Let' start with a specific example:
I have $N$ coins (biased or not), and I assume that they are independent. Let $(X_i)_{1\le i\le N}$ be the independent random variables that associate with the coin flipping events of each coin, and let $X_i \sim \mathrm{Bernoulli}(p_i)$ where $p_i$ is the probability that the coin $i$ shows head.
I toss the coins $M$ times each, and let $\left(\hat{X}_i^j\right)_{1\le j \le M}$ be the $M$ observations for coin $i$. I can then give an estimation of each $p_i$: let's call them $\left(\hat p_i\right)$. I then happen to find that there's a relation among $\left(\hat p_i\right)$: $$\hat p_1\le\hat p_2\le\cdots\le\hat p_N,$$ so I come up with a theory says: $$p_1\le p_2\le\cdots\le p_N.\tag{$H_1$}$$ My question is, with a certain confidence level, under which conditions of the observations $\left(\hat{X}_i^j\right)$, can I reject the null hypothesis $H_0:=\neg H_1$?
(Edit: The question above is just an example to describe the problem.)
More generally, $(X_i)$ may follow some different distributions with some unknown parameters $(p_i)$, and we want to see if we can suggest that a theory $R(p_1, p_2, \dots, p_N)$ exists under the observations $\left(\hat{X}_i^j\right)$.
I searched around but didn't find anything useful. The multivariate hypothesis testing might be related but still, I didn't find anything similar to the problem.
Any help is appreciated :)
Update
In fact, the $N$ coins one with $p_1 \le p_2 \le \cdots \le p_N$ is just an example to describe the problem, my question here is how to deal with the general situation, where $R$ can be any function that maps $N$ parameters to $\{ True, False \}$.
The multivariate hypothesis testing description says for $i=1,\dots, n$, let $H_{0 ,i}$ be a null hypothesis, and $\delta_i$ a level $\alpha_{0, i}$ test of $H_{0, i}$. If the combined null hypothesis $H_0$ is that $H_{0, i}$ are simultaneously true, and if $\delta$ is a test that rejects $H_0$ if at least one $\delta_i$ rejects $H_{0, i}$, then $\delta$ is a level $\sum_{i=1}^n \alpha_{0, i}$ test of $H_0$.
Let the $N-1$ null hypotheses be $p_i\le p_j$ for $i+1=j$ and $i=1,\dots,N-1$. So $H_0:p_1\le p_2\le\dots\le p_N$ is the null hypothesis. The alternative hypothesis is then $H_1: p_1>p_2$, $p_2>p_3$, ..., and/or $p_{N-1}>p_N$.
For each $i$ create a level $\alpha_0/(N-1)$ test of the hypothesis $H_{0,i}:p_i\le p_{i+1}, H_{1,i}:p_i>p_{i+1}$. It is possible to create a uniformly most powerful test of each of these hypotheses because the joint distribution of Bernoulli observations has increasing monotone likelihood ratio in the statistic $Y=\sum_{i=1}^M X_i$. Then the test that rejects $H_0$ if at least one of $H_{0,i}$ is rejected is a level of significance $\alpha_0$ test.