Discrete or continuous modeling ?
Tuesday, we got our conference “Insurance, Actuarial Science, Data & Models” and Dylan Possamaï gave a very interesting concluding talk. In the introduction, he came back briefly on a nice discussion we usually have in economics on the kind of model we should consider. It was about optimal control. In many applications, we start with a one period economy, then a two period economy, and pretend that we can extend it to n period economy. And then, the continuous case can also be considered. A few years ago, I was working on sports game as an optimal effort startegy (within in a game – fixed time). It was with a discrete model, I was running simulations to get an efficient frontier, where coaches might say “ok, now we have enough (positive) difference, and we get closer to the end of the game, so we can ‘lower the effort’ i.e. top players can relax a little bit” (it was on basket-ball games). I asked a good friend of mine, Romuald, to help me on some technical parts of proofs, but he did not like so much my discrete-time model, and wanted to move to continuous time. And for now six years, we keep saying that someday we should get back to that paper….
My initial thoughts were that the difference was really “cultural”: you are either a continuous-time sort of guy, or a discrete-time one (or maybe none of the two, but that’s another problem). He works with stochastic processes, I work with time series. Of course, we can find connections, but most of the time, the techniques are very different. And tuesday, Dylan mentioned a very nice illustration that it’s not necessarily a cultural difference, and sometimes, it is great to move to continuous time. So I wanted to illustrate that idea.
Consider for instance the following curve.
So, indeed, it looks like computational time to find the maximum in a list of n elements is linear in n, i.e. O(n). And R code is faster than home-made code. But also, interestingly, using continus time (based on analysis techniques) can be much faster. So, sometimes, considering continuous time models can be much easier to solve, from a numerical perspective.
i.e. if we choose , and if , we choose .
Why not consider a test based on ? The problem is that we need the distribution (more specifically the survival distribution) of . We can compute it, numerically. But that might be painful. An alternative is to consider some approximation, based on the central limit theorem, i.e.
Observe here that the test based on the maximum is not more powerful than the one based on the average (I just wonder if it could be due to the Gaussian approximation…).
For the first homework, there was a tricky question in problem 29, chapter 5. Here is the maximum of n random variables i.i.d. uniformly distributed on the unit interval . I gave a hint last week about the cumulative distribution function for the maximum, i.e.
Then, we use independent to obtain that this probability is a product, of equal quantities since all random variables are identically distributed, i.e.
Actually, the question is not that simple (see here for the history of that question).
is a standardized (or normalized) version of , in the sense that it is centered, i.e.
and with a unit variance, i.e.
This is the kind of standardization (or normalization) that is used in the central limit theorem i.e. it is interesting when we study the core of our distribution (i.e. the mean).
since if , then (see the prof of the central limit theorem we got a few days ago).
where is the cumulative distribution of the ‘s (the random variables used to build up the maximum). This work since the ‘s have a finite support, i.e. the are bounded, with an upper limit (here 1).