[Math] Method of Moments on a Uniform distribution (a,b)

algebra-precalculusprobability distributionsstatistics

Given that $x_1,x_2,…,x_n$ are i.i.d. random variables drawn from a Uniform distribution over $(a,b)$ (with $a<b$ and both are unknown), I hope to estimate $a$ and $b$ using the method of moments. My problem (embarrassingly enough) comes when I attempt to solve the system of equations obtained.

First, set $\bar{x}=\frac{a+b}{2}$, as that is the expected value of a uniform distribution. (Where $\bar{x}=\frac{x_1+x_2+…+x_n}{n}$)

Then, the second moment $\sum_{i=1}^{n}\frac{[E(x_i)^2]}{n}$$=\frac{(b-a)^2}{12}+(\frac{b+a}{2})^2$. (Just the variance plus the expected value squared).

So now, two systems of equations, two unknowns (as I'm hoping to solve for a and b in terms of $\bar{x}$ and $x_i$.

Foiling out the second equation and letting $a=2\bar{x}-b$, I obtain the following equation:

$4b^2-8b\bar{x}+14\bar{x}^2=\sum_{i=1}^{n}\frac{[E(x_i)^2]}{n}$.

I think I'm missing two steps from here: first, I'm not sure how to deal with the summation on the right side. Second, my grasp of algebra is not nearly what it once was, and I'm not really sure how to solve for b at all. Any help would be greatly appreciated!
Thanks!

Best Answer

I need to make a correction to your equation. First of all, your notation is off, as @PatrickLI noted. Let's call your mean $E(X)$ and your average of data squared (i.e. $\frac{1}{n} \sum_i x_i^2$) $E(X^2)$, where $X$ is the random variable associated with the above uniform distribution.

Using similar manipulations as you made, I get

$$b^2 - 2 E(X) b -[3 E(X^2)-4E(X)^2]=0$$

which may be solved using the quadratic formula:

$$b=E(X) + \sqrt{3} \sqrt{E(X^2) - E(X)^2}$$

Then $a=2 E(X)-b = E(X) - \sqrt{3} \sqrt{E(X^2) - E(X)^2}$

I chose the $+$ solution for $b$ because $b>a$. I hope this helps.

NOTE

You can verify this solution for $a$ and $b$ with random data in a program that generates uniform random variates such as Mathematica. For example, here is some Mathematica code that generates values of $(a,b)$ from an input interval and a number of data points to generate:

devCheck[left_, right_, n_] := Module[{m1, m2, q},
q = RandomVariate[UniformDistribution[{left, right}], n];
m1 = Mean[q];
m2 = Mean[q^2];
a = m1 - Sqrt[3] Sqrt[m2 - m1^2];
b = m1 + Sqrt[3] Sqrt[m2 - m1^2];
{a, b}];
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