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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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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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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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
$$f_0(x)=\frac{1}{2^{x+1}}\mathbf1_{x\in\{0,1,...
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Proof of Neyman Pearson Lemma
I am trying to understand the proof of Neyman Pearson Lemma as Uniformly Most Powerful test from here (Page 3).
It says the following:
Let $H_0: \theta = \theta_0$ and $H_a: \theta = \theta_1$. ...
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Chi-square test for image encryption
I have a cipher image $C$ that has intensity levels between $0-255$. I want to check this cipher image for uniformness. For this, I calculated the Chi-square
tests. The $\chi^2=270.2112$ and i know ...
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Neyman Pearson Lemma and most powerful test
This is a homework question. I was given a random sample of independent and identically distributed $X_i$'s and wish to test the hypotheses:
$$H_0: \theta = \theta_0$$
$\text{vs}$
$$H_A: \theta = \...
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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 ...
user163580
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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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Neyman Pearson lemma for Bernoulli variables
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 test for a sample of Bernoulli variables?
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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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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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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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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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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 & \...
user72621
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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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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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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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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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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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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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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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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 ...
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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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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 ...
user72621
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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
...
user72621
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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 ...
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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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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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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 ...
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Asymmetry of the Kullback-Leibler distance in hypothesis testing
My question is related to the asymmetry of the Kullback-Leibler distance. I'm using the discrete definition of the Kullback-Leibler distance, so we have:
$$
KL(p,q) = \sum_{s \in S} p(s) \log\left( \...
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Hypothesis testing with Neyman–Pearson Lemma
I have hypothesis:
$H_0: X\ \sim\ \mathcal{N}(-1,\,1)$;
$H_1: X\ \sim\ \mathcal{N}(-9,\,9)$.
Also I have significance level $\alpha = 0.2$.
I'm checking them by Neyman-Pearson Lemma.
1) Find ...
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UMP for $U(0,\theta)$ (simple x simple hypothesis)
Let $X_1,...,X_n $ be iid $U(0,\theta)$. Find the UMP to test $H_0: \theta = \theta_0$ versus $H_1: \theta=\theta_1$, for $\theta_1 < \theta_0.$ Obtain the power of the test.
My attempt:
We know ...
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False alarm in Neyman-Pearson Test. A doubt in text?
Neyman-Pearson Lemma says that the most powerful test, $\phi(x)$, of size $\alpha$ (probability of false alarm), for testing $H_0:\theta=\theta_0$ versus $H_1:\theta=\theta_1$ is the likelihood ratio ...
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Neyman-Pearson lemma: critical region and hypothesis testing
Let $X_1,X_2,...,X_n$ be i.i.d r.v's with common p.d.f.
$$
\mbox f(x)=\frac{x^5e^{-x/\theta}}{5!\theta^6}
$$
where $\theta$ > 0. Show that the Neyman-Pearson lemma produces a test of $H_0: \...
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Obtain a Neyman-Pearson Test of H0: p1=p2=2/5 against the alternative p1=1/2 and p2=1/5 when the type 1 error is 0.05
I kinda have some trouble trying to start with this question.
I know how to derive a Neyman-Pearson Test and its critical region. My main concern is deriving the Likelihood function of the sample ...
user42668
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Hypothesis testing and convergence of LRT statistic
This may be a very simple question, but I am not sure about my logic.
I have a standard point hypothesis testing scenario, where I collect a sequence of $n$ independent observations $\{x_1,\ldots,x_n\...
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Consequences of the Neyman-Pearson lemma?
Suppose $X_{1},...X_{n}$ are independently, identically distributed Bernoulli random quantities with parameter $k$. Consider the hypothesis test:$H_{0}: k = k_{0}$ vs $H_{1} : k = k_{1} \ \ $where $k_{...
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