I have two empirical distribution A and B (i.e. two continuous datasets in $\mathcal{R}$) and I am trying to set up a hypothesis test to test if a new data point is from distribution A. Naturally this calls for a likelihood ratio test. I was planning on setting the test as $$ H_0: \text{the new data point is from distribution A} $$ $$ H_1: \text{the new data point is from distribution B} $$ The likelihood ratio can be computed as $$ \frac{P(H_0|x)}{P(H_1|x)} = \frac{P(H_0)}{P(H_1)}\frac{P(x|H_0)}{P(x|H_1)} $$ However, I am not sure how I would compute the likelihood $P(x|H_0)$ and $P(x|H_1)$ here, given that A and B are empirical distribution in continuous space. I could compute the empirical CDF but I am not sure how to convert that to an empirical PDF. Should I consider fitting some kind of kernel density and use them as the likelihood?
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