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# Cramer-Rao Lower Bound

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1. Hi everyone

Just a quick question... you know the Cramer Rao Lower bound gives a lower bound for the variance of the statistic T(X) (being greater than or equal to 1/fisher info number).

Why is this useful?

Surely if anything, we want an upper bound to see the highest it's going to vary?

I know my logic is incorrect so please can someone explain what's going on?
2. I don't know anything about this thing in particular, but it's always useful to know bounds (upper and lower) for things when you can't work out their value exactly. If you have a lower bound on the variance, then you know that the data has to be spread out at least to some extent (i.e. if the lower bound is greater than zero, then you know that the data is spread out over some range and isn't just concentrated at a point).
3. (Original post by nuodai)
I don't know anything about this thing in particular, but it's always useful to know bounds (upper and lower) for things when you can't work out their value exactly. If you have a lower bound on the variance, then you know that the data has to be spread out at least to some extent (i.e. if the lower bound is greater than zero, then you know that the data is spread out over some range and isn't just concentrated at a point).
Thanks for the advice. That was my vague intuition, however I'da thought it'd be more useful to know the maximum possible variance an estimator could achieve as opposed to the lowest. If we only have a lower bound does this provide us with "useful" information?

For example if it was an upper bound then couldn't you intuitively tell that it was estimating accurately to within some range?

I don't know - I'm sorry for being silly it's all just beyond me if I'm entirely honest!
4. (Original post by jamesair99)
Thanks for the advice. That was my vague intuition, however I'da thought it'd be more useful to know the maximum possible variance an estimator could achieve as opposed to the lowest. If we only have a lower bound does this provide us with "useful" information?

For example if it was an upper bound then couldn't you intuitively tell that it was estimating accurately to within some range?

I don't know - I'm sorry for being silly it's all just beyond me if I'm entirely honest!
It doesn't really matter which is more useful than the other, because they're both useful As a completely off-the-whim example, suppose you're working for a fire department in a hot country, and you're looking at the incidence of forest fires over the course of a year. You want to know where to concentrate your funding so that your resources are used efficiently (e.g. you don't need too much water to put out fires in January, but you do in July). If you have a mean and a lower bound for the variance of the dates when the forest fires occur, then you can work out an absolute minimum amount of resources you put in (e.g. providing enough water to put out fires between 11th June and 3rd September) and so you can budget accordingly. Having an upper bound in this case is also useful because it will tell you when the resources you provide are unnecessary, but if you didn't have a lower bound you'd run the risk of your whole country burning to a crisp.
5. (Original post by nuodai)
It doesn't really matter which is more useful than the other, because they're both useful As a completely off-the-whim example, suppose you're working for a fire department in a hot country, and you're looking at the incidence of forest fires over the course of a year. You want to know where to concentrate your funding so that your resources are used efficiently (e.g. you don't need too much water to put out fires in January, but you do in July). If you have a mean and a lower bound for the variance of the dates when the forest fires occur, then you can work out an absolute minimum amount of resources you put in (e.g. providing enough water to put out fires between 11th June and 3rd September) and so you can budget accordingly. Having an upper bound in this case is also useful because it will tell you when the resources you provide are unnecessary, but if you didn't have a lower bound you'd run the risk of your whole country burning to a crisp.
Thank you very much, you're brilliant at explaining things and that example makes a lot of sense!

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