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Two uniformly most powerful unbiased tests and normal distribution
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I'm currently working my way through lectures notes concerning mathematical statistics.
Let's look at the family of normal distributions $N^n(mu,1)$ where $mu$ ranges in the real numbers. Suppose we want to test the null hypothesis $mu=0=:mu_0$ against the alternative $muneq0$ with significance level 50%.
Uniformly most powerful unbiased tests $phi$ are given by tests of the form $phi(x)=mathbb{1}_{(c_1,c_2)}(frac{sum_{i=1}^nX_i}{sqrt{n}})$, where $c_1,c_2$ are chosen so that $$(1):mathbb{E}_{mu_0}phi = alpha = 0.5$$ and $$(2):mathbb{E}_{mu_0}(phi cdot frac{sum_{i=1}^nX_i}{sqrt{n}})=0.5mathbb{E}_{mu_0}frac{sum_{i=1}^nX_i}{sqrt{n}}$$.
That's kind of the main takeaway as I understand it. (See for example the Book by Lehmann Testing statistical hypotheses)
Now since the normal distribution is symmetric I can chose $c_1=c_2$ as the 1-1/4 quantile of a standard normal distribution, right? (that's also in the notes)
Now the professor makes the following case: These best unbiased tests are not unique, because you could also chose $c_1=-infty$ and $c_2=0$.
While I see that (1) holds, I can't figure out why the second euation is supposed to hold as well.
Any ideas on that?
Thanks in advance!
probability-theory statistics statistical-inference
asked Jan 24 at 22:59
VanillaThunderVanillaThunder
582215
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add a comment |
$begingroup$
I'm currently working my way through lectures notes concerning mathematical statistics.
Let's look at the family of normal distributions $N^n(mu,1)$ where $mu$ ranges in the real numbers. Suppose we want to test the null hypothesis $mu=0=:mu_0$ against the alternative $muneq0$ with significance level 50%.
Uniformly most powerful unbiased tests $phi$ are given by tests of the form $phi(x)=mathbb{1}_{(c_1,c_2)}(frac{sum_{i=1}^nX_i}{sqrt{n}})$, where $c_1,c_2$ are chosen so that $$(1):mathbb{E}_{mu_0}phi = alpha = 0.5$$ and $$(2):mathbb{E}_{mu_0}(phi cdot frac{sum_{i=1}^nX_i}{sqrt{n}})=0.5mathbb{E}_{mu_0}frac{sum_{i=1}^nX_i}{sqrt{n}}$$.
That's kind of the main takeaway as I understand it. (See for example the Book by Lehmann Testing statistical hypotheses)
Now since the normal distribution is symmetric I can chose $c_1=c_2$ as the 1-1/4 quantile of a standard normal distribution, right? (that's also in the notes)
Now the professor makes the following case: These best unbiased tests are not unique, because you could also chose $c_1=-infty$ and $c_2=0$.
While I see that (1) holds, I can't figure out why the second euation is supposed to hold as well.
Any ideas on that?
Thanks in advance!
probability-theory statistics statistical-inference
asked Jan 24 at 22:59
VanillaThunderVanillaThunder
582215
$endgroup$
add a comment |
$begingroup$
I'm currently working my way through lectures notes concerning mathematical statistics.
Let's look at the family of normal distributions $N^n(mu,1)$ where $mu$ ranges in the real numbers. Suppose we want to test the null hypothesis $mu=0=:mu_0$ against the alternative $muneq0$ with significance level 50%.
Uniformly most powerful unbiased tests $phi$ are given by tests of the form $phi(x)=mathbb{1}_{(c_1,c_2)}(frac{sum_{i=1}^nX_i}{sqrt{n}})$, where $c_1,c_2$ are chosen so that $$(1):mathbb{E}_{mu_0}phi = alpha = 0.5$$ and $$(2):mathbb{E}_{mu_0}(phi cdot frac{sum_{i=1}^nX_i}{sqrt{n}})=0.5mathbb{E}_{mu_0}frac{sum_{i=1}^nX_i}{sqrt{n}}$$.
That's kind of the main takeaway as I understand it. (See for example the Book by Lehmann Testing statistical hypotheses)
Now since the normal distribution is symmetric I can chose $c_1=c_2$ as the 1-1/4 quantile of a standard normal distribution, right? (that's also in the notes)
Now the professor makes the following case: These best unbiased tests are not unique, because you could also chose $c_1=-infty$ and $c_2=0$.
While I see that (1) holds, I can't figure out why the second euation is supposed to hold as well.
Any ideas on that?
Thanks in advance!
probability-theory statistics statistical-inference
asked Jan 24 at 22:59
VanillaThunderVanillaThunder
582215
$endgroup$
I'm currently working my way through lectures notes concerning mathematical statistics.
Let's look at the family of normal distributions $N^n(mu,1)$ where $mu$ ranges in the real numbers. Suppose we want to test the null hypothesis $mu=0=:mu_0$ against the alternative $muneq0$ with significance level 50%.
Uniformly most powerful unbiased tests $phi$ are given by tests of the form $phi(x)=mathbb{1}_{(c_1,c_2)}(frac{sum_{i=1}^nX_i}{sqrt{n}})$, where $c_1,c_2$ are chosen so that $$(1):mathbb{E}_{mu_0}phi = alpha = 0.5$$ and $$(2):mathbb{E}_{mu_0}(phi cdot frac{sum_{i=1}^nX_i}{sqrt{n}})=0.5mathbb{E}_{mu_0}frac{sum_{i=1}^nX_i}{sqrt{n}}$$.
That's kind of the main takeaway as I understand it. (See for example the Book by Lehmann Testing statistical hypotheses)
Now since the normal distribution is symmetric I can chose $c_1=c_2$ as the 1-1/4 quantile of a standard normal distribution, right? (that's also in the notes)
Now the professor makes the following case: These best unbiased tests are not unique, because you could also chose $c_1=-infty$ and $c_2=0$.
While I see that (1) holds, I can't figure out why the second euation is supposed to hold as well.
Any ideas on that?
Thanks in advance!
probability-theory statistics statistical-inference
probability-theory statistics statistical-inference
asked Jan 24 at 22:59
VanillaThunderVanillaThunder
582215
asked Jan 24 at 22:59
VanillaThunderVanillaThunder
582215
asked Jan 24 at 22:59
VanillaThunderVanillaThunder
582215
asked Jan 24 at 22:59
VanillaThunderVanillaThunder
582215
asked Jan 24 at 22:59
VanillaThunderVanillaThunder
582215
582215
add a comment |
add a comment |
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