# 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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### 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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### Neyman-Pearson’s Lemma a to define the rejection region of the type nx > κ Bernoulli [duplicate]

I'm working through the following question: I understand that the formula is: Likelihood(Theta_0) / Likelihood(Theta_A) As its bernoulli, I think it shoudl work out as below but I am at a loss on how ...
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### Most powerful test for hypothesis testing with uniform and exponential distributions

Given sample of single observation $x_{[1]}$, we are checking hypothesis $H_0 \sim U[0, 1]$ versus $H_1 \sim Exp(1)$. I need to find most powerful test for hypothesis checking with given Type I error ...
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### Most Powerful Test for random variable with different distributions

The exercise We have $\Theta = \{0,1\}$ and let $X$ be a random variable with density function $f(x;0)=1$ and $f(x;1)=3x^2$ for $x\in (0,1)$. I want to find the most powerful test of size $\alpha=0.2$ ...
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1 vote
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### randomized Neyman-Pearson lemma for a discrete distribution

We let $\Theta=\{0,1\}$, and $X$ be a discrete R.V with the following probability distribution: x 1 2 3 4 5 6 7 8 $f(x;0)$ 0.02 0.02 0.02 0.02 0.02 0.02 0.02 0.86 $f(x;1)$ 0.14 0.12 0.10 0.08 0.06 ...
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1 vote
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### Using Neyman Pearson lemma on a two sample test for normal means

Say I have two groups. I collect $n$ data points from the first ($A$) group ($x_i$) and $m$ data points from the second ($B$) group ($y_j$). The null hypothesis is that the means of the two groups are ...
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### Which to Use: Likelihood Ratio Test or Uniformly Most Powerful Test?

I've recently been learning about MPTs (most powerful tests), UMPTs (uniformly most powerful tests) and LRTs (likelihood ratio tests), and do not totally understand in which context the different ...
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### UMP for Poisson distribution

Let $X_1, \ldots, X_n$ be an iid sample from a Poisson distribution with pmf $f(x; \theta) = \theta^x/x! \cdot e^{-\theta}$ for $x = 0, \ldots$ where $\theta \geq 1$. I want to come up with an ...
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1 vote
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### How to combine two independent likelihood ratio tests?

Let us know that a patient has one of disease A or B. Suppose that we run an experiment to find that the patient has disease A or disease B. The null hypothesis is that the patient has disease A and ...
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### Neyman-Pearson Lemma for Pareto Distribution [duplicate]

I have the following problem. Let $X_1, ..., X_n$ represent a random sample taken from a population with CDF given by $$F(x;\beta) = 1 - \frac{\beta}{x}, ~~ x \geq \beta > 0.$$ Based on the this ...
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1 vote
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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? [duplicate]

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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