1. METHOD OF MOMENTS

x₁, ... ,x_n are independent random variables. Take the functions g₁, g₂, ... , g_n and look at the new random variables

then the y's are also independent. If all the g are the same, then the y are iid.

Now suppose x₁, x₂, ... are iid. Fix k a positive integer. Thenare iid and by the weak law of large numbers

which are the k^th moments about the origin.

Define m_k = Ex^k as the k^th moment of x so

Suppose you wish to estimate some parameter,.

We know m_k = E(x^k) for k = 1, 2, ... and suppose

and g is continuous.

The sample moment is

Idea: If n is large thenshould be close to m_k for k = 1, 2, ... , N so should be close to.

Example:

The method of moments estimator foris

Example:

1

X is a continuous r.v. distributed uniformly on the interval [a, b]. We wish to estimate a and b. From experience we know

Suppose = 5 and S² = 2.5