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In this question I would like to ask you to choose between two simple scenarios of testing differences of rates between two random variable of Poisson distribution over different time periods.

We have two arrays $\mathfrak{A}$ and $\mathfrak{B}$ of lengths $L_A$ and $L_B$, respectively. In these arrays, there're allocated some blue and red balls and I'm interested in particular pairs of blue and red balls in this specific order and two cell spacing (marked with green arcs).

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The actual (empirically measured) numbers of these pairs in arrays are $A_S$ and $B_S$, respectively. Let's assume that occurrences of these pairs follow Poisson distributions: $$ A_S \sim Poisson(\mu_A \cdot L_A) ~~~~ B_S \sim Poisson(\mu_B \cdot L_B) $$

Without going into unnecessary details, I'm able to test equality of the rates $\mu_A$ and $\mu_B$ that is hypotheses: $$ H_0: \mu_A = \mu_B ~~ \text{against} ~~ H_A: \mu_A > \mu_B $$

And here, the main question arise: I'm interested in comparing only pairs of such specific order and spacing as described/counted by $A_S$ and $B_S$ and I can simply do that.

But isn't it more proper to take into account and enclose some information about pairs that don't meet the requirements so strictly? For example, if strict spacing is 2 then let $A_U$ and $B_U$ stand for pairs of blue and red balls with any order and let's say spacing ranging from 0 to 7. I already have a method to incorporate these values in the final test, but I'm asking if the below justification is correct and sound:

How can I expect many structured (strict orde + spacing=2) pairs $X_S$ in a certain array $\mathfrak{X}$, when in this array there very few pairs even with more loose requirements (any orde and distance $\in [0, 7]$) $X_U$? Should I levarage such influence?

Once again, I'm not asking method how to do that (I know how), but wheter my rationale is convincing and you would do the same.

I would appreciate all opinions which approach is more sound to you. We could even do a fast voting by statting your answer or comment with either YES or NO:

  • YES, Such adjustment taking into consideration also more loosely structured pairs reflects more appropriately the overall situation.

  • NO, I should rather stick to only pairs preserving order and spacing precisely.

What do you think, which option is better?

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