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a quick question about the NP lemma.

So that NP Lemma is tool to find a critical region which minimizes the type $2$ error in hypothesis testing for a fixed size of the test. We usually end up with a critical region of the form $\{ \underline{x}:\overline{x} \geq c\}$ or $\{ \underline{x}:\overline{x} \leq c\}$ where $c$ is some constant related to the size of the test.

What I am interested in is, is there a quick way to determine the sign in the critical region, i.e. make a sanity check? I expect it to be related to the expectation of the distribution the random variables are from.

For example: Suppose $X_1, . . . , X_n$ is a random sample from $N(µ, σ^2)$ where the variance is known.Find the most powerful test of size α of $H_0 : µ = 0$ against $H_1 : µ = µ_1$, where $µ_1 > 0$.

The critical region will be of the form $\{ \underline{x}:\overline{x} \geq c\}$ and if we change $H_1$ to $µ_1 < 0$ then the region will be $\{ \underline{x}:\overline{x} \leq c\}$

Cheers

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In an elementary case such as the one you mention, you can let intuition be your guide. You are dealing with data $\mathbf{x}$ from $\mathsf{Norm}(\mu,\sigma)$ with $\mu$ unknown and $\sigma$ known. The sample mean $\bar x$ gives information about the population mean $\mu.$

Therefore, if you are testing $H_0: \mu = 0$ vs. $H_1: \mu = \mu_1 > 0$, then you would favor $H_1$ (reject) for values of $\bar x$ sufficiently far above $0.$ This means that the critical region is $\{\mathbf{x}: \bar x > c\}$ (and you will have critical value $c > 0$).

Notes: (a) The reason I specified an elementary case is that you have to be careful to work through the logic of the N-P Lemma--especially in more advanced situations. For example, suppose data are from $\mathsf{Exp}(rate = \lambda),$ and you are testing $H_0: \lambda = 1$ vs. $H_1: \lambda = \lambda_1 > 1.$ Then the sample mean $\bar x$ gives you information about the population mean $\mu = 1/\lambda.$ Then you would reject for small values of $\bar x$ because small $\mu$ implies large $\lambda.$

(b) I hope you noticed that I changed Neumann to Neyman in the title of your question. I am old enough to remember lectures by Jerzy Neyman. A famous mathematician and early user of computers for simulation in physics and probability was John von Neumann. If you are interested in the history of the mathematical sciences you might want to google both names.

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  • $\begingroup$ That's exactly what I was looking for! Thanks a lot $\endgroup$
    – asdf
    Jun 24, 2017 at 23:12

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