I'm having $(X_1,...,X_N)$ and $(Y_1,...,Y_M)$. $X$ and $Y$ are supposed to have normal distribution. How can I check statisticaly if $X_n$ in most cases is greater then $Y_n$. What criterion should I use? What criterion could be used if distribution is not normal?
$\begingroup$
$\endgroup$
Add a comment
|
1 Answer
$\begingroup$
$\endgroup$
I'm assuming you're talking about repeated sampling from univariate distributions and not single draws from multivariate distributions, in which case the notion of "order" is less clear. In the former case it seems like you're asking a question about stochastic domination which means the probability that one draw is "large" is greater than the probability that another value is "large."
Formally we could define this as meaning $P(X > t) > P(Y > t)$ (or vice versa) for all $t$, and a nonparametric method for testing this is by using a Wilcoxon rank sum test. If $X$ and $Y$ are known to be normal and have equal variance (this is necessary) then a test for stochastic domination is the same as a test about means, so a $t$-test is a valid thing to do.