A Type I flower by any other name

I must have been one of many students who chose maths over medicine, because I have a terrible memory, and medics have to memorize books by the kilogram. In maths, if you understand how to do something, there is nothing to remember.  Right?

Up to a point.  Here are three mathematical relations where the order or direction matters, that I can never remember, no matter how often I have encountered them.

1. Mathematics:  The binary operator \(\wedge\) means 'and' and \(\vee\) means 'or'.
2. Probability: Jensen's Inequality says that if \(f(x)\) is a convex function and \(X\) is a random variable, then \({\rm E}[f(x)] \ge f({\rm E}[X])\).
3. Statistics: A hypothesis test that says 'yes' when the answer is 'no' (false positive) yields a Type I error.  If it says 'no' when the answer is 'yes' (false negative) it yields a Type II error.

So, when I was trying, not long ago, to think of names for two kinds of adverse selection (see Haçarız et al. (2020)), what catchy titles did I come up with?  The answer: 'Type I adverse selection' and 'Type II adverse selection'.  It was co-author Guy Thomas who conjured the much better names 'Precautionary adverse selection' (exploiting an information advantage to obtain necessary insurance cover) and 'Speculative adverse selection' (exploiting said advantage to buy outrageous amounts of cover as a financial gamble).  No doubt about which names are easier to remember!

While drafting Richards and Macdonald (2024), we had to distinguish between two different kinds of right-censoring.  Both had names in the survival modelling literature.  One was 'random censoring' — not too bad.  The other was 'Type I censoring'.  If I tell you that 'Type I censoring' means ceasing to observe a subject at a predetermined time, like the end of a year, its meaning is reasonably clear.  Actuaries meet it all the time. But they almost never meet Type II censoring (keep observing until a pre-determined number of subjects have failed), and if Types III and IV censoring exist, I have forgotten about them.

Lesson learned, we decided to keep 'random censoring' to mean that a poliyholder lapses a policy, at a time that couldn't possibly be predicted; but we chose the name 'obligate censoring' for ceasing to observe at a pre-determined time, known to the investigator in advance (in a sense that the paper makes precise).

I hope that Richards and Macdonald (2024) will contribute more to actuarial science than changing a name from 'Type I' to 'obligate'.  Others will be the judge of that.  But, if I were to guess that most readers of this blog have already forgotten the difference between Type I and Type II adverse selection, and if my guess were wrong, would I have made a Type I or Type II error?  Come on, come on, I'm going to have to hurry you here…

Postscript: How to remember Jensen’s Inequality — variances are non-negative, so \({\rm E}[X^2]\ge {\rm E}[X]^2\). Now you just have to remember whether \(x^2\) is convex or concave.