Answered: Let {Xi}ni be IID from Gamma(α, λ).…

Algebra & Trigonometry with Analytic Geometry

13th Edition

ISBN:9781133382119

Author:Swokowski

Publisher:Swokowski

Chapter2: Equations And Inequalities

Section2.7: More On Inequalities

Problem 44E

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Question

Let {X_i}ⁿ_i be IID from Gamma(α, λ). Find the method of moments estimator for α and λ and use simulations to check whether the estimator is biased or not. Use α = 10 and λ = 2 for your simulations.

Expert Solution

Step 1

From the given information,

$X_{1}, X_{2}, . . . . . X_{n}$ are iid from Gramma $(α, λ)$ .

For the given sample,

$E (X) = α λ E (X^{2}) = λ α^{2} (λ + 1)$

For the given population,

$M_{1} = α λ M_{2} = λ α^{2} (λ + 1)$

For the given sample,

$M_{1} = \bar{x} M_{2} = \frac{1}{n} \sum x_{i}^{2}$

By comparison,

$α λ = \bar{x} λ α^{2} (λ + 1) = \frac{1}{n} \sum x^{2} λ^{2} α^{2} + λ α^{2} = \frac{1}{n} \sum x^{2} {\bar{x}}^{2} + α \bar{x} = \frac{1}{n} \sum x^{2} \hat{α} = \frac{\frac{1}{n} \sum x^{2} - {\bar{x}}^{2}}{\bar{x}}$

And,

$α λ = \bar{x} \frac{(\frac{1}{n} \sum x^{2} - {\bar{x}}^{2})}{\bar{x}} * \hat{λ} = \bar{x} \hat{λ} = \frac{{\bar{x}}^{2}}{\frac{1}{n} \sum x^{2} - {\bar{x}}^{2}}$