Man, it is tough to find a discussion about a specific problem from Mahler's exams.
I think I have a mental block when it comes to calculating bias. I set up the equation without a problem and think I know where Im going but then there is some assumption, or some step, logic, etc. that I just don't see. Happens on all but the most simple bias problems I try (and those Im likely getting by just being lucky).
Take this problem for example. The extimator is Xbar*e^-.54. We are estimating the mean of the lognormal. We know that. It's given in the tables.
I set up: Bias = E(extimator) - parameter_being_estimate. We are given sigma^2 = .36
So Bias = E(Xbar*e^-.54) - e^(u-sigma^2).
First term = e^-.54E(Xbar)
Second term = e^-.36 * e^u
Ok, now here is where things get weird for me. The problem gives us u=8 and n=1000. If I plug in what I have, I am left with:
Bias = e^-.54(E(xbar)) - e^7.64.
The solution says that the estimator is an unbiased estimator (how, i don't see, I can't hang with the steps of the solution). But OK. That means that the terms on the right are equal. So I solve for what's remaining, E(xbar), which comes out to 3568.85. So? what does that tell me - pretty much nothing because without the solution saying the bias is 0 I wouldn't have determined E(xbar), and it is apparently irrelevant anyway.
In the solution it says the E(xbar) = E(x). Why? How do we know that? isn't x bar = sum of our samples / n? What allows us to make this assumption? I think if I could understand this part (why E(xbar) = E(x)) I might be able to figure out what is going on with this problem.
Anyone got any hints?
Thanks!
I think I have a mental block when it comes to calculating bias. I set up the equation without a problem and think I know where Im going but then there is some assumption, or some step, logic, etc. that I just don't see. Happens on all but the most simple bias problems I try (and those Im likely getting by just being lucky).
Take this problem for example. The extimator is Xbar*e^-.54. We are estimating the mean of the lognormal. We know that. It's given in the tables.
I set up: Bias = E(extimator) - parameter_being_estimate. We are given sigma^2 = .36
So Bias = E(Xbar*e^-.54) - e^(u-sigma^2).
First term = e^-.54E(Xbar)
Second term = e^-.36 * e^u
Ok, now here is where things get weird for me. The problem gives us u=8 and n=1000. If I plug in what I have, I am left with:
Bias = e^-.54(E(xbar)) - e^7.64.
The solution says that the estimator is an unbiased estimator (how, i don't see, I can't hang with the steps of the solution). But OK. That means that the terms on the right are equal. So I solve for what's remaining, E(xbar), which comes out to 3568.85. So? what does that tell me - pretty much nothing because without the solution saying the bias is 0 I wouldn't have determined E(xbar), and it is apparently irrelevant anyway.
In the solution it says the E(xbar) = E(x). Why? How do we know that? isn't x bar = sum of our samples / n? What allows us to make this assumption? I think if I could understand this part (why E(xbar) = E(x)) I might be able to figure out what is going on with this problem.
Anyone got any hints?
Thanks!
Mahler PE 3.6