All Questions
Tagged with likelihood-ratio mathematical-statistics
43
questions
3
votes
1
answer
182
views
Karlin-Rubin theorem: relationship between test statistic having the MLR property vs being sufficient
Let's suppose we are trying to compare two hypotheses for a single parameter $\theta$. The null hypothesis $H_0$ is that $\theta = \theta_0$, and the alternative is that $\theta ≥ \theta_0$.
The ...
- 435
1
vote
0
answers
69
views
Likelihood ratio as minimal sufficient statistics in infinite parameter space
I just read a question from here (Likelihood ratio minimal sufficient) and have some thoughts. Let me restate the question first:
Consider a family of density functions $f(x|\theta)$ where the ...
- 31
3
votes
1
answer
180
views
Likelihood Ratio Testing for Binomial Distributions
I have a feeling this is a silly question. I am working on a research paper, at some point in it we perform a likelihood ratio test. The first guess would be to apply Wilks's theorem. However, if we ...
2
votes
1
answer
53
views
Generalized Likelihood Ratio Test - Why is the denominator a union
For GLRT, the ratio is:
$$
\Lambda^* = \frac{\max_{\theta \in \omega_0} L(\theta)}{\max_{\theta \in \omega_1}L(\theta)}
$$
but we instead use:
$$
\Lambda = \frac{\max_{\theta \in \omega_0}L(\theta)}{\...
- 23
3
votes
1
answer
128
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How does this explanation of likelihood ratio make sense?
I am currently studying the textbook In All Likelihood by Yudi Pawitan. In chapter 2.4 Likelihood ratio, the author says the following:
$$\dfrac{L(\theta_2; y)}{L(\theta_1; y)} = \dfrac{L(\theta_2; x)...
- 2,096
7
votes
1
answer
93
views
Why not always use CI's from LRT since they don't require symmetry?
I'm confused on why anyone would appeal to asymptotic normality of mle,
$$\hat{\theta} - \theta_0 \rightarrow^D N(0,I^{-1}(\theta))$$
When we can simply invert the likelihood ratio test
$$L(\hat{\...
- 151
6
votes
2
answers
205
views
Does the likelihood ratio test violate the likelihood principle?
I've been going over Berger's famous example of negative binomial vs binomial sampling leading to two different p-values conditional on the same observed data. To summarize, suppose we observe 9 tails ...
- 151
1
vote
1
answer
558
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GLRT of exponential distribution and critical region
Find the Generalized likelihood ratio test (GLRT) for $H_0: \lambda = \lambda_0$ when $H_A: \lambda \ne \lambda_0$ for $X_1 ... X_n$ taken from $X \sim Exp(\lambda;x)$, with a test size of $0.06$, ...
- 131
3
votes
2
answers
4k
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Likelihood Ratio Test for Common Variance from Two Normal Distribution Samples
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Problem Statement: Let $S_1^2$ and $S_2^2$ denote, respectively, the variances of
independent random samples of ...
- 5,942
4
votes
3
answers
2k
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Binomial Distribution: Likelihood Ratio Test for Equality of Several Proportions
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Problem Statement: A survey of voter sentiment was conducted in four midcity
political wards to compare the fraction ...
- 5,942
1
vote
1
answer
780
views
Critical region for an uniform distribution
Let $X_1, X_2, ... , X_n$ be a random sample from the uniform distribution over $[0, \theta]$. Suppose we wish to test $H_0 : \theta = 5$ versus $H_A : \theta < 5$ at significance level $\alpha = 0....
- 121
6
votes
1
answer
501
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Likelihood ratio test for $H_0:(\mu_1,\mu_2)=(0,0)$ vs $H_1:(\mu_1,\mu_2) \neq (0,0)$
There are $X_1, X_2$ where $X_i \sim N(\mu_i,1), i=1,2$. They are independent. The question is
Find the likelihood ratio test with $H_0:(\mu_1,\mu_2)=(0,0), H_1:(\mu_1,\mu_2) \neq (0,0)$. The ...
- 478
2
votes
0
answers
55
views
Interpretation of the ratio of two pdfs evaluated at a certain point?
What is the interpretation of the ratio of two pdfs evaluated at a certain point?
Is that a statistical distance?
Is there any applications of it?
I only know one application of differential privacy, ...
- 213
4
votes
1
answer
943
views
Chi square approximation of the likelihood test ratio
I wasn't able to find any satisfying answer about that topic. I hope someone who understand correctly the subject could enlighten this shadow.
This is not very important, just for the sake of ...
- 333
4
votes
2
answers
1k
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How can I show that the LRT of $H_0:\theta_0=\theta_1=...=\theta_k$ is given by the F test under one-way ANOVA assumptions?
I am a bit curious as to how I can show this. I am aware that you can find this by using the union-intersection test, but it has been hinted that one can use the LRT test to find that LRT for $H_0$ is ...
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