# 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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### UMP test for $H_0:p=0.5$ vs $H_1:p\neq0.5$?

Let $X_1,\dots, X_n$ Bernoulli trials. I know that the UMP tests for $$H_0:p=0.5 \quad\text{vs}\quad H_1:p>0.5$$ and $$H_0:p=0.5 \quad\text{vs}\quad H_1:p<0.5$$ can be obtained with the Neyman ...
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### Why is the Neyman-Pearson lemma a lemma and not a theorem?

This is more of a history question than a technical question. Why is the Neyman-Pearson lemma'' a Lemma and not a Theorem? link to wiki: https://en.wikipedia.org/wiki/Neyman%E2%80%93Pearson_lemma ...
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### Reproduce figure of “Computer Age Statistical Inference” from Efron and Hastie

The summarized version of my question (26th December 2018) I am trying to reproduce Figure 2.2 from Computer Age Statistical Inference by Efron and Hastie, but for some reason that I'm not able to ...
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### Understanding Uniformly Most Powerful vs Uniformly Most Powerful Unbiased tests

I am struggling a little to understand the difference between these two classes of tests. Suppose we were testing a simple null hypothesis and a composite two sided alternative hypothesis. I am ...
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### Most powerful test of simple vs. simple in $\mathrm{Unif}[0, \theta]$

Say $X \sim \mathrm{Unif}[0, \theta]$. Denote the observations as $x_i$ $(i=1, \cdots, n)$. Show that any test $\phi$ that satisfies the following two conditions is most powerful test of level $\alpha$...
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Suppose $X$ is a random variable and two hypothesis defined as: $$H_0:f(x;\lambda_{0}) = ‎\lambda_0 \exp(−\lambda_0 x)$$ $$H_1:f(x;\lambda_{1})$$ $$x \geq 0 \quad \text{and} \quad \lambda_1>\... 1answer 70 views ### Neyman Pearson Lemma I've been reading up on Neyman Pearson Lemma but don't understand it to its full extent. Could someone please explain to me how to obtain the most powerful size in a bernoulli distribution? 1answer 164 views ### UMP test for a distribution and calculating alpha My f(x) = \theta x ^{ (\theta -1)} for 0 < x < 1 and \theta > 0. I found that the UMP to test H_0: \theta = 1 \mbox{ vs } H_a: \theta = \theta_a is \sum(ln(x_i)) \geq c. I have a ... 0answers 36 views ### How to detect signal when threshold is not known for this communication problem? I am trying to work on a nano scale communication problem. The transmission media is considered to be fluid and the molecules are the one that communicates information from transmitter to receiver. I ... 0answers 539 views ### example of uniformly most powerful test let X be a single observation from the density f(x;\theta) = \theta x^{\theta -1} I_{(0,1)}(x) is there a UMP size-\alpha test for testing H_0 :\theta \ge \frac{1}{2}  V/S  H_1 : \theta <... 0answers 1k views ### Finding UMPT for uniform distribution with varying support \textbf{Problem} Let X_1,\dots,X_n be a random sample from f(x;\theta) = 1 / \theta, where 0 < x < \theta. We want to test H_0: \theta \leq \theta_0 versus H_1: \theta > \theta_0. ... 0answers 62 views ### Uniformally most powerful test let x be a random variable with density function f(x)= \frac{2\theta x + 1}{\theta + 1} if 0\le x \le 1,\theta > -1 and 0 otherwise consider the problem of testing H0 :\theta \le 1    ... 1answer 852 views ### UMP test of size \alpha for H_0: \theta=0 versus H_1: \theta >0 with X_1,X_2,\dots,X_n \stackrel{iid}{\sim} \mathcal{U}(\theta,\theta+1) (Note - This is also on MSE but I thought I might have better luck here). I was posed the following question: Let X_1,X_2,\dots,X_n \stackrel{iid}{\sim} \mathcal{U}(\theta,\theta+1). Consider ... 0answers 206 views ### What is the general methodology for constructing a UMP test for a simple hypothesis versus a composite one? I think I understand the Neyman-Pearson lemma, but I'm really struggling to understand the reasoning with which it's used as a building block to build tests for composite hypotheses. Take this worked ... 2answers 309 views ### Support of likelihood ratio test statistic Say I'm testing H_0: Y \sim \text{Exp}(1) against H_1: Y \sim \text{U}(0, 1). I believe this gives me the following likelihood ratio test:$$ t^*(y) = \frac{p_1(y)}{p_0(y)} = \frac{1}{e ^ {-y}} ...
In the context of hypothesis test $$H_0:\theta\in \Theta_0$$ $$H_1:\theta\notin \Theta_0$$. Find the relationship between the 0-1 loss defined by L(\theta,\delta)= \begin{cases} 1-\delta & \...