Timeline for Are these MLE estimates biased?
Current License: CC BY-SA 3.0
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| May 2, 2014 at 18:04 | history | edited | Scortchi♦ | CC BY-SA 3.0 |
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| May 2, 2014 at 17:47 | comment | added | Scortchi♦ | @Joe: You're right, but I'm not sure what you're getting at. Asymptotically, the MLE's median-unbiased & mean-unbiased, but that doesn't imply the MLE for a finite sample is either mean- or median-unbiased, which was the confusion that prompted the OP's question. | |
| May 2, 2014 at 17:34 | comment | added | Joe | @Scortchi It will converge to the Normal asymptotically... but for the finite sampling distribution of the parameter, I don't see any reason to expect that the mean and median will be equal. | |
| May 2, 2014 at 14:38 | comment | added | Scortchi♦ | @Joe: Good point - bias is defined in terms of the expected value of the sample estimate. All the same the asymptotic convergence at issue here is to a normal distribution whose mean & median coincide at the true value of the parameter. | |
| May 2, 2014 at 14:28 | comment | added | Scortchi♦ | For the general ideas read the chapter on point estimation in a theoretical statistics book. Unbiased estimates sound good but ...(1) They may have higher variance; enough to make the overall MSE higher. (2) Unlike MLEs, they're not in general invariant to reparameterization. So you can get a unbiased estimator of the scale parameter for your Weibull distribution, but its reciprocal won't be an unbiased estimator of the rate. (3) If you choose an estimate that isn't the MLE, your observations were less probable (roughly speaking) if that estimate were right than if the MLE were. | |
| May 2, 2014 at 14:25 | comment | added | Joe | What makes you think the median of the shapes will be equal to the shape? That doesn't mean that the method is biased for finite samples... What's the sampling distribution of the shape parameter? | |
| May 2, 2014 at 14:06 | comment | added | sqrt | OK, so are there ways of correcting this consistent bias for the finite sample size case, or is there a general statistical topic name for this? Thanks | |
| May 2, 2014 at 13:34 | vote | accept | sqrt | ||
| May 2, 2014 at 13:33 | comment | added | Scortchi♦ | That's right. In statistics "asymptotically" means "as the sample size goes to infinity" unless otherwise qualified. You might also be interested in looking at the variance of your estimates, & hence the overall mean-square error. | |
| May 2, 2014 at 13:30 | comment | added | sqrt | So you're saying that even if I had infinite simulations, but still a finite sample size, it would always be biased? I was originally interpreting asymptotically as meaning "If I simulate enough, it will eventually match the original parameters" | |
| May 2, 2014 at 13:17 | history | answered | Scortchi♦ | CC BY-SA 3.0 |