How do you determine the best critical region?

How do you determine the best critical region?

Let C be a critical region of size α; that is, α=P(C;θ0). Then C is a best critical region of size α if, for every other critical region D of size α=P(D;θ0), we have that P(C;θ1)≥P(D;θ1).

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Fragile States Index 2021

Which is the best critical region of α?

Let C and D be critical regions of size α, that is, let: Then, C is a best critical region of size α if the power of the test at θ = θ a is the largest among all possible hypothesis tests. More formally, C is the best critical region of size α if, for every other critical region D of size α, we have:

Which is the best critical region for h 0?

Then, if C is a critical region of size α and k is a constant such that: then C is the best, that is, most powerful, critical region for testing the simple null hypothesis H 0: θ = θ 0 against the simple alternative hypothesis H A: θ = θ a. See Hogg and Tanis, pages 400-401 (8th edition pages 513-14).

Which is the best Test of the null hypothesis h 0?

Consider the test of the simple null hypothesis H 0: θ = θ 0 against the simple alternative hypothesis H A: θ = θ a. Let C and D be critical regions of size α, that is, let: Then, C is a best critical region of size α if the power of the test at θ = θ a is the largest among all possible hypothesis tests.

Which is the uniformly most powerful critical region?

The critical region C is called a uniformly most powerful critical region of size α. Let’s demonstrate by returning to the normal example from the previous page, but this time specifying a composite alternative hypothesis.