1. METHOD OF MOMENTS
x1, ... ,xn are independent random variables. Take the functions g1, g2,
... , gn 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 kth moments about the origin.
Define mk = Exk as the kth
moment of x so
Suppose you wish to estimate some parameter,.
We know mk = E(xk) for k = 1, 2, ... and suppose
and g is continuous.
The sample moment is
Idea: If n is large thenshould be close to mk 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