Hey guys, in this video I'm going to do another
video on method of moments estimation.
I previously have done one other video on
method of moments estimation
and I'll put a link in the description box below for that video.
In my previous video, I
show an example of how you can do method of moments
estimation--or how you can find an estimator
based on the method of moments--using the first moment.
So I thought I'd make another video showing how you can find
a method of moments estimator based on any moment.
Not just the first moment, but any Kth
moment. So based on the Kth
moment, where K is just a number.
I'm going to do a very simple example using a
random variable X that's uniformly
distributed from 0 to theta and we're going to find the
method of moments estimator for theta.
Based on this probability distribution,
the pdf f(x) is going to be
equal to 1 over theta minus
0, which is just 1 over theta.
And as I went over in the previous video,
if you're doing
the method of--if you're finding the method of moments
estimator for this distribution using the
first moment, then that's just finding the expected value
of the random variable X. In this case,
the formula for that is just X times the pdf,
which in this case is 1 over theta dx.
And then this is from negative infinity to infinity.
For this particular problem, the pdf only
exists from 0 to theta, so that's what you would plug into your
anti-derivative. You would go 0 to theta
and then just solve this using integration.
If you do that, you would get X squared over 2
times 1 over theta from 0 to theta.
And then if you plug in theta, you would get theta squared over
2 theta minus 0, which would just give you
theta over 2. And so
to find the method of moments estimator, you would
say that the sample mean is equal to the expected value,
which is equal to theta over 2
and then just solve for--basically put a tilde over that
and then solve for theta. That would give you
that 2 times the sample mean
is the method of moments estimator based
on the first moment. That's kind of just
another example using the first moment and
if you wanted to find the method of moments estimator based
on any other moment. So say the second moment
or third moment or any Kth moment,
then it's just a very simple extension of this idea
of finding the expected value of X
but now instead of finding the expected value
of X, you're going to find the expected value of X to the K.
Implicitly what we were doing for the first moment
was this was X to the 1
but now it's going to be to any integer K.
And then the only difference for the formula is it's
still going to be the pdf times X but instead of having
X to the 1, it's going to be X to the K. We're going to integrate
X to the K times the pdf,
which is 1 over theta dx.
And this will still be from 0 to theta.
If we integrate this, that's going to be X to the
K plus 1 divided by K plus 1
times 1 over theta from 0 to theta.
And if we plug in theta and 0, that's going to give us
theta to the K plus 1 over K
plus 1 times theta,
which is equal to theta to the K
over K plus 1.
And the advantage of doing it this way using
K instead of a particular number is
now, if you wanted to find the method of moments estimator
based on
several different moments, it's very easy to just plug in
a number K and find the method of moments
estimator. For example if we wanted to find
the method of moments estimator
for the second moment, then we're just going to plug in
2 for K. If we plug that in here,
that's going to be theta squared over 3.
And then to find the actual estimator
for theta, we're going to do theta tilde
and we're going to say that this is equal
to the sample mean of X squared
which is also equal to the sum of Xi squared
divided by n. This is just what
the definition of sample mean is and we're going to
set that equal to theta tilde squared divided
by 3. And same thing as last time, we're just going to
solve for theta tilde.
We're going to get theta tilde on one side
and then move everything else to the other side.
That's going to be 3 times the sum of Xi
squared divide by n and
take the square root of that and then that would give you
your method of moments estimator for
this probability distribution based on the
second moment and you can just plug in any K to get
the third or fourth or any Kth moment
as well. And that's all I wanted to go over in this
problem. And thank you guys for watching!
*chiptune music*