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Can you suggest me a paper that proposes an optimal stoping rule based on an empirical distribution?

I am looking for some like this:

  1. I observe a random variable $x$ for a given time (this is usually the variable we need to build the rule and it is usually a proportion of the possible observation time window) that comes from an unknown empirical distribution;

  2. I find the threshold $T$;

  3. If $x>T$, I stop.

It may eventually propose other solutions.

DanielTheRocketMan
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    What do you mean by "optimal"? – cdalitz Aug 07 '21 at 18:52
  • For instance, suppose that you want to sell a stock that you have the empirical distribution. Is there an optimal rule to sell it for the maximal price? How long should you wait to sell it? Or what kind of stop rule you should implement. This ideia is similar to the secretary problem, but there you do not have any information and here I am supposing that you have the empirical distribution. – DanielTheRocketMan Aug 08 '21 at 02:50

1 Answers1

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From your explanation in the comment, I conclude that your question is about speculation strategies on finance markets. This is off-topic on CV, but allow me to answer nevertheless.

Speculation strategies on finance markets have been analyzed in great detail by Stephan Schulmeister. For a summary, see

Schulmeister: "Short-term Asset Trading, long-term Price Swings, and the Stabilizing Potential of a Transactions Tax." IMF Seminar, Nov 2010

Beware that these strategies are in direct contradiction to "efficient market theory" and are one of the reasons of finance market instability.

Schulmeister gives a gentle introduction in chapter 9 of his book "Der Weg zur Prosperität" (2018, ecowin), which puts these strategies in a broader context and also discusses the obvious question who actually pays the gains achieved with these strategies.

Edit: From your comment below, it is still unclear what your criterion actually is, that is to be maximized, and the question ist thus still unanswerable. The general apporach will be:

  1. Define your criterion $J$ that is to be maximized.
  2. Derive a formula for computing $J(T)$, i.e., how $J$ is computed when the parameter $T$ and the emirical distribution is given.
  3. Use or find some algorithm for finding $arg max\{J(T)\}$.

As your criterion only seems to be a scalar value (the threshold $T$), a simple algorithm for finding the maximum is golden section search.

cdalitz
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  • Thank you! I will read, but I was actually expecting an idea similar to the secretary problem when you have an empirical distribution of the random variable. – DanielTheRocketMan Aug 08 '21 at 13:23
  • @danieltherocketman In the secretary problem, the crterion is a *probability*, in your comment under your question, you ask for optimizing an *expectation value*. As it might not be clear for yourself what you are actually looking for, I have edited my answer to include a generic answer. – cdalitz Aug 09 '21 at 08:09
  • Actually, it is clear for me what I am looking for. I am looking for problems/solutions of optimal stoping times where we have partial information and this partial information is provided by the empirical distribution. – DanielTheRocketMan Aug 09 '21 at 12:03
  • In other words: you are looking for examples for applications with different definitions of "optimal"? Apparently my answer is not what you meant, and I will therefore delete it. I would suggest to clarify what you are looking for in the question. – cdalitz Aug 09 '21 at 13:20
  • I think your answer contribute to the question +1 but it is not exactly what I am looking for. I tried to find solutions based on empirical distribution and I was not able to find. Usually, people assume a specific distribution. – DanielTheRocketMan Aug 09 '21 at 13:29