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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Proving a test is UMP for Uniformly distributed random variable

Let $X_1, X_2,..., X_n$ be a sample of size n from the PMF $$P_N(x) = {1 \over N},\ \ \ \ \ \ \ \ \ x = 1,2,...,N;N \in \mathbb{N} $$ Show that $$ \varphi(x_1, x_2, ..., x_n) = \begin{cases} 1 & ...
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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 ...
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Uniformly Most Powerful (UMP) statistic for N Poisson distributions

For samples of $M$ Poisson-distributed datapoints $X_{1, r},...,X_{M,r}$ $\sim$ $Pois(\lambda_{r})$, and $N$ such distributions ($1 < r < N$), I have defined a likelihood function to describe my ...
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32 views

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 ...
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87 views

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 ...
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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 to choose from a group of parameters for every single estimation?

I have done a series of SNR-estimation with the ground-truth SNR from 0 to 15 dB at a step of 0.1 dB, 1000 samples each time. So there are 151 distributions and they all follow ExtremeValue ...
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435 views

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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How can I further reduce my MP size $\alpha$ test given a random sample from a shifted exponential distribution?

Let $X_1,\cdots,X_n$ be a random sample from $f(x|\theta)=e^{-(x-\theta)},x>\theta,$ where $\theta$ is an unknown real number. Find an MP size $\alpha$ test for $H_0:\theta=\theta_0$ v. $H_1:\theta=...
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Finding Uniformly Most Powerful test

My Attempt Comparing $f(x;\theta)$ with the form $a(\theta)b(x)exp[c(\theta)d(x)]$ , we get $d(x) = log (1-x)$ and $ c(\theta ) = \theta -1 $ as monotone , increasing function in $\theta$ and ...
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Is the $p-$value uniformly distributed in this case?

Let $(Ω, A,P)$ be a statistical model, $H_{0} = \{P_{0}\}\subseteq P$ a simple null hypothesis, and $H_{1} = \{P_{1}\} ⊂ P$ a different simple alternative, so that $P_{1}$ with respect to $P_{0}$ has ...
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Basic hypothesis testing homework question

Suppose that random variable $X$ has the probability density function $$f_X(x)=\sqrt{\frac{\theta}{\pi x}}\exp(-x\theta), \quad x>0,$$ where $\theta$ is a positive parameter. It can be shown that $...
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Suggesting the Most Powerful test based on NP lemma- Bernoullie distribution

A country that gives the UK points in the Eurovision Song Contest with a probability of $p=0.5$ would be called "a UK supporter", whereas a country that gives the UK points with a low probability of $...
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Hypothesis testing - Neyman-Pearson Lemma

While studying for my exam and practicing with old exams I came across this question. In the answer to part d) they mention that both coefficients are positive and hence for some c the test in part b) ...
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Is a best critical region unique?

For testing a simple hypothesis $H_0:\theta=\theta_0$ against another simple hypothesis $H_1: \theta=\theta_1$, a best critical region or a most powerful test of size (aka, significance level) $\alpha$...
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Finding UMP test when testing a simple hypothesis against a composite hypothesis

Hi all I have question regarding the following when reading the notes on Page 5 here: http://www.ams.sunysb.edu/~zhu/ams571/Lecture8_571.pdf The question that I have is when the author showed how to ...
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Finding the NP test

I am really interested on solving the following problem I found in the Casella and Berger. Suppose we have the following pdf: 2$\theta x + 2(1-\theta)(1-x)$ where $ 0<x<1$ and $0<\theta<...
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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 $X_i \sim f(;\theta)$. ...
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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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Likelihood ratio test and sample statistics

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 ...
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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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283 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 ...
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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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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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Proof of Neyman Pearson Lemma

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 $H_a: \theta = \theta_1$. ...
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Chi-square test for image encryption

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 ...
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773 views

Neyman Pearson Lemma and most powerful test

This is a homework question. I was given a random sample of independent and identically distributed $X_i$'s and wish to test the hypotheses: $$H_0: \theta = \theta_0$$ $\text{vs}$ $$H_A: \theta = \...
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530 views

When to use Neyman Pearson or Likelihood ratio

question about Neyman-Pearson lemma vs the likelyhood ratio. From my textbook it says that if you want to test: $H_0: \theta \in \Theta_0$ vs $H_1: \theta \in \Theta_0 ^ c$, then you can use a LRT $...
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By using Neyman-Pearson, how do I know if the most powerful test is also UMP?

I understand that the Neyman Pearson Lemma gives us the most powerful test for a certain alternative simple hypothesis. I also understand by definition, a most powerful test is also UMP if it gives us ...
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In plain English what is the difference between a most powerful test and a uniformly most powerful test?

I'm having trouble understanding the two concepts of a powerful test and a uniformly powerful test. I'm reading about these tests in context of the Neyman Pearson Lemma and it seems like they're ...
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A question on Likelihood Ratio testing

In part (a) of the given question I have derieved the UMP level alpha test using Neyman Pearson Lemma. In part (b) it is asked whether LR level alpha test will match with UMP test. Below is my ...
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Most powerful test and its size

am getting halfway to the answer but not being able to get the final result in terms of chi square values
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Building Neyman-Pearson test in 2D-space having only data for $H_0$ and $H_1$

Suppose we have some statistical data which is points in 2D-space. More precise sample space is upper right quarter of $\mathbb{R^2}$: $$\Omega = \{X | X=(x_1, x_2),\, x_1, x_2\in \mathbb{R_+}\}$$ ...
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Show that the test always terminates

$p_0(\mathbf{x})$ and $p_1(\mathbf{x})$ are two distinct density functions on $R^d$. $E_i, i \in \{0,1\}$ is the expectation when density of $\mathbf{X}$ is $p_i(\mathbf{x})$ $L_n = \frac{\prod_{i=...
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Neyman-Pearson test

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>\...
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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?
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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 ...
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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 ...
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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 <...
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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 $H_1: \theta > \theta_0$. ...
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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 $ $ $ ...
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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 ...
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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 ...
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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}} ...
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Hypothesis testing to discriminate between two renewal processes

We have time [0,T] to observe a renewal point process, where the inter-renewal timings are i.i.d, and then decide whether the observation is according to a renewal process in which the pdf of inter-...
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218 views

Relationship between 0-1 Loss and error Type I and II in Neyman Pearson

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 & \...