Method of Moment Estimators

Question

A sample of 4 observations (0.4, 0.7, 0.7, 0.9) collected from a continuous distribution with density,

f(x) = θ*(x^(θ−1)) ; 0 < x < 1

Estimate θ by the method of moment.

Answer 1

Question: A sample of 4 observations (0.4, 0.7, 0.7, 0.9) collected from a continuous distribution with density, f(x) = θ*(x^(θ−1)) ; 0 < x < 1. Estimate θ by the method of moment.

Answer: We're estimating one parameter, so we'll end up with one equation to solve for θ. Set the first sample moment equal to the first theoretical moment and solve for θ. Here, the first sample moment will be xbar, the sample average. The theoretical first moment, or E(X), is a different matter to be approached by the definition of expectation (here, for a continuous random variable):

E(X) = ∫x*θ*(x^θ−1) dx (from x = 0 to x = 1)

E(X) = θ ∫x*(x^θ−1) dx (from x = 0 to x = 1)

E(X) = θ ∫x^θ dx (from x = 0 to x = 1)

E(X) = [θ/(θ+1)] (x^θ+1), evaluated from x=0 to x=1

E(X) = [θ/(θ+1)] (1^θ+1 - 0^θ+1)

E(X) = [θ/(θ+1)] (1 - 0)

E(X) = θ/(θ+1) <----- the first theoretical moment

Now we can return to the essence of this method: set the first sample moment equal to the first theoretical moment and solve for θ.

M₁ = E(X) <--- again, setting the first sample moment equal to the first theoretical moment...

xbar = θ/(θ+1) <----- ...where the first sample moment is xbar.

xbar*(θ+1) = θ

xbar*(θ) + xbar = θ

xbar*(θ) - θ = - xbar

θ(xbar - 1) = - xbar

θ = (- xbar) / (xbar - 1)

θ = xbar / (1 - xbar)

^

θ_MOM = xbar / (1 - xbar) is the method of moments estimator for θ.

To get your final answer, take the average of your sample data, call it xbar, and plug it into the MOM estimator above and evaluate. I hope this helps!