1. ( (20 %) ) Let ( X{1}, X{2}, ldots, X{n} ) be an i.i.d....

1. ( (20 %) ) Let ( X_{1}, X_{2}, ldots, X_{n} ) be an i.i.d. sample from a uniform distribution with pdf ( f(x ; heta)= ) ( 1 /(2 heta) ) for ( x in[- heta, heta] ), zero elsewhere, where ( heta>0 ) is an unknown parameter. (a) Find the MME (method of moments estimation) ( hat{ heta} ) of ( heta ). (b) Is ( hat{ heta} ) consistent? Why?

Transcript

00:01 We have these independent samples, x1 to xn.

00:02 They come from this distribution, 1 over 2 theta on the interval x being between negative theta up to theta, closed interval, and theta is positive.

00:16 And we're going to find the method of moments estimator of theta, theta hat, and we're going to show whether or not theta hat is consistent.

00:23 So one thing to note as we're looking at this is this is a uniform distribution.

00:32 And that's going to help us figure out the method of moments estimator.

00:35 So if we take the method of moments estimator of theta, to be theta hat, well, the first moment, the mean, is equal to the expected value of x, which in this case is just the center of our interval here, which is the, you know, if we could take this as the, we could also take it as x times our function, 2 theta, 3 theta, 2 theta, dx.

01:10 And when we evaluate this, we get x squared on 4 theta, negative theta to theta.

01:18 But when we do this, we end up with theta squared over 4 theta minus theta squared over 4 theta, which is 0.

01:25 So we don't get anything.

01:27 We don't get any information about this.

01:29 But we can find the second moment, which is the expected value of x squared.

01:38 And this will give us our second moment.

01:40 Which ends up being the variance of our uniform distribution.

01:48 We'll go through the process here just so you can kind of see no matter what kind of function you have, if it's uniform or not.

01:55 So x squared, 1 over 2 theta dx, x cubed on 6 theta, theta to theta, you get theta cubed on 6 minus negative theta cubed on 6, or actually that'd be on 6 theta.

02:13 Sorry about that.

02:14 On 6 theta, which is 2 theta squared on 6, which is thirds.

02:25 There we go.

02:26 And something to note, this is our theoretical.

02:36 It's our theoretical second moment.

02:40 So there's that.

02:41 But we haven't found our method of moment estimator.

02:42 Well, that's fine.

02:42 This is what we have.

02:45 But to find the estimator, we need our sample second moment, which is s squared...

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