# How can I compare the MLE by standard error?

In the paper Extreme value theory based on the r largest annual events (page 32), the idea is that he wants to fit the Gumbel distribution using a dataset. In this dataset, we have the largest ten values. He wants to compare if we will get a better estimate of the parameters in Gumbel using the ten largest values than only one annual maximal. He got the maximal likelihood estimators of $$\mu$$ and $$\sigma$$ in Gumbel (consistency and asymptotic normal).

Question: The authors want to compare which is the better estimate of the estimated quantity under different $$r.$$ How can I get this conclusion by standard error?

For example, for MLE of $$\alpha$$, why is $$r=5$$ better than $$r=1?$$ Since we know that $$\rm MSE=Bias^2+Variance,$$ do not we also need the bias is smaller?

It seems that MLE are consistent, so is the bias asymptotically small?

• What does "$r$" represent?
– whuber
Nov 7, 2022 at 22:44
• I have attached a link to the paper. The article wants to fit the Gumbel distribution, r is r largest order statistics. He wants to compare the fit of a maximum (r=1), and the first r maximums of the MLE which is better. Nov 8, 2022 at 4:07
• Because the article is behind a paywall, you probably won't get as much help until you fully explain the issues in your question. But I could be wrong.
– JimB
Nov 8, 2022 at 18:10
• Yes, that makes sense. I'm only suggesting that more details be given in the question rather than in comments. And I could access the article previously through my work. I was concerned for others.
– JimB
Nov 8, 2022 at 18:30
• @SextusEmpiricus Yes, they use the asymptotic joint distribution for fitting the data. In paper, authors thought that there is a increasing trend. Well, this is my another question. Why the author set a linear regression model for $\mu_n=\alpha+\beta(n/N)$. Nov 15, 2022 at 20:25

1. The question in this paper is stated in the bottom of page 28: "Suppose we are given , not just the maximum value for each year, but the largest ten (say) values. How might we use this data to obtain better estimates than could be made with just the annual maxima?".

2. With $$r$$ being the number of largest annual values, they are ordered (descending) and assumed to be iid. This is what enables eqn (2.1).

3. They don't just use the linear trend model introduced in equation (2.4) but also a quadratic model (eqn 3.1) and two different periodic models (11 and 19 years) (eqn 2.6).  Question: The authors want to compare which is the better estimate of the estimated quantity under different $$r.$$ How can I get this conclusion by standard error?

For example, for MLE of $$\alpha$$, why is $$r=5$$ better than $$r=1?$$ Since we know that $$\rm MSE=Bias^2+Variance,$$ do not we also need the bias is smaller?

You can see in the comments here the reasons for using a biased estimator in order to get a lower MSE. Furthermore, with $$r$$ increasing from 1 to 5 or 10, we are bound to get a lower MSE: the $$r$$ observations are independent, hence the information function grows and we get a better Cramér–Rao lower bound.

Another question: for this non-stationary dataset, why did the author set $$\mu=\mu_n?$$ This seems to be very similar to the residual in linear regression model.

As you can see, they used different trend models. They do note just below eqn (3.3) the results obtained for the likelihood functions. You can see the comparison of the 4 models in fig. 2 where they compare the annual predicted maxima and medians. They further explain, under the summary section, that "In the Venice data, the main contribution to trend was a linear trend from 1931 onwards" and then explain that other trends had lesser effect.

• But does the method of using $r$ maxima increase the bias or not? Nov 17, 2022 at 9:15
• The use of $r$ maxima decreases the variance there's no doubt about that, but the Cramer Rao bound that you mention does not relate to the MSE, it relates to the variance. It is nice that we can reduce the variance. A constant estimator will even have zero variance. But, we also need to place it into perspective in relation to bias. Sure a biased estimator can be better, but is it better in this case? Nov 17, 2022 at 9:19
• The authors do not use MSE anywhere, their comparison is based mainly on information and covariance matrices. I seriously cannot stress enough how much the OP needs to read the whole article. Nov 17, 2022 at 9:41
• But for a biased estimator... Does a smaller SD mean a smaller MSE? What if the biased with r=1 is much smaller than the biased with r=5? Nov 21, 2022 at 6:12
• @SextusEmpiricus Do you mean how to get the likelihood function of the $r-$largest order statistics? That is a result in this paper google.com.hk/…. Nov 22, 2022 at 23:49