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

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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.

Expert Solution
Step 1

From the given information, X1,X2,....,Xn form a random sample from the uniform distribution on the interval θ1,θ2.

For the given sample,

X¯=Xii=1nnS2=(Xii=1n-X¯)2n-1

For the given distribution Uθ1,θ2,

μ=θ1+θ22σ2=(θ2-θ1)212

Here, the method of moment estimators for θ1 and θ2 are obtained by equating X¯=μ and S2=σ2

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