Type II Error in Likelihood Ratio Tests

Good afternoon,

I'm currently working on my master thesis. I'm trying to recommend some improvements to the likelihood ratio tests that already exist to assess the quality of Value at Risk (VaR) Models, so I'd like to calculate the Type II error of a likelihood ratio test.

I know that theoretically, the Type II error is expressed as P(Non reject H0|H0 is false), nevertheless I'm having trouble to express it mathematically for a LR Test.
I tried working with the non-rejection regions, in order to determine if there is any overlapping between the regions under the null hypothesis and an alternate hypothesis, but I don't get satisfactory results.

If someone have an idea that could help me, I would be very pleased.

Thank you very much,
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