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I'm working on a Pareto distribution function, where I have to find the method of moments estimate of $\theta$. The function is:

$f(x|x_0, \theta) = \theta \cdot x^{\theta}_0 \cdot x^{-\theta-1}$

When $x > x_0$ and $\theta > 1$. Assume that $x_0 > 0$ is known. $\theta$ is unknown and a random sample $(X_1, X_2,...,X_n)$ satisfying a Pareto distribution is given.

At the moment I'm mostly a bit confused about the notation and variables, especially the $\theta$.

My approach so far is to use the following from my textbook:

That the $k$th moment of a random variable taken about the origin is

$\mu_k' = E(X^{k})$ and the corresponding $k$th sample moment is the average.

Relating the above to my pareto function, does $k$ corresponds to $\theta$?

bestmate21
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    No. You need to find the theoretical first moment (expectation) of the Pareto distribution in terms of $x_0$ and $\theta$, then compare this to the sample first moment (average), and then solve this to find "method of moments" estimator of $\theta$ in terms of the sample average and $x_0$. Here $k=1$, though you might want to consider other values if for example you were trying to estimate both $\theta$ and $x_0$ – Henry Dec 17 '21 at 11:11

1 Answers1

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Assume that $x_0>0$ is known.

The job is to express the desired parameter ($\theta$) as a function of $\mu$ than substitute $\mu$ with the sample mean $\overline{X}_n$

You know that

$$\mu=\frac{\theta x_0}{\theta-1}$$

that is

$$\theta=\frac{\mu}{\mu-x_0}$$

and thus, applying MoM:

$$\hat{\theta}_{\text{MM}}=\frac {\overline{X}_n}{\overline{X}_n-x_0}$$

tommik
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  • Thank you for your reply! I know it just me been staring at this problem for too long, but did you get from step 1 to step 2? – bestmate21 Dec 22 '21 at 09:02
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    @bestmate21 : step 1 is to define the first population's moment. Step 2 is to solve the equation w.r.t. the unknown parameter. Step 3 is to substitute the first population's moment with the first sample moment. By the way did you forget to accept my answer? – tommik Dec 22 '21 at 09:50
  • Thank you! That makes sense. Yes I did, thanks for reminding me. – bestmate21 Dec 22 '21 at 10:22