̄ = α(1-X̄)/X̄
**Final Answer:**
The method of moments estimators for α and β are:
* α̂ = X̄(1 - X̄) / S²
* β̂ = (1 - X̄) / X̄ * α̂ = (1 - X̄) / X̄ * [X̄(1 - X̄) / S²] = (1 - X̄)² / S²
Where:
* X̄ is the sample mean.
* S² is the sample variance.
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