# Tagged Questions

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### Likelihood-ratio test in the big picture

I am trying to understand the big picture of statistical tests. That is how to choose the correct statistical test under different situations. I found this table where choosing a statistical test is ...
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### Challenging Likelihood Ratio Test

Derive Likelihood Ratio Test of size $\alpha$. H$_0$: $\theta=\theta_0$ H$_1$:$\theta \neq \theta_0$ $${f}(x,\theta , c) = \theta(x-c)^{\theta-1} \ {c<x<c+1}$$ I ...
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### The test statistic in the likelihood ratio test for nested linear models

Imagine that we have a family of probability disributions with p.d.f $f_{\theta}(z)$ where $\theta \in \Theta$. We also know that there is a linear dependence between parameters. As a consequence we ...
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### Likelihood-ratio test for different distributions

I aim to discriminate between three populations by using a maximum likelihood classifier. The idea is that every population has a unique distribution $\hat{f_i} (x)$. The pdf $\hat{f_i}(x)$ has been ...
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### Is likelihood ratio test the only way to build hypothesis tests?

Usually we can construct likelihood ratio for testing the Null hypothesis and alternative hypothesis: The likelihood ratio test $P( l(\beta_{1}) / l(\beta_{2}) ) < \alpha$ is the rejecting ...
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### Likelihood ratio test for a random variable following the Gamma Distribution

Assuming we have a random variable $X\sim \operatorname{Gamma}\left(\alpha,\beta \right )$:$\frac{1}{\Gamma (\alpha )\beta ^{\alpha}}x^{\alpha-1}e^{\frac{-x}{\beta }}$ I'd like to test the ...
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### Understanding Sequential Probability Ratio Test (SPRT) Likelihood Ratio

I am a software developer looking to develop an alternative for the simple hypothesis testing scheme described here. In short, the test works as follows: Two URLs are compared for their ability to ...
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### Likelihood ratio test: $f(x)=2x$ vs $f(x)=3x^2$: $2n$ degrees of freedom?

Suppose $X_1, . . . , X_n$ are i.i.d. with pdf $f(ยท)$. We want to test the hypotheses \begin{align} H_0 &: f(x) = 2x , \;\, \text{ for } 0 \le x \le 1, \text{ against}, \\ H_1 &: f(x) = 3x^2 , ...
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### Comparing tests for staging cancer

I am comparing 4 different tests for the evaluation/identification of cancer. One is a continuous variable yielding cut-off values based on both the mean and the median of 5 consecutive measurements ...
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### Can I use a likelihood ratio test when the error distributions differ?

This question might sound simple, but in fact I am absolutely unsure whether it is allowed to use an LRT for model comparison when the error distributions differ. For example, can I compare a model ...
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### How to find the exact and likelihood ratio p-value for a hypothesis test that a population sex ratio is 1:1?

Skulls were excavated from a dig. The sex of each skull was determined by anatomical appreciation. For Sites A, the data was Male=162, Female=110. a) Find the exact p-value for a ...
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### How to choose between z and t in 2 sample (non-paired) tests?

I understand that in the one sample case, z is employed where population variance is known and t for where it is unknown. In the two sample case: we are trying to test $H_0: \mu_x=\mu_y$ against ...
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### How to interpret this criterion of logistic regression result

I run a backward variable selection logistic regression and find out the SAS program selected 12 variables and give me the output like this: It is funny that my configuration of the SAS ...
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### What is the outline for the procedure of model selection, with different models based on likelihood functions?

I have two basic models $M_1$ and $M_2$. They each have a likelihood function; $L_{M_1} = f(\mathbf{X}|\mathbf{\theta_1})$ and $L_{M_2} = f(\mathbf{X}|\mathbf{\theta_2})$ (here $\mathbf{X}$ is the ...
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### Practical definition of a UMP test?

I'm trying to make sense of these two statements about UMP (uniformly most powerful) tests: If $g(t\mid\theta)~$ is a UMP then $~g(t\mid\theta_1)>k g(t\mid\theta_0)~\forall~t\in C$ and ...
### Likelihood ratio test vs. $\chi^2$/Z-test for comparing binomial datasets
When does/can one use the likelihood ratio significance test instead of Fisher's exact test or its Pearson $\chi^2$ approximation for comparing two binomial datasets? Given two binomial datasets ...