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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Prove that a test is most powerful when $X_1,\cdots,X_n\sim U(0,\theta)$
Let $X_1,\cdots,X_n\sim U(0,\theta),\theta >0$ be independent random variables. I want to prove that $\phi :\mathbb{R}^n\to [0,1]$ given by $\phi (x):=\begin{cases}1,&\theta _0<x_{(n)}\vee ...
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Proof of the statement "the best test is unbiased"
There is a corollary from Hogg and McKean's textbook titled "Introduction to Mathematical Statistics" and I have miserably failed to understand the proof. Unfortunately, my question requires ...
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is the likelihood ratio test "best" for finite samples?
Wikipedia says
The Neyman–Pearson lemma states that this likelihood-ratio (lr) test is the most powerful among all level α alpha tests for this case.
Is this only true for infinite sample sizes? Is ...
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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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Finding P-value and power of the Most Powerful Test
You observe a sample $X_1, \quad, X_{20}$ with the density
$$
f(x, \vartheta)=2\left(x / \vartheta^2\right) I_{[0 \leq x<\vartheta]}
$$
with an unknown parameter $\vartheta>0$, yielding
$$
\min \...
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UMP two sided tests for exponential families
Consider a random variable $X$ with density $$f(x : θ) = C(θ)e^{η(θ)T(x)}h(x), θ ∈ Θ$$.
Assume that $η(θ)$ is strictly increasing in $θ$ and that the family is full rank. Show that there will not be ...
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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 ...
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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.
...
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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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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).[6] LRTs have several desirable ...
user318514
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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(...
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The Test Statistic of the Neyman - Pearson Lemma
I cannot understand this statement from the book Robust Statistics:
The most powerful tests between two densities $p_0$ and $p_1$, are based on a statistic of the form
$$\int \psi F_n(dx) = \frac{1}{n}...
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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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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
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$f(x;0)$
0.02
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$f(x;1)$
0.14
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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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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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Most Powerful test for indicating exponential or Weibull distribution [closed]
Let $X_1,...,X_n$ be iid distribution function of $F(x)$. I want to test whether $F$ is exponential or Weibull. This means that either $F(x)=1-exp(-x), x>0$ (exponential) $F(x)=1-exp(-x^{\theta}), ...
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How would you find a p threshold for a binary classification prediction? [duplicate]
Lets say that there's a binary classification problem where $X$ ∈ $R_p$ and $Y ∈ \{0,1\} $ and $Pr(Y = 1 | X = x) = p$ for $p$ in $[0,1]$. There is a loss function $L_{falseneg} > 0$ for false ...
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Neyman-Pearson hypothesis testing and composite alternative hypothesis
I am in love with the idea of setting up a statistical test à la Neyman-Pearson when possible, because it is just so intuitive. Most of times, $H_0$ is some kind of point hypothesis, but $H_1$ is ...
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Finding the likelihood ratio to test which distribution has the largest mean
The problem: Let $X_1, \ldots, X_n$ and $Y_1, \ldots, Y_m$ be two i.i.d. samples drawn from $\mathcal{N}(\mu_x, \sigma^2)$ and $\mathcal{N}(\mu_y, \sigma^2)$, respectively. I wanna test $H_0: \mu_x \...
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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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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
$$
\varphi(x_1, x_2, ..., x_n) = \begin{cases}
1 & ...
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Restricting sets of alternative and null hypotheses to just two values
I have encountered this following question quite a few times in different exercises, and have seen some examples using it, however, in the notes and book that I am following, I am unable to find a ...
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How does the effect size inform the design (or analysis) of an NHST?
Consider this answer on how to design an NHST. I don't quite understand what exact process one is supposed to follow to determine the minimum sample size once we have:
A null hypothesis that is ...
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A basketball probability question using Neyman–Pearson lemma
It is known that the probability of a basketball player to make his first shot is $p=0.6$
A player argues that it does not matter if he made the previous shot or not his odds stays the same. We say if ...
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Neyman pearson on discrete distribution
I have found the ratio of h1 to h0 and the ratio is increasing .So we should reject H0 for large values of x.How should i find the critical region for such type of questions.It seems to me the ...
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UMP test equivalence of definitions
I've been revising the past couple of days and have come across $2$ definitions of a UMP test.
Suppose we want to test $H_0: \theta=\theta_0$ vs $H_1: \theta>\theta_0$. Then the test is UMP if:
...
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How can I apply the Neyman-Pearson Lemma for $f(x|\theta)=\frac{1}{2\theta}\exp[-|x|/\theta]$?
Let $X_1,\cdots,X_n$ be a random sample from:
$$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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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,$$
where $n$ is the sample size used to test the hypothesis, and the sample used for this is $X_i \sim f(;\theta)$. ...
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Understanding the general theory proposed by Neyman & Pearson
I'm reading Neyman & Pearson, 1933, i.e.
Neyman and Pearson. On the problem of the most efficient tests of statistical hypotheses. Philosophical Transactions of the Royal Society of London. ...
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Constant value in Neyman Pearson lemma
To know the k value in Neyman Pearson lemma, do we need to know the alternate hypothesis. To what I understood (from articles like PenStateNotes), we could get value of k using null hypothesis and the ...
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What is the NP?
Suppose $X_1, X_2, X_3,\ldots, X_n$ are i.i.d. variables Poisson $(\lambda)$
and $g(λ)=\lambda(c - e^{-cλ})$
c:constant
What is the NP for
$H_0:g(λ)=c1$ vs $H_1:g(λ)=c2$ ??
My thought:
...
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Likelihood ratio test and sample statistics
Given a sample $\mathbf X =(X_1,...,X_n)$ from a parent random variable $X$, Neyman-Pearson's test for two point hypotheses $H_0$ and $H_1$ is the one defined by the critical region
$$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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