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# 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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### 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 ...
259 views

### 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. ...
1 vote
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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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### Is this decision rule well-known/optimal in some setting?

First, you'll have to forgive me if my exposition of this is not the best, I am a computer scientist, not statistician. I have a certain classification task where I am given two (say discrete for ...
1 vote
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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). LRTs have several desirable ... 1 vote
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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(...
1 vote
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1 vote
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### Restricting sets of alternative and null hypotheses to just two values

I have encountered this following question quite a few times in different exercises, and have seen some examples using it, however, in the notes and book that I am following, I am unable to find a ...
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### How does the effect size inform the design (or analysis) of an NHST?

Consider this answer on how to design an NHST. I don't quite understand what exact process one is supposed to follow to determine the minimum sample size once we have: A null hypothesis that is ...
1 vote
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### A basketball probability question using Neyman–Pearson lemma

It is known that the probability of a basketball player to make his first shot is $p=0.6$ A player argues that it does not matter if he made the previous shot or not his odds stays the same. We say if ...
1 vote
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### Neyman pearson on discrete distribution

I have found the ratio of h1 to h0 and the ratio is increasing .So we should reject H0 for large values of x.How should i find the critical region for such type of questions.It seems to me the ...
1 vote
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### UMP test equivalence of definitions

I've been revising the past couple of days and have come across $2$ definitions of a UMP test. Suppose we want to test $H_0: \theta=\theta_0$ vs $H_1: \theta>\theta_0$. Then the test is UMP if: ...
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### How can I apply the Neyman-Pearson Lemma for $f(x|\theta)=\frac{1}{2\theta}\exp[-|x|/\theta]$?

Let $X_1,\cdots,X_n$ be a random sample from: $$f(x|\theta)=\frac{1}{2\theta}\exp[-|x|/\theta] \quad \quad \quad x \in \mathbb{R},$$ where $\theta>0$ is unknown. How can I find an MP size $\alpha$...
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### Can the alternative hypothesis depend on the sample size?

Suppose that we want to test: $$H_0: \theta = 0 \,\,\, vs. \,\,\, H_1:\theta = 1/n,$$ where $n$ is the sample size used to test the hypothesis, and the sample used for this is $X_i \sim f(;\theta)$. ...
1 vote
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### Understanding the general theory proposed by Neyman & Pearson

I'm reading Neyman & Pearson, 1933, i.e. Neyman and Pearson. On the problem of the most efficient tests of statistical hypotheses. Philosophical Transactions of the Royal Society of London. ...
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### Constant value in Neyman Pearson lemma

To know the k value in Neyman Pearson lemma, do we need to know the alternate hypothesis. To what I understood (from articles like PenStateNotes), we could get value of k using null hypothesis and the ...
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### What is the NP?

Suppose $X_1, X_2, X_3,\ldots, X_n$ are i.i.d. variables Poisson $(\lambda)$ and $g(λ)=\lambda(c - e^{-cλ})$ c:constant What is the NP for $H_0:g(λ)=c1$ vs $H_1:g(λ)=c2$ ?? My thought: ...
1 vote
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Given a sample $\mathbf X =(X_1,...,X_n)$ from a parent random variable $X$, Neyman-Pearson's test for two point hypotheses $H_0$ and $H_1$ is the one defined by the critical region $$C=\left\{\mathbf ... 1 vote 0 answers 104 views ### UMP test for H_0:p=0.5 vs H_1:p\neq0.5? Let X_1,\dots, X_n Bernoulli trials. I know that the UMP tests for$$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$$can be obtained with the Neyman ... 11 votes 2 answers 1k views ### Why is the Neyman-Pearson lemma a lemma and not a theorem? [duplicate] This is more of a history question than a technical question. Why is the Neyman-Pearson lemma'' a Lemma and not a Theorem? link to wiki: https://en.wikipedia.org/wiki/Neyman%E2%80%93Pearson_lemma ... 9 votes 1 answer 641 views ### Reproduce figure of "Computer Age Statistical Inference" from Efron and Hastie The summarized version of my question (26th December 2018) I am trying to reproduce Figure 2.2 from Computer Age Statistical Inference by Efron and Hastie, but for some reason that I'm not able to ... 2 votes 0 answers 727 views ### Understanding Uniformly Most Powerful vs Uniformly Most Powerful Unbiased tests I am struggling a little to understand the difference between these two classes of tests. Suppose we were testing a simple null hypothesis and a composite two sided alternative hypothesis. I am ... 4 votes 1 answer 847 views ### Most powerful test of simple vs. simple in \mathrm{Unif}[0, \theta] Say X \sim \mathrm{Unif}[0, \theta]. Denote the observations as x_i (i=1, \cdots, n). Show that any test \phi that satisfies the following two conditions is most powerful test of level \alpha... 1 vote 0 answers 445 views ### To find the Most Powerful Test (MP test) of the given hypothesis problem A friend of mine asked me the question below on testing: Let X be a single observation from one or other member of the family \{f_0(x),f_1(x)\} where$$f_0(x)=\frac{1}{2^{x+1}}\mathbf1_{x\in\{0,1,...
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I am trying to understand the proof of Neyman Pearson Lemma as Uniformly Most Powerful test from here (Page 3). It says the following: Let $H_0: \theta = \theta_0$ and $H_a: \theta = \theta_1$. ...
I have a cipher image $C$ that has intensity levels between $0-255$. I want to check this cipher image for uniformness. For this, I calculated the Chi-square tests. The $\chi^2=270.2112$ and i know ...