Methods of Moments Estimation Watch

Chittesh14
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Question is at the bottom of the image really.
If I have a formula for my population parameter in terms of the population moments.
Can I just say the estimate of that population parameter is the same as: substituting the estimates of those population moments i.e. the sample moments, into the formula?
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Gregorius
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(Original post by Chittesh14)
I'm afraid I couldn't follow your written work. But it looks like you're making heavy weather of it! The point of the method of moments is this: In the population you have E[X] = \mu and E[X^2] = \mu^2 + \sigma^2 . In the sample, you have first and second moments given by  \frac{1}{n}\Sigma X_{i} and  \frac{1}{n}\Sigma X_{i}^{2}, respectively. You get the method of moments estimators \hat{\mu} and \hat{\sigma} by equating these to get a set of simultaneous equations.

\displaystyle \hat{\mu} = \Sigma X_{i}

\displaystyle \hat{\sigma}^{2} + \hat{\mu}^2  = \Sigma X_{i}^{2}

The rest is algebra.
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Chittesh14
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(Original post by Gregorius)
I'm afraid I couldn't follow your written work. But it looks like you're making heavy weather of it! The point of the method of moments is this: In the population you have E[X] = \mu and E[X^2] = \mu^2 + \sigma^2 . In the sample, you have first and second moments given by  \frac{1}{n}\Sigma X_{i} and  \frac{1}{n}\Sigma X_{i}^{2}, respectively. You get the method of moments estimators \hat{\mu} and \hat{\sigma} by equating these to get a set of simultaneous equations.

\displaystyle \hat{\mu} = \Sigma X_{i}

\displaystyle \hat{\sigma}^{2} + \hat{\mu}^2  = \Sigma X_{i}^{2}

The rest is algebra.
OK, that makes much more sense - the way you've written it. I was always confused which to estimate and all that stuff.
So, can I say: \sigma^2 = E[X^2] - \mu^2, and hence the estimator of \sigma^2 is given by the sum of the estimators of E[X^2] and (\mu)^2.
So, \displaystyle \hat{\sigma}^{2} + \hat{\mu}^2 = \hat{E[X^2]}?
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Gregorius
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(Original post by Chittesh14)
OK, that makes much more sense - the way you've written it. I was always confused which to estimate and all that stuff.
This is one of those instances where it's easy to get lost in notation (I can remember being confused by this forty years ago!) and it's best, perhaps, to put it in words. In the population, the moments are functions of the parameters that specify the population distribution. You have to find these functions. Then turn to the sample: the sample moments are set equal to these functions applied to the estimators that you're trying to find. Now solve these equations for the estimators.

So, can I say: \sigma^2 = E[X^2] - \mu^2, and hence the estimator of \sigma^2 is given by the sum of the estimators of E[X^2] and (\mu)^2.
So, \displaystyle \hat{\sigma}^{2} + \hat{\mu}^2 = \hat{E[X^2]}?
I think you're getting muddled - but why would you want to do any more theory? You have two equations in the two unknown estimators \hat{\mu} and \hat{\sigma}, solve them.
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Chittesh14
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(Original post by Gregorius)
This is one of those instances where it's easy to get lost in notation (I can remember being confused by this forty years ago!) and it's best, perhaps, to put it in words. In the population, the moments are functions of the parameters that specify the population distribution. You have to find these functions. Then turn to the sample: the sample moments are set equal to these functions applied to the estimators that you're trying to find. Now solve these equations for the estimators.
Sorry, got carried away with sports a bit.
Thank you, yeah the notation is awful and tough to deal with until you really understand it properly.
OK makes sense.

I think you're getting muddled - but why would you want to do any more theory? You have two equations in the two unknown estimators \hat{\mu} and \hat{\sigma}, solve them.
Regarding this, I was just saying it in general. Of course, after reading your method - I would follow the method.
But, in general - I was just saying that if I know the variance \sigma^{2} is equal to E[X^2] - (E[X])^2 = E[X^2] - \mu^{2}. So, let's say I didn't read your method - and I am just simply wondering:
I already know that the variance \sigma^{2} is a function of the population moments i.e. 2nd population moment - (1st population moment)^2.... so, can I say that: if I put an estimator on each of the variables, the equation is still valid.
I.e. an estimator for Var(X) is equal to the (estimator for the 2nd population moment) - (estimator for the 1st population moment)^2 i.e. \displaystyle \hat{\sigma}^{2} = \hat{E[X^2]} - \hat{\mu}^2 ?

OR: Is this not the general case and you recommend following the normal method of finding population moments and sample moments, equating them and then rearranging for the parameter estimators?
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Gregorius
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(Original post by Chittesh14)
I already know that the variance \sigma^{2} is a function of the population moments i.e. 2nd population moment - (1st population moment)^2.... so, can I say that: if I put an estimator on each of the variables, the equation is still valid.
I.e. an estimator for Var(X) is equal to the (estimator for the 2nd population moment) - (estimator for the 1st population moment)^2 i.e. \displaystyle \hat{\sigma}^{2} = \hat{E[X^2]} - \hat{\mu}^2 ?
This gets things back-to-front! The point here is that you're trying to find estimators for the population parameters; in general, things won't come out anything like as easily as they do here. The estimator for variance that you're using in the sample hasn't popped out of thin air, and in fact here it's a biased (although consistent) estimator.

Perhaps the source of your confusion is that in this example, the population parameters for which you're trying to find estimators, have a particularly simple relationship to the population moments. In particular, here we have \sigma^2 equal to the population variance.

I suggest that you try the exercise of finding the method of moments estimators for the paramaters a and b of uniform distribution on the closed interval [a, b], using the first two population moments. Then mess around with the estimates for a and b using some possible samples; you'll get a better idea of how this all works, and how it sometimes doesn't work.
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Chittesh14
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(Original post by Gregorius)
This gets things back-to-front! The point here is that you're trying to find estimators for the population parameters; in general, things won't come out anything like as easily as they do here. The estimator for variance that you're using in the sample hasn't popped out of thin air, and in fact here it's a biased (although consistent) estimator.

Perhaps the source of your confusion is that in this example, the population parameters for which you're trying to find estimators, have a particularly simple relationship to the population moments. In particular, here we have \sigma^2 equal to the population variance.

I suggest that you try the exercise of finding the method of moments estimators for the paramaters a and b of uniform distribution on the closed interval [a, b], using the first two population moments. Then mess around with the estimates for a and b using some possible samples; you'll get a better idea of how this all works, and how it sometimes doesn't work.
Yes, thank you! This is what I wanted to know hehe, because it is done in the way I described in my notes, which caused my confusion.
I just wanted to know if it'd work everytime like this, and I'm now satisfied because I know it won't lol, so I will stick to the original method you described.

Finally got it, so in cases where the relationship is not so obvious, this method would not work. Thank you!!!!
I will try the exercise that you have said
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