Suppose that X1, . . . , Xn form a random sample from the uniform distribution on the interval [θ1, θ2], where both θ1 and θ2 are unknown (−∞ < θ1 < θ2 < ∞). Find the MOM estimates of θ1 and θ2.

Answered: Suppose that X1, . . . , Xn form a…

A First Course in Probability (10th Edition)

10th Edition

ISBN:9780134753119

Author:Sheldon Ross

Publisher:Sheldon Ross

Chapter1: Combinatorial Analysis

Section: Chapter Questions

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Suppose that X₁, . . . , X_n form a random sample from the uniform distribution on the interval [θ₁, θ₂], where both θ₁ and θ₂ are unknown (−∞ < θ₁ < θ₂ < ∞). Find the MOM estimates of θ₁ and θ₂.

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Step 1

From the given information, $X_{1}, X_{2}, . . . ., X_{n}$ form a random sample from the uniform distribution on the interval $[θ_{1}, θ_{2}]$ .

For the given sample,

$\bar{X} = \frac{{\sum X_{i}}_{i = 1}^{n}}{n} S^{2} = \frac{{\sum (X_{i}}_{i = 1}^{n} - \bar{X})^{2}}{n - 1}$

For the given distribution $U [θ_{1}, θ_{2}]$ ,

$μ = \frac{θ_{1} + θ_{2}}{2} σ^{2} = \frac{{(θ_{2} - θ_{1})}^{2}}{12}$

Here, the method of moment estimators for θ₁ and θ₂ are obtained by equating $\bar{X} = μ a n d S^{2} = σ^{2}$

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