Finding the right σ-algebra. Question on uncertainty related to the secretary problem

Question

Assume a number of iid. items is presented and the task was to stop under the objective of picking the best item.

In this setting it is relevant what is the distribution of the values of the presented items $(X_i)_{i \leq n}$ the problem is to what $\sigma$-algebra filtration $(\natural_i)_{i \leq n}$ the best stopping time $T$ has to be adapted to. If one has full information on the distribution it would be $(\sigma(X_1,...,X_i))_{i \leq n}$. The winning chance then is asymptotically $0.58$ if $X_1$ has a continuous c.d.f.

For no information on the possible distributions the right filtration is: $(\sigma(R_1,...,R_i))_{i \leq n}$ with $R_j = \sum^{j}_{k=1}1_{X_k \leq X_j}$, that is the rank of $X_j$.

What is the right $\sigma$-algebra to model the case where for an example one knows $X_1$~$N(\mu,\sigma^2)$ but nothing on $\mu$ and $\sigma^2$?

This relates here, and I have made proposals and submitted an example to deal with.

https://math.stackexchange.com/questions/1798208/finding-the-right-sigma-algebra-question-on-uncertainty-related-to-the-secre

Answer 1

I am not sure exactly what your question is. What do you mean by "the right filtration"? In each example, $\sigma(X_1,\dots, X_i)$ would do the job fine.

Maybe you are looking for the smallest sigma-algebra in each case. But in fact the forms given in your examples are not the smallest possible:

(1) for the case where you know the exact distribution, for each $i$ there is some threshold $c_i$ such that you take item $i$ iff it exceeds the threshold $c_i$. So you do not need to see all the values $X_i$; all you need is the information about whether each $X_i$ exceeds its threshold or not.

(2) for the case where nothing is known about the distribution, famously you never take any of the early items; if $i<n/e$ you never take item $i$. So no information at all about the relative ranks of the first $n/e$ items is necessary. $\natural_i$ may as well be empty for $i<n/e$. Thereafter, you take any item which is the best so far. So you still don't need to know the exact ranks of the later items, just the information about whether they are the largest so far.

So, this is not an "answer" but here is a formulation in the spirit of your examples, for the case where $X_i\sim N(\mu,\sigma^2)$ with unknown $\mu$ and $\sigma^2$. You can let $\natural_i$ be the sigma algebra generated by all events which depend on $X_1,\dots, X_i$ and which are invariant under the transformation $(X_1, \dots, X_i)\to(aX_1+b, \dots, aX_i+b)$ for all constants $a>0$ and $b$. Such transformations preserve both the $N(\mu, \sigma^2)$ family and the ordering of the random variables.