Questions tagged [neyman-pearson-lemma]
A theorem stating that likelihood ratio test is the most powerful test of point null hypothesis against point alternative hypothesis. DO NOT use this tag for Neyman-Pearson approach to hypothesis testing, this tag is for the lemma only.
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is the likelihood ratio test "best" for finite samples?
Wikipedia says
The Neyman–Pearson lemma states that this likelihood-ratio (lr) test is the most powerful among all level α alpha tests for this case.
Is this only true for infinite sample sizes? Is ...
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General Lower Bound of Power in Neyman-Pearson
Let $X$ be an $\sigma$-finite space $(\mathcal{X}, F_{\mathcal{X}}, \nu)$ valued absolutely continuous random variable whose distribution is one of $P_0 = f_0(x)d\nu(x)$ or $P_1 = f_1(x)d\nu(x)$. We ...
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Understanding proof of Neyman Pearson Lemma
I am trying to understand Neyman Pearson Lemma's proff from Rice's book. The lemma is intuitive, however I am not able to understand the reasoning for the first inequality in the proof. I highlighted ...
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Neyman-Pearson Testing: Swapping the main and alternative hypotheses to ensure P(Type I) < P(Type II)
I have been reading up on hypothesis testing, and realized I misunderstood something, which made me mix Fisher's p-values with Neyman-Pearson's critical regions. I am going to amend that situation, so ...
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Finding P-value and power of the Most Powerful Test
You observe a sample $X_1, \quad, X_{20}$ with the density
$$
f(x, \vartheta)=2\left(x / \vartheta^2\right) I_{[0 \leq x<\vartheta]}
$$
with an unknown parameter $\vartheta>0$, yielding
$$
\min \...
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UMP two sided tests for exponential families
Consider a random variable $X$ with density $$f(x : θ) = C(θ)e^{η(θ)T(x)}h(x), θ ∈ Θ$$.
Assume that $η(θ)$ is strictly increasing in $θ$ and that the family is full rank. Show that there will not be ...
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Show a composite test is the most powerful after deriving a similar most powerful simple test
Let $X$ be a real-valued random variable with density $f(x) = (2\theta x + 1 - \theta) \mathbb{1}(x \in [0,1])$ where $1$ here is the indicator function and $-1 < \theta < 1$. I am trying to ...
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Is there a uniformly most powerful yet exact test for independence of two categorical variables?
I know that uniformly most powerful tests have to be based on the likelihood ratios as test statistic, which is not the case for the Fisher exact test. Nevertheless couldn't I use the G2 test metric, ...
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Why is Neyman-Pearson lemma a lemma or is it a theorem?
A classical result in statistical theory is the Neyman-Pearson lemma, which not only shows the existence of tests with the most power that return a pre-specified level of Type I error, but also a way ...
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Biased coin game
Assume, there's a 50% chance I get a fair coin and 50% I get a biased coin with 0.6 chance of getting heads.
Then, I get to toss the coin I got as many times as I want, but each toss costs a dollar.
...
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How to Justify this Two-Sided Test is UMP with NP Lemma?
UMP tests generally do not exist for two sided tests, ie $H_0: \theta = \theta_0$ vs $H_a: \theta \neq \theta_0$. However, if we observe $n$ iid observations of $X\sim Unif(0,\theta)$, we can ...
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Given a UMP test, why does NP lemma deliver the same critical region for all $\theta_1\in\Omega_1? $
I'm unsure why, given a uniformly most powerful test exists, that the Neyman-Pearson lemma delivers the same critical region for all $\theta_1\in \Omega_1.$ Is it because this is the smallest critical ...
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Does there always exist for n small, a non-chi-squared test-statistic for the likelihood-ratio (neyman-pearson, karlin-rubin), score, and wald-tests?
An additional reason that the chi-squared distribution is widely used is that it turns up as the large sample distribution of generalized likelihood ratio tests (LRT).[6] LRTs have several desirable ...
user318514
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Uniformly most powerful test
Suppose we have Xi~Exp(λ), and we want to construct a most powerful test for
H0 : λ = λ0, H1 : λ = λ1
I then proceed to use the Neyman Pearson lemma : reject H0 when the likelihood ratio L(λ1;X)/L(...
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The Test Statistic of the Neyman - Pearson Lemma
I cannot understand this statement from the book Robust Statistics:
The most powerful tests between two densities $p_0$ and $p_1$, are based on a statistic of the form
$$\int \psi F_n(dx) = \frac{1}{n}...
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