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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2
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3answers
55 views

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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0answers
23 views

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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1answer
23 views

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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44 views

Hypothesis Testing and Neyman Pearson Lemma

Are there any general steps for hypothesis testing, in terms of what is a one-sided and two-sided test? If the null hypothesis H_o, then there are 3 cases. Is this correct? If H_o:θ=θ_0 3 cases: <...
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0answers
52 views

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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0answers
45 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 ...
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2answers
701 views

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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1answer
201 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 ...
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0answers
122 views

Critical region of distribution $f(x;\theta) = \theta x ^{\theta -1}$

I have been asked to find the critical region of the distribution given by $f(x;\theta) = \theta x ^{\theta -1}$ under the hypotheses $H_0: \theta = \theta_0 $ and $H_1:\theta < \theta_0.$ Show ...
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0answers
191 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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0answers
22 views

MAP/MLE/Neyman Pearson Example Problem

I'm currently attempting a practice problem and wanted help checking my work: The random variable X is such that P(X=1) = 2/3 and P(X=0) = 1/3. When X = 1, the random variable Y is exponentially ...
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1answer
65 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 $\...
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0answers
136 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\...
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1answer
427 views

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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0answers
188 views

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 ...
2
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1answer
602 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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0answers
313 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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2answers
737 views

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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0answers
36 views

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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41 views

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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0answers
63 views

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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2answers
60 views

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=...
2
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1answer
135 views

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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1answer
57 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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1answer
143 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 ...
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0answers
35 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 ...
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0answers
495 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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0answers
900 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 $H_1: \theta > \theta_0$. ...
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0answers
52 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 $ $ $ ...
2
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1answer
537 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 ...
2
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0answers
159 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 ...
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2answers
201 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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0answers
27 views

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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1answer
152 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 & \...
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0answers
568 views

Neyman - Pearson criterion: most powerful but not consistent?

I am having a bit of confusion with hypothesis testing basics, could you please help me clear it? Suppose we are testing a simple hypothesis $H_0:\theta=\theta_0$ against another simple hypothesis $...
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1answer
2k views

Best critical region and Neyman-Pearson lemma?

I'm reading an online course and I'm really confused about the Neyman-Pearson lemma. It states From my understanding, does it mean that critical region could be any region like (a,b) and (c,+00)? ...
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2answers
2k views

Calculate type I and II error - solution verification

There are 7 balls in urn. $Q$ of them are white and the rest are black. We have hypothesis $H_0:Q=3$ and $H_1:Q=5$. To test this we draw 2 balls (balls don't come back to the urn - i.e. they are drawn ...
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0answers
110 views

Calculate power of a test - solution verification

Let $X_1,X_2,\cdots,X_n$ simple random sample from gaussian distribution $\mathcal{N}(m,4)$. Calculate power of a test ($1-\beta$) for $H_0:m=0$, $H_1:m=2$, when significance level of a test is $\...
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1answer
48 views

Do we need to have MLR (in same direction) for whole parameter space?

Suppose $X$ is a single observation from a pdf, and I want to test some hypothesis: $H_0:\Theta=\Theta_0$ vs. $H_1:\Theta=\Theta_0^c$. I want to check whether there exists a UMP level $\alpha$ test ...
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0answers
101 views

Neyman-Pearson method for model selection

I am studying about the competing model fittings, i.e., selection of one model. How can the Neyman-Pearson method be used for model selection in general?
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1answer
912 views

What are the ''desirable'' statistical properties of the likelihood ratio test?

I am reading an article whose method is fully based on the likelihood ratio test. The author says that the LR test against one sided alternatives is UMP. He proceeds by claiming that "...even when ...
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0answers
169 views

Neyman-Pearson lemma equivalent for undefined pdf

The Likelihood Ratio Test is the most powerful test for the simple hypotheses $H_0:\theta=\theta_0$ against $H_1:\theta=\theta_1$ by virtue of the Neyman-Pearson lemma. However, the implicit ...
0
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1answer
782 views

UMP test for geometric distribution at $\alpha=0.05$

Let the pmf of a population $X$ be $P$($X=k$)=$p^k(1-p)$ with $0<p<1$. We take a sample of size $n=1$. The hypothesis $H_0$ says $p\leq0.95$ and $H_1$ is $p>0.95$. Find a level $\alpha=0.05$ ...
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1answer
2k views

Ways to find a UMP test

I'm studying for my final exams and the subject of proof will basically test hypotheses, I will try to summarize here my doubts. For found the UMP test the ways are 1) Use Neyman–Pearson lemma ...
4
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1answer
582 views

Neyman-Pearson Lemma and hypothesis-testing

Consider testing $H_0:\theta=\theta_0$ vs $H_1:\theta=\theta_1$, where the pdf or pmf corresponding to $\theta_i$ is $f(x|\theta_i)$, $i=0,1$ using a test with rejection region R that satisfies ...
2
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1answer
727 views

Eliminating a nuisance parameter in likelihood ratio test

I am having an argument with a co-author about how to eliminate a nuisance parameter in a simple likelihood ratio test and am hoping that the community helps us settle it. Our data $\mathbf{x}$ can ...
2
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1answer
3k views

how to get the critical region for a uniformly most powerful test for mean of normal?

I need help in understanding how to construct a uniformly most powerful test using the Neyman-Pearson lemma. Here is an excerpt in my text that I have trouble following: I have no idea how to get $(5....
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0answers
99 views

Need to clarify certain doubts regarding Testing Hypotheses

I have some doubts in Hypothesis Testing which I would want to clarify. We know, by Neyman Pearson Lemma that the test which rejects $H_0$ when $\dfrac{f_1(x)}{f_0(x)}>\dfrac{k}{1}$ minimises the ...
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0answers
126 views

Uniformly Most Poweful Hypothesis test for the difference between two means

I have data for two groups, $G_1$ of n trials and $G_2$ of m trials, of which datasets $x_1$ and $x_2$ have been collected. These can be binomially distributed, $X_1$ ~ B(n,p), $X_2$ ~ B(m,q). I ...