Minimal sample size for determining change in unkown success probabilities of independent Bernoulli random variables

Question

Let $x_i$ be independent Bernoulli random variables with unknown success probabilities $p_i$. I want to estimate the probabilities $p_i$ depending on a number of Bernoulli trials made for each $x_i$. With rising $i$ the number of trials made for each variable ($N_i$) gets smaller.

How do I best determine the minimal $N$ for making a meaningful statement about the change in $p_i$ with rising $i$?

As the question may be unclear, here is a minimal example of the data I am talking about:

10 trials were made for variable $x_0$
7 trials were made for variable $x_1$
3 trials were made for variable $x_2$
1 trial was made for variable $x_3$

$p_i$ is unknown and may or may not be the same for $x_0$ to $x_3$. My assumption is, that there may be some clear correlation between each $p_i$ like $p_0 > p_1 > p_2 > p_3$. The problem is that I can't consider $x_3$ as only one trial was made for this variable. But can I consider $x_2$?
In the actual experiment there are 270 variables with a maximum of 99 trials.

Answer 1

See Hoeffding's inequality and Chernoff's bound here for different ways you can bound the $p_i$ values as dependent on $N_i$ (note that the estimate would just be the empirical average, but your confidence in how near that is to the actual value would improve). Note that as your $N_i$ values decrease, the estimate is less accurate (at the same confidence level). In order to make a statement regarding your certainty that $p_i > p_{i+1}$ would thus be dependent on both $N_i$ and $N_{i+1}$.