# Can you use a test statistic like Anderson–Darling for parameter estimation?

I was looking at the Wikipedia article for maximum spacing estimation and this got me thinking. The idea behind this method is that if you know the true distribution of the data, then its CDF should transform the data to samples from a uniform distribution. Maximum spacing estimation exploits this by finding the parameters of a parametrized distribution that make the data look most uniform after applying the CDF to them. This uniformity is measured as the logarithm of the geometric mean of the spacings.

Now the Anderson-Darling and Cramér-von Mises tests do something similar to determine data could have been generated by a certain distribution (when the parameters have fixed values). They both measure the uniformity of the data points are after the CDF has been applied to them in one way or another. So now I'm wondering: could you use the Anderson-Darling or Cramér-von Mises test statistic as a parameter estimation metric in some way similarly to the geometric mean from maximum spacing estimation? I would expect that that would invalidate the test as a goodness-of-fit metric of the fitted distribution, but that might not always be a problem.

I suspect that the answer to my question is probably "no", but I can't really convince myself one way or another yet so I was hoping someone here could shed some light on this.

• Sure you can--it's not a bad idea. The right question to ask, though, is what properties would such an estimator have? I cannot think of any property that wouldn't be inferior to the Maximum Likelihood estimator. Such estimators, or variants of them, come to the fore with censored data, where they are known as "Regression on Order Statistics" estimators (Google them). Many ROS procedures are less sophisticated about weighting the values than your proposals, though. – whuber Jul 29 '20 at 19:57
• @whuber I will admit that I'm not really that knowledgeable about estimator properties, but the MLE is know for being problematic in some cases and the wiki article specifically mentions that maximum spacing estimation is more robust in several ways. I would imagine that an estimator based on, e.g., Anderson-Darling would share that advantage since it would be conceptually similar to the maximum spacing estimator. – Sjoerd Smit Jul 29 '20 at 20:21
• I suspect the A-D could be related asymptotically to the MLE. – whuber Jul 29 '20 at 20:37