I read all the posts in the sticky for the soa sample problems for #49, did Wikipedia, google and now hunting around in youtube for an explanation I still don't get it. Little bummed that nobody but me seems to have gotten hung up on this problem in a long while. Maybe Im just being thick.
Can someone straighten me out?
x = 0, 1, 2, 3 with probability 0.5, 0.3, 0.1, 0.1, respectively.
method of moments is used to estimate the population mean and variance by Xbar and Sn2=Sigma(Xi-Xbar)^2/n.
calculate the bias of Sn2, when n=4. (ans: -0.24)
Here's what I got:
Equation for Bias: E(Theta-hat) - Theta = Bias.
We're given that we calculate u and variance (sigma square) by method of moments. Ok. I got Xbar = 0.8, Sigma^2 = 0.96. I calculated with MOM, i.e., E(x^2) - E(x)^2.
They gave us what the estimator is, what we are using to estimate Theta. They define that by Sn^2.
Question1 - If Sn^2 that is the estimator for sigma^2 then the expected value that is the first term in the equation for bias, correct?
Question2 - Is E(Sn^2) = E(whatever that equation, the "estimator")? I took this to mean that the expected value of Sn^2 is E(value we calculate with given equation) = E(1.74). This does not appear to be correct based on the solutions I've seen but I don't see why. The official solution says E(Sn^2) = n-1 / n * sigma^2. I can buy that but only if the problem said "Sn^2 is used to estimate sigma^2", because by definition Sn^2 = sigma^2HAT * (n/n-1). In other words, Sn^2 = n/n-1 * sample variance we calculate because Sn^2 is the unbiased sample variance. But the question did not say that this was the estimator. They said the estimator is Sn^2 = SUM(xi-Xbar)^2/n which, given by the sample data (x1-x4) = 1.74.
Question3 - Since they gave us that expression for the estimator, why don't we use it to find the first term for the bias equation?
Question4 - Ok, so say that instead of using what we were given for the estimator I was able to read their mind and found our traditional Sn^2 instead. I would have calculated Sn^2 the same way as sigma^2 except divided by n-1, not n. So I would then have 0.96 * 4/(4-1) = 1.28. So my equation for bias = E(1.28) - 0.96 = 030. Obviously not correct - not even an answer choice.
So what gives? What am I not seeing here.
Thanks for any insight. Who knows what it will take to snap me out of whatever it is that's keeping me from understanding this problem.
Can someone straighten me out?
x = 0, 1, 2, 3 with probability 0.5, 0.3, 0.1, 0.1, respectively.
method of moments is used to estimate the population mean and variance by Xbar and Sn2=Sigma(Xi-Xbar)^2/n.
calculate the bias of Sn2, when n=4. (ans: -0.24)
Here's what I got:
Equation for Bias: E(Theta-hat) - Theta = Bias.
We're given that we calculate u and variance (sigma square) by method of moments. Ok. I got Xbar = 0.8, Sigma^2 = 0.96. I calculated with MOM, i.e., E(x^2) - E(x)^2.
They gave us what the estimator is, what we are using to estimate Theta. They define that by Sn^2.
Question1 - If Sn^2 that is the estimator for sigma^2 then the expected value that is the first term in the equation for bias, correct?
Question2 - Is E(Sn^2) = E(whatever that equation, the "estimator")? I took this to mean that the expected value of Sn^2 is E(value we calculate with given equation) = E(1.74). This does not appear to be correct based on the solutions I've seen but I don't see why. The official solution says E(Sn^2) = n-1 / n * sigma^2. I can buy that but only if the problem said "Sn^2 is used to estimate sigma^2", because by definition Sn^2 = sigma^2HAT * (n/n-1). In other words, Sn^2 = n/n-1 * sample variance we calculate because Sn^2 is the unbiased sample variance. But the question did not say that this was the estimator. They said the estimator is Sn^2 = SUM(xi-Xbar)^2/n which, given by the sample data (x1-x4) = 1.74.
Question3 - Since they gave us that expression for the estimator, why don't we use it to find the first term for the bias equation?
Question4 - Ok, so say that instead of using what we were given for the estimator I was able to read their mind and found our traditional Sn^2 instead. I would have calculated Sn^2 the same way as sigma^2 except divided by n-1, not n. So I would then have 0.96 * 4/(4-1) = 1.28. So my equation for bias = E(1.28) - 0.96 = 030. Obviously not correct - not even an answer choice.
So what gives? What am I not seeing here.
Thanks for any insight. Who knows what it will take to snap me out of whatever it is that's keeping me from understanding this problem.
Little help? SOA 305 #49