There are two distribution functions and a series of numbers. How do I determine which distribution is more likely to generate the given numbers?
I'm working on financial data where I'm trying to determine when to shift to another distribution. For example bear market and bull market. I guess ML is required but still want to know if there are classical stat methods.
You could compute the likelihood of the numbers relative to the two distributions. The likelihood of $\{x_1, x_2, \dots, x_n\}$ relative to a distribution having probability density function $p(x)$ is $$ \prod_{i=1}^n p(x_n) \text{.} $$ since this usually produces ridiculously small numbers (and since products are less familiar than sums), it is common to compute the log-likelihood: $$ \sum_{i=1}^n \log p(x_n) \text{.} $$
In the absence of any other consideration, the larger likelihood corresponds to a better model fit. (See also "Likelihood ratio and relative likelihood" at the "likelihood" link above.) A hazard in practice is that a model with more parameters is naturally able to produce a better fit. There is theory (example, list) for how to penalize multi-parameter models so that such models have to fit even better than fewer-parameter models.